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\title{Expositio problematum Aristotelis, Particula XIX: De harmonia}
\author{Pietro d’Abano}\makeatletter 
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{\LARGE\uppercase{Expositio problematum Aristotelis, Particula XIX: De harmonia}} % Title
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%------------------------------------------------
%	Editor(s)
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Edited By

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{\Large Caleb Mutch} % Editor list
        
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{\Large Medieval Academy of America}\\
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{\large Library of Digital Latin Texts}\\                      
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{\large Volume ?}\\
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{\normalsize 202?}        
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{\LARGE The Digital Latin Library}\\650 Parrington Oval, Carnegie Building 101, Norman, OK  73019, USA
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\small
{The Digital Latin Library} publishes the \textit{Library of Digital Latin Texts} under the authority of The University of Oklahoma. Individual volumes are reviewed and sponsored by the Society for Classical Studies, the Medieval Academy of America, or the Renaissance Society of America, depending on the era of the text(s).

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\textcopyright \thinspace Caleb Mutch, 202?.

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{\LARGE Acknowledgments}
    
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\item TEI XML encoding:: Caleb Mutch.
\item TEI XML encoding:: Samuel J. Huskey.

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\section{Preface}

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\subsection{I. Pietro d’Abano’s \textit{Expositio problematum Aristotelis}}

\subsubsection{1.1 General}
\par
The pseudo-Aristotelian \textit{Problems} is an extensive collection of questions and proposed answers concerning a wide range of subjects, including biology, mathematics, and physics, divided topically into thirty-eight sections. The nineteenth of these sections preserves the first substantial account of ancient Greek music making and theorizing to have been rediscovered in the medieval period. This material held great promise for medieval thinkers seeking to learn more about the foreign world of ancient Greek music, whose legendary emotional power had been lamentedly lost, and the text’s attribution to Aristotle imbued it with special authority. Yet the \textit{Problems} was also a challenging text to decipher. Not only was the subject manner often unfamiliar, the manuscript tradition of the Greek text is also corrupt in places. Furthermore, the first complete Latin translation, made by Bartholomew of Messina (fl. 1258–66), observed the common medieval practice of verbatim, indeed servilely literal, word-for-word translation. For all these reasons, potential readers of the \textit{Problems} were in sore need of orientation and explanation.\par
Into the breach strode Pietro d’Abano (c. 1257–1316), who was a professor of medicine with wide-ranging interests and expertise. Pietro composed his \textit{Expositio}, the first commentary on the entirety of the \textit{Problems} in the first decade of the fourteenth century, and there are more than twenty extant manuscripts of it, as well as multiple early print editions. His commentary also attracted the attention of his contemporaries, most notably Jean de Jandun (c. 1285–1328), a philosopher and theologian of considerable importance based at the University of Paris, who produced a revision of the entire \textit{Expositio}. Moreover, Pietro’s commentary on the problems of section XIX, in particular, was the first widely circulated medieval attempt to interpret and assimilate rediscovered knowledge about music in the ancient world, and his explanations consequently became a touchstone for later generations of music scholars who referred to the \textit{Problems}, including Franchino Gaffurio, Girolamo Mei, and Francisco de Salinas. Most recently, Claude Palisca has called attention to Pietro’s anticipation of Renaissance modes of scholarship (1985), and David E. Cohen has demonstrated that Pietro’s idiosyncratic interpretations of some of the problems concerning music played an important role in the development of the music-theoretical concept of harmony (2012). Yet despite the inherent interest of Pietro’s commentary as a site of cross-cultural engagement and its historical significance to later music theorizing, his commentary on section XIX, on music, has been inaccessible to scholars, for until recently it had not been published in any form since the sixteenth century’s sadly unreliable print editions. Even that recent publication (Meyer 2022), however, does not provide a proper critical edition of the text, as it is based on a sixteenth-century print edition, only considers half the extant manuscripts, and fails to distinguish between Pietro's first version of his commentary and his later revision (see section 2.1). The present critical edition rectifies that situation, establishing a base text informed by a stemmatic hypothesis of the complete manuscript tradition. Furthermore, this edition is openly accessible, and it presents a semantically encoded version of the text, allowing readers to select the types of variants they wish to consider. As a result, it provides a better informed version of Pietro's commentary and makes it more widely available than ever before.
\subsubsection{1.2 Structure}
\par
Each of the pseudo-Aristotelian problems follows a set structure. First, a question is posed, such as at the start the first problem: ``Why do those who are suffering and those who are rejoicing have pipes played for them?'' (\textit{Propter quid dolentes et gaudentes fistulantur}?) Then one or more possible reasons are proposed, often accompanied by additional explanation. For instance, the first problem continues: ``Is it so that the former mourn less, but the latter rejoice more?'' (\textit{Aut vt hii quidem minus tristentur, alii autem vt magis gaudeant}?) In his commentary Pietro follows a similar pattern. He first paraphrases the opening question, and then addresses the remainder of the problem, dividing it into between one and seven parts, depending on its length. He signals the beginning of each successive division of the text with its first word or two, followed by a partial or complete paraphrase. (These paraphrases are enclosed by targeted "gloss" elements in the xml format, and they are italicized in the rendered pdf.) Given the unidiomatic Latin of Bartholomew of Messina’s translation and the common occurrence of obscure vocabulary, Pietro’s paraphrases provide an important means of ingress to the text on their own. He often supplements these paraphrases with additional explanatory material, both his own and by authors from antiquity. In the latter case, he often provides cross-references to other pertinent passages within the \textit{Problems} (from both section XIX and other sections), and he also inserts abbreviated quotations, most often from the Aristotelian and medical corpora.
\subsection{II. The Textual Tradition}

\subsubsection{2.1 Introduction}
\par
Establishing the best text of Pietro d’Abano’s commentary is no simple task, for it survives in several different forms. By analyzing variants in the introduction with which Pietro starts his commentary on the \textit{Problems}, Gijs Coucke has demonstrated that Pietro produced a revised version of his commentary after the initial version was complete (2008, vol. 2, xlii–xliv). About a quarter of the extant manuscripts derive from the initial version, another quarter from the revision, and the remaining half of the manuscripts descend from the revision made by Jean de Jandun (concerning which see Swaenepoel 2009). While these latter manuscripts do not aid in the reconstruction of the text(s) Pietro himself wrote, they provide important testimony as to the work’s reception in the Parisian university scene, and consequently I have recorded variants from five selected manuscripts in the critical apparatus. Coucke notes that “all manuscripts appear to have the identical explicit, which refers to 1310 as the year in which Peter finished his \textit{Expositio}” (2008, vol. 2, xliv)—that presumably being the date when the initial version was completed.\par
Further complicating matters, some of the manuscripts change affiliations partway through the lengthy work. As Joan Cadden has observed, manuscript P\textsuperscript{z} ``contains the first half of Pietro’s version and the second half of Jean’s,'' and manuscript E\textsuperscript{a} has the converse arrangement (2013, 228). Additionally, in his edition of section IV of the \textit{Expositio}, Coucke identifies manuscript V\textsuperscript{b} as deriving from Pietro’s revised version, whereas by section XIX it hews more closely to the initial version. (Furthermore, manuscript P\textsuperscript{q} contains Jean’s version in sections IV and XIX, but by section XXVII it appears to have changed to Pietro’s version [Coucke 2008, vol. 2, xxvi].) As a result, I have preserved Coucke’s manuscript sigils, but not his labeling of families and conjectured sub-types. For instance, Coucke labels Pietro’s first and second versions \textit{P\textit{a}} and \textit{P\textit{b}}, respectively whereas I call them α and κ. (This change is made in part to avoid the confusing use of letters in Coucke’s \textit{Conspectus siglorum}, where the manuscripts elsewhere labeled as \textit{P\textit{b}} are listed under heading A, and those as \textit{P\textit{a}} under heading B.)\par
In classifying the manuscripts I used Coucke’s findings on section IV as a point of departure. I collated all the manuscripts’ versions of section XIX’s tenth problem to test the validity of Coucke’s findings to section XIX. Then I collated all the manuscripts representing solely either Pietro’s first or his second version (with the exception of M\textsuperscript{b}, to be discussed below), along with manuscripts E\textsuperscript{a} and P\textsuperscript{z} (since their classification differs between sections IV and XIX), and four representative manuscripts from the Jean de Jandun redaction.
\subsubsection{2.2 List of Sources}

\paragraph{Manuscripts}\label{bibliography-manuscripts}

\vspace{1.5em}

{\large{Pietro's first version (α)}}
\begin{msitemlist}{1}
\bibitem [β]{β}  Consensus of N\textsuperscript{u} and P\textsuperscript{y}, where N\textsuperscript{u} is extant 
\begin{msitemlist}{1}
 \bibitem [N\textsuperscript{u}]{Nu}  Nuremberg, Stadtbibliothek, Cent. III.38, \textbf{ff. 61v–64v}:  before 1389
 \bibitem [P\textsuperscript{y}]{Py}  Paris, Bibliothèque Nationale man. lat. 6541 (accessed 2022-01-03), \textbf{ff. 179v–91v}:  fourteenth century. \url{http://gallica.bnf.fr/ark:/12148/btv1b9068411m.r=6541}
\end{msitemlist}

\bibitem [γ]{γ} \textit{Consensus of V\textsuperscript{a} and V\textsuperscript{b}}  
\begin{msitemlist}{1}
 \bibitem [V\textsuperscript{a}]{Va}  Vatican, Biblioteca Apostolica lat. 2174 (accessed 2022-01-03), \textbf{ff. 72r–77r}:  fourteenth century. \url{https://digi.vatlib.it/view/MSS\textunderscore Vat.lat.2174}
 \bibitem [V\textsuperscript{b}]{Vb}  Vatican, Biblioteca Apostolica lat. 2175 (accessed 2022-01-03), \textbf{ff. 107–16r}:  fifteenth century. \url{https://digi.vatlib.it/view/MSS\textunderscore Vat.lat.2175}
\end{msitemlist}

\bibitem [E\textsuperscript{a}]{Ea}  Erfurt, Bibliothek Amploniana fol. 15, \textbf{ff. 136v–47r}:  early fourteenth century
\end{msitemlist}

\vspace{1.5em}

{\large{Pietro's revised version (κ)}}
\begin{msitemlist}{1}
\bibitem [P\textsuperscript{x}]{Px}  Paris, Bibliothèque Nationale man. lat. 6540 (accessed 2022-01-03), \textbf{ff. 154v-64v}:  Fourteenth century. Italy. \url{http://gallica.bnf.fr/ark:/12148/btv1b90684121.r=6540}
\bibitem [λ]{λ}  Consensus of M\textsuperscript{a}, V\textsuperscript{m}, and V\textsuperscript{z} 
\begin{msitemlist}{1}
 \bibitem [M\textsuperscript{a}]{Ma}  Cesena, Malatestiana D.XXIV.2 (accessed 2022-01-03), \textbf{ff. 96v–104r}:  before 1381. \url{http://catalogoaperto.malatestiana.it/elenco-libri/libro/?saggioid=DX.24.02}
 \bibitem [V\textsuperscript{m}]{Vm}  Venice, Biblioteca Nazionale Marciana, lat. VI. 126, \textbf{ff. 95r–100r}:  fifteenth century
 \bibitem [V\textsuperscript{z}]{Vz}  Venice, Biblioteca Nazionale Marciana, lat. Z. 263, \textbf{ff. 48v–70r}:  ca. fourteenth century, or after 1451 on the basis of watermark. On paper, in one column
 \bibitem [M\textsuperscript{b}]{Mb}  Cesena, Malatestiana S.VI.2/3 (accessed 2022-01-03), \textbf{S.VI.3, ff. 38v–56r}:  1381. \url{http://catalogoaperto.malatestiana.it/elenco-libri/libro/?saggioid=SX.06.03}A copy of M\textsuperscript{a}
\end{msitemlist}

\end{msitemlist}

\vspace{1.5em}

{\large{Jean de Jandun's redaction}}
\begin{msitemlist}{1}
\bibitem [π]{π}  Collated manuscripts 
\begin{msitemlist}{1}
 \bibitem [G\textsuperscript{u}]{Gu}  Ghent, Bibliothéek van de Rijksuniversiteit, ms. 72 (accessed 2022-01-03), \textbf{ff. 234r–49v}:  fourteenth century. \url{http://lib.ugent.be/catalog/rug01:000802948}
 \bibitem [P\textsuperscript{p}]{Pp}  Paris, Bibliothèque Nationale man. lat. 6542 (accessed 2022-01-03), \textbf{ff. 183v–94v}:  1385. \url{http://gallica.bnf.fr/ark:/12148/btv1b10721259d/f1.item.r=6542}
 \bibitem [P\textsuperscript{u}]{Pu}  Paris, Bibliothèque de l'Université 122, \textbf{114v–23r}:  fifteenth century
 \bibitem [P\textsuperscript{z}]{Pz}  Paris, Bibliothèque Nationale man. lat. 6541A (accessed 2022-01-03), \textbf{ff. 179r-92r}:  fourteenth century. \url{http://gallica.bnf.fr/ark:/12148/btv1b90680829/f2}

\textit{Hands}
\begin{itemize}
\item P\textsuperscript{z2} is responsible for the header, and writes in an ink that has faded significantly. It may be the same scribe who wrote the large initials which begin each problem.
\end{itemize}
 \bibitem [V\textsuperscript{o}]{Vo}  Vatican, Biblioteca Apostolica Ottobon. lat. 1764, \textbf{ff. 147r–57v}:  fourteenth century
\end{msitemlist}

\bibitem Other manuscripts from the Jean de Jandun redaction 
\begin{msitemlist}{1}
 \bibitem [C\textsuperscript{a}]{Ca}  Cambridge, Peterhouse 79, \textbf{ff. 76r–81v}:  fourteenth century
 \bibitem [P\textsuperscript{a}]{Pa}  Paris, Bibliothèque Arsenal, 723, \textbf{ff. 183v–97r}:  fourteenth century
 \bibitem [P\textsuperscript{q}]{Pq}  Paris, Bibliothèque Nationale man. lat. 15454, \textbf{ff. 127r–36r}:  fifteenth century
 \bibitem [S\textsuperscript{c}]{Sc}  Seville, Biblioteca Colombina, 7.7.9, \textbf{ff. 121r ff.}:  fourteenth to fifteenth century
 \bibitem [V\textsuperscript{c}]{Vc}  Vatican, Biblioteca Apostolica lat. 2176 (accessed 2022-01-03), \textbf{ff. 62r–65r}:  fourteenth century. \url{https://digi.vatlib.it/view/MSS\textunderscore Vat.lat.2176}
\end{msitemlist}

\end{msitemlist}

\paragraph{Editions: Witnesses}\label{bibliography-early-editions}

\begin{bibitemlist}{1}

 \bibitem [m]{m} Illarius, Stephanus. \textit{Expositio succinta problematum Aristotelis quam Petrus edidit Paduanus} (accessed 2022-01-03). Mantua: Paulus de Butzbach, 1475. \url{http://daten.digitale-sammlungen.de/\textasciitilde db/0006/bsb00068444/images/}
 \bibitem [v]{v} Anonymous. \textit{Expositio preclarissimi atque eximii artium ac medicine doctoris Petri de Ebano Patauini in librorum problematum Aristotelis} (accessed 2022-01-03). [Venice]: Johannes Herbort, 1482. \url{http://daten.digitale-sammlungen.de/\textasciitilde db/0005/bsb00058463/images/}\end{bibitemlist}
\subsubsection{2.3 Pietro d’Abano’s First Version}
\par
The first version of Pietro’s commentary on section XIX of the \textit{Problems}, which I label α, is found in five manuscripts: N\textsuperscript{u}, P\textsuperscript{y}, V\textsuperscript{a}, V\textsuperscript{b}, and E\textsuperscript{a}. Manuscripts N\textsuperscript{u} and P\textsuperscript{y} usually carry the best version of this text. Unfortunately, N\textsuperscript{u}’s codex has lost a number of pages, resulting in total loss of the text from a point partway into problem 9 through problem 15, and again from the end of problem 38 through the end of the section.\par
Manuscripts V\textsuperscript{a} and V\textsuperscript{b} are associated by shared error. The most salient of these occurs in problem 20, where they repeatedly mistake discussion of the Greek language (\textit{oratio greca}) for “grammatical speech” (\textit{oratio gramatica}). The scribe of manuscript V\textsuperscript{a} evidently prioritized speed over neatness of script and also had very idiosyncratic orthography, regularly doubling single consonants or omitting one where they are normally doubled. Given the ubiquity of these variants, I have omitted them from the critical apparatus unless they are shared by other manuscripts. In his edition of section IV, Coucke finds that V\textsuperscript{b} contains the second version of Pietro’s commentary, but by section XIX there is little evidence of V\textsuperscript{b} sharing readings characteristic of that second version. In addition, the sub-family represented by V\textsuperscript{a} and V\textsuperscript{b} is particularly important for the commentary’s reception history since the incunable editions of the text align most closely with them.\par
The situation is more complicated with manuscript E\textsuperscript{a}. Coucke assigns manuscript E\textsuperscript{a} to the Jean de Jandun redaction, and in section XIX the manuscript adopts many (though not all) of the omissions that characterize that redaction, and it sometimes shares readings with P\textsuperscript{z}. Nonetheless, in non-omitted passages E\textsuperscript{a}’s text more closely resembles the manuscripts transmitting Pietro’s first version. As a result, I primarily affiliate it with that family of texts, with the proviso that the scribe of E\textsuperscript{a} evidently consulted a P\textsuperscript{z}-related manuscript from the Jandun redaction as well.
\subsubsection{2.4 Pietro d’Abano’s Second Version}
\par
The second version of Pietro’s commentary, which I label κ, is also found in five manuscripts: P\textsuperscript{x}, V\textsuperscript{m}, V\textsuperscript{z}, M\textsuperscript{a}, and M\textsuperscript{b}. With respect to the fourth section, Coucke hypothesized that manuscripts P\textsuperscript{x} and V\textsuperscript{b} (which in section IV transmits Pietro’s second version) most closely follow the original form of the second version, that a further revised form of the second version was the model for manuscripts M\textsuperscript{a}, M\textsuperscript{b}, and P\textsuperscript{z}, and that manuscript V\textsuperscript{m} shares common error with these latter three (Coucke 2008, vol. 2, xxxix). Only some of Coucke’s observations hold true in section XIX, however. I agree that a revised version, which I call λ, was the model for many of these manuscripts. As mentioned above, however, manuscript V\textsuperscript{b} has switched its affiliation and now follows Pietro’s first version, and P\textsuperscript{z} likewise has shifted to Jean of Jandun’s redaction. Additionally, Coucke does not consider V\textsuperscript{z}, since it does not contain section IV.\par
The resulting picture is that P\textsuperscript{x} follows the original form of the second version, and that M\textsuperscript{a}, M\textsuperscript{b}, V\textsuperscript{m}, and V\textsuperscript{z} derive from λ, a revised version thereof. Coucke’s hypothesis of a further revised source for M\textsuperscript{a}, M\textsuperscript{b}, and P\textsuperscript{z} (but not V\textsuperscript{m}) is not supported in section XIX, since there is good reason to believe that M\textsuperscript{b} is copied either directly from M\textsuperscript{a} or from another copy of M\textsuperscript{a}. For instance, in problem 2 the scribe of M\textsuperscript{a} erroneously wrote ``unaa'' instead of ``una,'' and M\textsuperscript{b}’s scribe wrote ``unaa'' before deleting the superfluous a. Still stronger evidence of copying is found in problem 3’s table of Greek note names, where M\textsuperscript{b}’s scribe was consulting a manuscript that breaks the word ``prollabamenos'' (\textit{recte: proslambanomenos}) into ``prolla'' and ``bamenos,'' as M\textsuperscript{a} does. We can tell this because M\textsuperscript{b}’s scribe evidently thought the line break divided two distinct words: in M\textsuperscript{b}, following ``prolla'' the rest of the line (about six characters’ worth) is left blank, before the table of note names begins at the top of the next column with ``bamenos.'' Furthermore, in problem 6 M\textsuperscript{b}’s scribe follows the word \textit{magnatis} with the superfluous phrase \textit{vel huius est inequale non eundem seruans tenorem} ahead of where it should occur in the text, and then repeats it in the correct place, following \textit{excellentis}. In this case of homeoteleuton the -\textit{tis} endings of \textit{magnatis} and \textit{excellentis} are nearly aligned in two successive lines of M\textsuperscript{a} (f. 97v), so it is very likely that M\textsuperscript{b}’s scribe’s eyes skipped from the end of \textit{magnatis} down a line to \textit{excellentis}, and then erroneously copied the words that followed. In sum, because my collation of M\textsuperscript{b} across the first ten problems revealed no meaningful discrepancies with M\textsuperscript{a}, but rather signs of direct copying, I conclude that it is not useful for establishing the best text of Pietro’s second version.\par
Of the remaining manuscripts in sub-family λ, V\textsuperscript{m} often preserves better readings than M\textsuperscript{a} and V\textsuperscript{z}. Zanetti dates V\textsuperscript{z} to about the fourteenth century (1741, 121), but it may have been copied later, as it is written on paper and is the only extant manuscript to be copied in a single column. I have examined its watermark, which is a bellflower with a five-petalled bell, two leaves that curl up, and a stem that curves sharply; it closely resembles watermark DE4620-PO-127158, which has been dated to 1447, in Rome, though its bell is straighter. Nonetheless, the manuscript's unique readings are not obviously inferior to those of M\textsuperscript{a}, so I have included its variants in the critical apparatus throughout.
\subsubsection{2.5 Jean de Jandun’s Redaction}
\par
The redaction undertaken by Jean de Jandun necessarily cannot help reconstruct the best text of Pietro’s commentary, but it is still of some interest. This is in part because ``Jandun is one of the most important thinkers in the Latin Averroist tradition ''(South 2002, 372), and consequently his understanding of the \textit{Problems} and Pietro’s commentary merits attention. Furthermore, due to the significant circulation of Jean’s redaction in the environs of the University of Paris it is valuable for understanding the \textit{Problems}’ reception. As a result, the xml version of this edition includes readings from five selected manuscripts of Jean’s redaction. In the interest of brevity, this pdf version usually presents only the best reading from that recension, unless other variants are of particular interest.\par
Three broad feature of Jean’s redaction identified by Coucke in section IV also occur in section XIX. First, the redaction is most saliently characterized by omissions that ``either cover well defined text-units (paragraphs, or entire (subordinate) clauses), or entail the removal of repetitive or superfluous speech'' (2008, vol. 2: xxvii). Second, as a whole the manuscripts of Jean’s redaction largely agree with those of Pietro’s first version, but on occasion they follow his second version instead, as Coucke found (xxx–xxxi). Third, as in section IV manuscript Vo appears to be most valuable for establishing the text of the redaction (xxxiii). It has almost no unique errors of any importance, but a brief passage in problem 9 (\textit{vtrobique existente. Et hoc quia in cantu}), which it alone omits, indicates that it cannot be the direct (or at least the sole) model for the rest of the redaction’s manuscripts.\par
Beyond these broad characteristics, however, further clarity is elusive. Whereas Coucke found that manuscripts P\textsuperscript{p} and V\textsuperscript{c} are “contaminated,” since in section IV they complete the omissions which are otherwise characteristic of Jean’s redaction (xxxi–xxxii), in section XIX P\textsuperscript{p} never does so and V\textsuperscript{c} only sometimes does. Coucke then goes on to note of section IV (and he observes that Delaurenti’s [then]-in-progress edition of section VII confirms the same), that: \begin{displayquote}The other representatives of J [Jean’s redaction], however, are hardly differentiated, and among them the relations are volatile and changing. Yet, the samples allow for a division of the manuscripts along two main branches: (1) one of them is built around V\textsuperscript{o}, C\textsuperscript{a} and E\textsuperscript{a}; (2) another unites the distinct pair P\textsuperscript{q} P\textsuperscript{u} (xxxii).\end{displayquote} Broad outlines of these relations among manuscripts are still present in section XIX: in the first branch, C\textsuperscript{a} and V\textsuperscript{o} are still closely related, but E\textsuperscript{a} now aligns better with Pietro’s first version. In the second branch, moreover, frequent agreements of P\textsuperscript{q} with G\textsuperscript{u} and P\textsuperscript{u} with P\textsuperscript{p} also emerge. The remaining manuscripts, V\textsuperscript{c}, P\textsuperscript{a}, and S\textsuperscript{c}, do not clearly align with either branch.\par
Given that Jean’s redaction cannot help to establish Pietro’s text, I have not collated all its manuscripts. In the interest of brevity, this pdf version usually presents only the best reading from that recension, unless other variants are of particular interest. I follow Coucke (xxxiv) in selecting manuscript V\textsuperscript{o}, due to its paucity of errors.\par
The xml version of this edition, by contrast, includes readings from four other selected manuscripts of Jean’s redaction. Again following Coucke, it includes G\textsuperscript{u}, because of that source's careful preparation and occasional unique variants. The xml version also collates manuscript P\textsuperscript{z}, on account of its interest as it switches from Pietro’s second version to Jean’s redaction. Because Coucke identified manuscripts P\textsuperscript{u} and P\textsuperscript{q} as constituting a differentiated branch in section IV, I selected manuscript P\textsuperscript{u} for inclusion, since by section XIX P\textsuperscript{q} closely matches G\textsuperscript{u}. And finally I included manuscript P\textsuperscript{p} to help establish the second branch’s readings. (Other manuscripts from Jean’s redaction are also occasionally cited when they support variants that would otherwise appear unique.)
\subsubsection{2.6 Stemmatic Hypothesis}
\par
\noindent\includegraphics[width=\textwidth]{images/Stemma.png}
\subsection{III. Pietro d’Abano’s Practices of Quotation}
\par
In section XIX of his commentary Pietro employs quotations in the same two ways observed by Coucke in section IV: more literal quotations, which come immediately after the source’s reference (usually cited by the work’s author, title, and book number, when applicable), and loose paraphrases, which precede the reference (2008 vol. 2, lxi). To this general pattern we may add a more specific observation: in section XIX Pietro’s more literal quotations draw exclusively on Boethius and Isidore, and he uses them to lay out background information about the discipline of music’s terminology and its received history. He does this most prominently in problem 1, but also later, too, for example in problems 3, 7, and 16. Pietro uses paraphrases, by contrast, largely to support or exemplify his interpretive claims, and they come from a significantly wider range of sources, as will now be addressed.\par
The sources with which Pietro engages in section XIX fall into three main categories: the Aristotelian corpus, medical texts, and late-Antique writings on music. Pietro cites sources from the first category most frequently. Cross-references abound to passages within the \textit{Problems}, from both section XIX and other sections, and Pietro also cites other Aristotelian works that were well known in his day, especially book VIII of the \textit{Politics}. (Other Aristotelian works with which Pietro engages in medical-focused sections of the \textit{Problems}, such as the \textit{De generatio animalium} and \textit{De historia animalium} [Coucke 2008, vol. 2, lxii], make no appearance here.)\par
Given section XIX’s dedication to the subject of music, the extent to which Pietro incorporates medical sources, the second category, undoubtedly speaks to his training as a physician. Three of these treatises are by Galen, whom Claude Palisca misidentified as ``an otherwise unknown author Balienus'' in a chapter on Pietro (1985, 55). The other medical sources range from the better known, such Avicenna’s \textit{Canon medicinae}, to the obscure, for instance the \textit{Aphorismi} of Yuḥannā ibn Māsawaiyh, which Pietro attributes to John of Damascus.\par
The number of texts in the third category, writings on music, is very small. While Pietro makes reference to Macrobius and Ptolemy, he demonstrates first-hand knowledge of only Boethius’s \textit{De institutione harmonica} and the section (III.15–23) of Isidore’s \textit{Etymologies} which concerns music. He also has at least passing familiarity with some aspects of medieval music theory, as he makes reference to the ``Guidonian tradition'' (\textit{traditio Guidonis}) in problem 1 and on occasion employs solmization syllables, but he displays no further awareness of music-theoretical writing from after late Antiquity. (He is, however, certainly aware of his day’s musical culture, as he discusses antiphonal singing and medieval instruments, which are unmentioned in Boethius or Isidore.)
\subsection{IV. Principles of Edition}

\subsubsection{4.1 Latin Translation by Bartholomew of Messina}
\par
The textual tradition of Bartholomew of Messina’s Latin translation of the pseudo-Aristotelian text is largely independent of Pietro’s commentary, as only six of the twenty extant medieval manuscripts of Pietro’s commentary include the entirety of Bartholomew’s translation. Furthermore, sixty manuscripts contain part or the entirety of Bartholomew’s text, compared to only around thirty for the commentary (Coucke 2008 vol. 1, xxxvi; vol. 2, xxiii). As a result, a critically established text of the Latin translation of section XIX is beyond the scope of this edition, though I am presently preparing one for publication elsewhere. For readers’ convenience, I have reconstructed a version of Bartholomew’s translation that draws on the base text of my draft critical edition with alterations to reflect readings found in Pietro’s commentary.
\subsubsection{4.2 The \textit{Expositio problematum} by Pietro d’Abano}
\par
To constitute the text and the critical apparatus I have accounted for all the manuscripts containing Pietro’s first and second versions of the commentary (except for manuscript M\textsuperscript{b}, since it is a copy of M\textsuperscript{a}). The xml version of the edition includes all those readings, while this pdf version trims variants that are only found in one manuscript or have been deemed immaterial to the text's reconstruction. In this pdf version of the edition I have also included best readings from Jean of Jandun’s recension, usually based on manuscript V\textsuperscript{o}. Furthermore, readings from the two incunable editions are mentioned when they agree on a unique reading of interest or with variants found in other manuscripts.\par
In the edition’s text I aim to reconstruct Pietro’s second version of the commentary, and consequently readings shared by manuscript P\textsuperscript{x} and family λ are preferred. Where their readings diverge, if either P\textsuperscript{x} or λ agree with multiple manuscripts from Pietro’s first version, I prefer that shared text. Where the surviving manuscripts of Pietro’s second version are corrupt, manuscript P\textsuperscript{y} (and N\textsuperscript{u}, when not lacunate) is generally preferred.\par
With respect to orthography, I have largely preserved medieval spellings, rather than changing them to classical Latin standards. (The most prominent exceptions are that words ending with -cio, -cia, and the like have been changed to –tia, etc., and the –ij and –ijs endings to –ii and –iis.) The xml version of this edition records most spelling variances in the apparatus, but they have been silently normalized in this pdf version unless they are of particular interest. All contractions have been silently expanded, including the ampersand and \textbackslash Tironian \textit{et}, and tall s has been changed to round s. I have ignored all changes of hand and/or ink for the marginal numbering of the problems.\par
With regard to numbers, the manuscripts use a hodgepodge of written-out words, Arabic, and Roman numerals. To present a representatively medieval version of the text I have followed the approach used by the scribe of P\textsuperscript{x}. Punctuation also varies widely among the manuscripts. I initially adopted manuscript P\textsuperscript{x}’s punctuation, which is relatively sparse. Thereafter I added further punctuation in keeping with modern practice, especially where it was lacking at points of significant syntactic divisions. I followed a similar practice with paragraph divisions, basing them on those found in P\textsuperscript{x}, and adding others at moments clearly signaled by the text (such as new thoughts introduced by \textit{nosce quod, notandum,} and the like).\par
Relatedly, it should be noted that in section XIX manuscript V\textsuperscript{a} and the incunable editions mark the beginnings of new paragraphs with \textit{deinde}. In Coucke's edition of section IV, however, he shows while that the manuscripts of Pietro’s second version (excepting V\textsuperscript{b}, which Coucke assigns to the second version) use \textit{deinde} in that way, the remaining manuscripts do not. The change of practice in V\textsuperscript{a} is easily explained: a new copyist takes over at f. 47r, at which point that use of \textit{deinde} to mark new paragraphs begins. In the manuscripts from Pietro’s second version, by contrast, that use of \textit{deinde} gradually fades away, and seems largely to disappear after problem 15 of section 11.
\subsection{V. Bibliography}

\begin{bibitemlist}{1}
\bibitem  [Cadden 2013]{Cadden2013} Cadden, Joan. \textit{Nothing Natural is Shameful: Sodomy and Science in Late Medieval Europe.}  Philadelphia: University of Pennsylvania Press, 2013.
\bibitem  [Cohen 2012]{Cohen2012} Cohen, David E.. \textit{Harmonic Mediation and the Triad: Gaffurio, Zarlino, Lippius and Pietro d’Abano’s Commentary on the Pseudo-Aristotelian \textit{Problems}.} Paper presented at the annual conference of the  {\orgName Music Theory Society of New York State}. New York, N.Y., March 31, 2012.
\bibitem  [Coucke 2008]{Coucke2008} Coucke, Gijs. \textit{Philosophy between Text and Tradition The Reception of Aristotle's \textit{Problemata} in the Middle Ages: Peter of Abano's \textit{Expositio Problematum}}. Leuven: Ph.D. dissertation, Katholieke Universiteit Leuven, 2008.
\bibitem  [Coucke 2009]{Coucke2009} Coucke, Gijs. \textit{Needle in the Haystack: In Search of the Model of Peter of Abano's \textit{Expositio Problematum}}. \textit{Revue d'histoire des textes} 4 (2009): 179–213
\bibitem  [Klek 1922]{Klek1922} Klek, Joseph, Knoellinger, Hermann, Ruelle, Charles Emanuel, Pseudo-Aristotle. \textit{Aristotelis quae feruntur problemata physica}  Leipzig: Teubner, 1922.
\bibitem  [Kuksewicz 1985]{Kuksewicz1985} Kuksewicz, Zdzislaw. \textit{Les \textit{Problemata} de Pietro d'Abano et leur 'Rédaction' par Jean de Jandun}. \textit{Medioevo} 11 (1985): 113–37
\bibitem  [Leemans and Coucke 2011]{LeemansCoucke2011} Leemans, Pieter De, and Coucke, Gijs. \textit{\textit{Sicut vidi et tetigi}...: Ego-statements and Experience in Pietro d'Abano's \textit{Expositio Problematum Aristotelis}}. In \textit{Expertus sum: l'éxperience par les sens dans la philosophie naturelle médiévale, ed. Thomas Bénatouïl and Isabelle Draelants, 405–26. Florence, SISMEL, Edizioni del Galluzzo, 2011.}
\bibitem  [Meyer 1922]{Meyer2022} Meyer, Christian, d'Abano, Pietro. \textit{Expositio problematum (XIX): Édition, introduction et notes critiques et explicatives}  Leuven: Leuven University Press, 2022.
\bibitem  [Palisca 1985]{Palisca1985} Palisca, Claude. \textit{The Earliest Musical Humanists: Pietro d’Abano}. Chap. 3 in \textit{Humanism in Italian Renaissance Musical Thought. New Haven, Yale University Press, 1985.}
\bibitem  [Siraisi 1970]{Siraisi1970} Siraisi, Nancy. \textit{The \textit{Expositio Problematum Aristotelis} of Peter of Abano}. \textit{Isis} 61, no. 2 (1970): 321–39
\bibitem  [South 2002]{South2002} South, James B. \textit{John of Jandun}. In \textit{A Companion to Philosophy in the Middle Ages, ed. Jorge Gracia and Timothy Noone, 372–76. Malden, Mass., Blackwell, 2002.}
\bibitem  [Swaenepoel 2009]{Swaenepoel2009} Swaenepoel, Tine. \textit{Les deux versions des «\textit{Problemata}» de Pierre d'Abano}. \textit{Filologia mediolatina} 16 (2009): 215–45
\bibitem  [Zanetti 1741]{Zanetti1741} [Zanetti, Anton Maria]. \textit{Latina et italica D. Marci Bibliotheca codicum manu scriptorum per titulos digesta}. [Venice]: Apud Simonem Occhi Bibliopolam, 1741.

\end{bibitemlist}

\mainmatter
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\section[{Expositio problematum Aristotelis, Particula XIX: De harmonia}]{\centering\uppercase{\so{Expositio problematum Aristotelis, Particula XIX: De harmonia}}}\label{edition-text}
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\fancyhead[CE]{Expositio problematum Aristotelis}
\fancyhead[CO]{Particula XIX: De harmonia}

\subsection{Primum problema}\label{part1}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DOLENTES ET DELECTANTES FISTULANTUR? \textsuperscript{2}AUT VT HII QUIDEM MINUS TRISTENTUR? ALII AUTEM MAGIS GAUDEANT}{\lemma{PROPTER \dots\ GAUDEANT}\Cfootnote{PROPTER QUID DOLENTES \textit{βE\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{z}} | PROPTER QUID DOLENTES ET GAUDENTES \textit{V\textsuperscript{b}V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{2} Determinatis problematibus circa \edtext{mathematicam}{\Cfootnote{mathematica \textit{M\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{m}}}} et orationes, determinat nunc problemata circa armonicam vel musicam, eo quod in ipsa est quidam amor sermonis, vt in priori particula. Et primitus ad huius \edtext{maiorem}{\Cfootnote{\textit{om}. \textit{π}}} expositionem premittantur quedam communia, \edtext{sicut}{\Cfootnote{scilicet \textit{πV\textsuperscript{b}}}} quid est musica et que ipsius? \edtext{Que}{\Cfootnote{\textit{om}. \textit{π}}} secundum boecium \textit{musice} \edtext{5°}{\Cfootnote{4° \textit{κ} | 1° \textit{π}}} \edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boethius \textit{De inst. mus.} V.2 ((Friedlein p. 352, ll. 4–5))}} ``\edtext{est armonica}{\Cfootnote{armonica est \textit{V\textsuperscript{o}} | est armonice \textit{γP\textsuperscript{x}}}} facultas differentias \edtext{acutorum}{\Cfootnote{acutorum sonorum \textit{π}}} et grauium sonorum \edtext{sensu et}{\Cfootnote{sensu ac \textit{γE\textsuperscript{a}N\textsuperscript{u}} | \textit{om}. \textit{π}}} ratione perpendens.'' Secundum vero ysidorum in \edtext{\textit{ethymologiis}}{\Cfootnote{ethicis \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} \edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Isidore of Seville \textit{Etym.} III.15.1}} ``est peritia modulationis, sono cantuque consistens.''
\pend

\pstart
\textbf{3} Dicitur etiam quod ``sit scientia numeri ad sonos \edtext{proportionati}{\Cfootnote{proportionatos \textit{π}}}.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{An. post.} 90a.14–23 }} Que secundum ysidorum ``dicitur a muso \textit{\edtext{quasi a querendo}{\Cfootnote{quasi acquirendo \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}} | id est acquirendo \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}P\textsuperscript{z}}}}}, quia per \edtext{eam}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{vis}{\Cfootnote{ius \textit{π}}} carminum \edtext{querebatur}{\Cfootnote{acquiritur \textit{π}}}. Vnde a poetis \hyperref[Ju]{iouis} et \hyperref[Mem]{memorie} filias \edtext{mutas}{\Cfootnote{musas \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} esse confictum est, nam \edtext{si}{\Cfootnote{\textit{om}. \textit{κπ}}} \edtext{memoria soni}{\Cfootnote{soni memoria \textit{π}}} non \edtext{teneantur, pereunt}{\Cfootnote{teneatur, pereunt \textit{κ} | teneatur, perit \textit{π}}}, cum non \edtext{possint}{\Cfootnote{possunt \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}v} | possit \textit{πE\textsuperscript{a}}}} scribi.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Isidore of Seville \textit{Etym.} III.15.1–2}} \edtext{A nonnullis vero}{\Cfootnote{Aut \textit{π}}} dicitur a \textit{moys} \textit{quod est aqua,} quia \edtext{olim primitus}{\Cfootnote{olim amicus \textit{C\textsuperscript{a}V\textsuperscript{o}} | olim tali primitus \textit{G\textsuperscript{u}P\textsuperscript{q}}}} \edtext{inuenta fuit}{\Cfootnote{fuit inuenta \textit{πV\textsuperscript{a}}}} in illa, et in ydriis siue instrumentis aquaticis similiter exercita. ``Refert autem \hyperref[Mos]{moyses} hanc inuenisse \hyperref[Tub]{tubal} de stirpe \hyperref[Cay]{cayn} ante diluuium.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Isidore of Seville \textit{Etym.} III.16.1}} ``Sed greci,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boethius \textit{De inst. mus.} I.10 ((Friedlein p. 196–97)); Macrobius \textit{Comm. in somn. Scip.} II.1 ((Eyssenhardt p. 572–575))}} vt apparet per macrobium et boecium, ``dicunt eam inuenisse \edtext{\hyperref[Pyth]{pitagoram}}{\Cfootnote{\hyperref[Pyth]{pictagori} \textit{βE\textsuperscript{a}P\textsuperscript{x}} | pitagorici \textit{V\textsuperscript{a}mv}}} \edtext{\hyperref[Pyth]{samium}}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}} sonitu malleorum, cum casu in publicum exiret ad fabros, et cordarum \edtext{demum}{\Cfootnote{deinde \textit{π}}} impulsione.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boethius \textit{De inst. mus.} I.10 ((Friedlein p. 196–7)); Macrobius \textit{Comm. in somn. Scip.} II.1 ((Eyssenhardt p. 572–575))}}
\pend

\pstart
\textbf{4} Diuiditur autem musica iuxta eosdem in mundanam, humanam, et organicam vel instrumentalem. Mundana \edtext{vero}{\Cfootnote{quidem \textit{π}}} attenditur in sono armonico, quem causant orbes celestes, secundum \edtext{\hyperref[Pyth]{pictagoram}}{\Cfootnote{pithagoricos \textit{γ}}} cum aliis musicis, differenter \edtext{commotos}{\Cfootnote{commoti \textit{π}}}. ``Sonum enim armonicum causant, sed eum non \edtext{percipimus}{\Cfootnote{percipimus vt \textit{π}}}, dicunt, \edtext{propter quod in auribus nostris a principio natiuitatis est complantatus}{\Cfootnote{propter hoc quod a principio nobis in auribus nostris natum est et complantatur \textit{π}}}, vt in \edtext{eris}{\Cfootnote{eis \textit{γE\textsuperscript{a}}}} apparet malleatoribus strepitum suum vix percipientibus consuetudine''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{De caelo} II.9, 290b.25}} \edtext{quos}{\Cfootnote{quod \textit{π}}} aristotelem impugnat \edtext{\textit{celi et mundi} 2°}{\Cfootnote{2° \textit{celi et mundi} \textit{γλπ}}}. 
\pend

\pstart
\textbf{5} Musica vero humana dicitur que circa \edtext{concordantiam elementorum}{\Cfootnote{elementa \textit{π}}} diuersorum ad inuicem \edtext{in vno composito}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} vel compositionem corporis versatur, per \edtext{eam}{\Cfootnote{ea \textit{P\textsuperscript{x}V\textsuperscript{b}}}} enim natura spiritualis coniungitur \edtext{corpori et rationalis}{\Cfootnote{rationalis \textit{P\textsuperscript{x}} | \textit{om}. \textit{πE\textsuperscript{a}}}} concorditer cum irrationali \edtext{coniungitur}{\Cfootnote{\textit{om}. \textit{π}}}. Quod videtur a platonicis habitum ponentibus animam quandam armonicam \edtext{fore consistentiam}{\Cfootnote{esse consistentiam \textit{π}}}. Vnde \edtext{datum in \textit{afforismis}}{\Cfootnote{dictum est in \textit{afforismis} \textit{γλ} | damascenus in \textit{afforismis} \textit{π}}} ``colligatio \edtext{anime et corporis}{\Cfootnote{corporis et anime \textit{απ}}} secundum ordinis naturalis consonantiam facta est, vt \edtext{substat}{\Cfootnote{subsistat \textit{απ}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Yuḥannā ibn Māsawaiyh \textit{Aph.,} f. 39v, ll. 7–8) }} quare \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}V\textsuperscript{o}}}} \edtext{musice}{\Cfootnote{musice naturali \textit{α} | numeratione naturali \textit{π}}} \edtext{videntur}{\Cfootnote{videntur hec \textit{π}}} scientie supponi.
\pend

\pstart
\textbf{6} Instrumentalis vero dicitur que quibusdam exercetur armonicis instrumentis, que \edtext{predicamentis videtur}{\Cfootnote{predicamentis videntur \textit{E\textsuperscript{a}V\textsuperscript{o}V\textsuperscript{z}}}} \edtext{fore}{\Cfootnote{esse \textit{π}}} considerationis \edtext{existens duplex}{\Cfootnote{\textit{om}. \textit{π}}}: nam quedam exercetur instrumento naturali, vt hominis voce, cum reliquis organis \edtext{locutionis}{\Cfootnote{locutiuis \textit{M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}, \edtext{aut}{\Cfootnote{an \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} artificiali. Et id multipliciter: aut tactu vt viellis, vel pulsu \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} cimbalis, vel \edtext{flatu}{\Cfootnote{flatio \textit{P\textsuperscript{x}V\textsuperscript{m}}}} \edtext{vt}{\Cfootnote{sicut \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}} | in \textit{E\textsuperscript{a}P\textsuperscript{x}}}} tubis, aut quasi \edtext{omni modo}{\Cfootnote{omnino \textit{V\textsuperscript{b}V\textsuperscript{z}}}} \edtext{sicut}{\Cfootnote{vt \textit{πP\textsuperscript{y}}}} fistulis.
\pend

\pstart
\textbf{7} \edtext{Que}{\Cfootnote{Qui \textit{P\textsuperscript{x}V\textsuperscript{m}}}} iuxta ysidorum ``\edtext{diuiditur}{\Cfootnote{dicitur \textit{P\textsuperscript{x}V\textsuperscript{m}}}} in armonicam metricam et rithimicam. \edtext{Quarum}{\Cfootnote{\textit{om}. \textit{π}}} armonica graue \edtext{in sonis et acutum}{\Cfootnote{et acutum in sonis \textit{π}}} discernit. Metrica \edtext{est}{\Cfootnote{etiam \textit{κ}}} que ratione probabili cognoscit mensuram diuersorum pedum vt heroycum, iambicum et heliacum. Rithimica \edtext{vero}{\Cfootnote{vero est \textit{β} | est \textit{V\textsuperscript{a}}}} que verborum incursionem \edtext{inquirit}{\Cfootnote{iungit \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}}, vt \edtext{aut}{\Cfootnote{an \textit{βE\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}} | \textit{om}. \textit{π}}} suauiter \edtext{vel}{\Cfootnote{an \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} non suauiter sonus cohereat.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Isidore of Seville \textit{Etym.} III.18.1–2}}
\pend

\pstart
\textbf{8} Que siquidem \edtext{tria}{\Cfootnote{tripliciter \textit{γ}}} ad genus \edtext{videntur}{\Cfootnote{videtur \textit{γM\textsuperscript{a}V\textsuperscript{m}}}} concurrere cantilene, armonica tamen, vt \edtext{tactum}{\Cfootnote{tactum est \textit{πM\textsuperscript{a}V\textsuperscript{m}}}}, \edtext{consonantiam}{\Cfootnote{consonantia \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} grauis et acuti attendit, \edtext{rithimica}{\Cfootnote{rithimica vero \textit{β}}} \edtext{consonantiam}{\Cfootnote{consonantia \textit{P\textsuperscript{x}V\textsuperscript{m}}}} terminationum, metrica vero temporum vt apparet in \edtext{traditione}{\Cfootnote{introductione \textit{γP\textsuperscript{z}}}} musice \hyperref[GAr]{guidonis}. Et circa has tres partes musice, et maxime duas priores, \edtext{dubitat}{\Cfootnote{dubitabit \textit{λ}}} aristoteles ``vtilitas \edtext{vero}{\Cfootnote{\textit{om}. \textit{π}}} musice est,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII.5, 1339a.11}} quia delectat \edtext{animos}{\Cfootnote{animas \textit{απ}}} \edtext{excitat}{\Cfootnote{exercitat \textit{γP\textsuperscript{y}} | \textit{om}. \textit{π}}}, et affectus mouet, et furiam placat, \edtext{siluestrem}{\Cfootnote{siluestre \textit{πP\textsuperscript{y}}}} mansuescere \edtext{facit}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{m}}}}, vt apparet clangore, namque tubarum et similium fortiores redduntur bellatores. Similiter lassatos \edtext{et}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{o}}}} exercitio \edtext{laborioso}{\Cfootnote{laborioso et \textit{π}}} detentos reddit \edtext{faciles}{\Cfootnote{facilius \textit{π}}} tollerare. Vnde \textit{politicorum} 8°: ``musica habet delectationem naturalem propter quod omnibus etatibus et omnibus moribus vsus ipsius est amicus.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII.5, 1340a.3–5}}
\pend

\pstart
\textbf{9} Recitat etiam \edtext{boecius}{\Cfootnote{boecium \textit{P\textsuperscript{x}P\textsuperscript{y}}}} ``\edtext{\hyperref[Pyth]{pictagoram}}{\Cfootnote{pictagoricas \textit{γ}}} iuuenem furentem in \edtext{combustionem}{\Cfootnote{combustione \textit{πP\textsuperscript{y}}}} domus riualis in qua \edtext{sortum}{\Cfootnote{scortum \textit{λP\textsuperscript{y}}}} continebatur melodiis \edtext{musicis}{\Cfootnote{musicum \textit{γ}}} eius animum sedasse.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} I.1 ((Friedlein p. 185, l. 3))}} Cuius \edtext{vtilitatem}{\Cfootnote{subtilitatem \textit{V\textsuperscript{b}mv}}} tangit \textit{politicorum} 8°, ``\edtext{ponitur}{\Cfootnote{ponuntur \textit{πV\textsuperscript{z}} | ponit \textit{β}}} enim \edtext{vnum}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}P\textsuperscript{z}} | ibi \textit{π}}} quatuor que debet adiscere puer: que sunt \edtext{littere}{\Cfootnote{lingue \textit{γ}}} \edtext{luctatiua}{\Cfootnote{luctatio \textit{M\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} musica et protractiua,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII, 1337b.23}} \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}P\textsuperscript{p}P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{a}V\textsuperscript{o}}}} \edtext{scribere}{\Cfootnote{scriptura \textit{π}}} et pingere.
\pend

\pstart
\textbf{10} Querit \edtext{igitur}{\Cfootnote{ergo \textit{P\textsuperscript{x}V\textsuperscript{a}}}} \hyperref[Ar]{aristoteles} more \edtext{consueto}{\Cfootnote{solito \textit{λ}}} \edtext{problema, dicens}{\Cfootnote{\textit{om}. \textit{π}}}: \textit{Quare dolentes \edtext{et contristantes et}{\Cfootnote{contristantes et \textit{E\textsuperscript{a}G\textsuperscript{u}N\textsuperscript{u}P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{b}V\textsuperscript{o}} | contristantes \textit{P\textsuperscript{y}V\textsuperscript{a}} | et tristantes et \textit{M\textsuperscript{a}P\textsuperscript{x}}}} delectantes gaudentesque fistulas et \edtext{vniversaliter}{\Cfootnote{vniuersaliter omnia \textit{π}}} instrumenta \edtext{delectationem et gaudium}{\Cfootnote{gaudium et letitiam \textit{π}}} inducentia expedit audire?} Vnde \edtext{alexander}{\Cfootnote{alexander in \textit{problematibus} \textit{β}}} et maxime quantum ad primam partem: ``Propter quid coram patientibus fistule et tympana.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.-Alex. Aph. \textit{Prob.} I.78 ((Vat. reg. lat. 747, f. 75v))}}
\pend

\pstart
\textbf{11} \edtext{Nosce quod ``\edtext{omnes passiones}{\Cfootnote{licet omnes passiones \textit{γP\textsuperscript{y}}}} \edtext{sint coniuncte}{\Cfootnote{sicut continetur \textit{κ}}},'' \edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{De an.} I.1.403a.25. ((Cf. 403a l. 16))}} primo \textit{de anima}; quedam tamen \edtext{se plus}{\Cfootnote{plus se \textit{β}}} \edtext{tenet}{\Cfootnote{tenent \textit{βγ}}} ex parte corporis, vt dolere et delectari, alie vero ex parte anime vt gaudere et tristari; ita vt per vnum expressum aliud \hyperref[Ar]{aristoteles} audiat recitatum}{\lemma{Nosce \dots\ recitatum}\Cfootnote{\textit{om}. \textit{π}}}.
\pend

\pstart
\textbf{12} \edtext{\textit{AUT.}}{\Cfootnote{AUT QUIA. \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{q}P\textsuperscript{u}} | Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse vt \textit{qui tristantur \edtext{ac etiam}{\Cfootnote{et \textit{π}}} dolent minus \edtext{contristentur}{\Cfootnote{tristentur \textit{π}}} et doleant \edtext{propter}{\Cfootnote{propter melodiam vel \textit{π}}} iocunditatem inductam ex sonitu instrumenti.} In \edtext{oppositis}{\Cfootnote{oppositis vero \textit{π}}} causa \edtext{extat}{\Cfootnote{extet \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}} | existit \textit{π}}} vt \textit{magis gaudeant et \edtext{delectentur}{\Cfootnote{iocundentur \textit{π}}}, si enim aliqui sunt iocundi addita causa iocunditatis, vt fistulationis \edtext{armonice}{\Cfootnote{armonia \textit{V\textsuperscript{o}}}} \edtext{amplius delectabuntur.}{\Cfootnote{delectabuntur amplius. \textit{λ}}}}
\pend

\pstart
\textbf{13} Nosce quod \edtext{id}{\Cfootnote{illud \textit{π}}} potest contingere quia idem potest esse causa diuersorum aut \edtext{ratione}{\Cfootnote{ratione diuersorum aut ratione \textit{πE\textsuperscript{a}}}} dispositionis susceptiui; vel quia diuersa \edtext{egeant}{\Cfootnote{agat \textit{π}}} \edtext{eadem}{\Cfootnote{eodem \textit{λN\textsuperscript{u}V\textsuperscript{b}}}} \edtext{vel}{\Cfootnote{\textit{om}. \textit{πM\textsuperscript{a}}}} ratione eius quod per se \edtext{aut}{\Cfootnote{an \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}} | et \textit{V\textsuperscript{o}}}} per accidens extat, ``vt nauta per sui presentiam \edtext{causa}{\Cfootnote{causa est \textit{γ}}} salutis nauis, eius vero absentia submersionis''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Phys.} II.3, 195a.13}} 2° \textit{physicorum}.
\pend

\pstart
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\begin{center}
    {\Large \ding{167}}
\end{center}
\pend
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\newpage

\subsection{Secundum problema}\label{part2}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID LONGIUS IDEM EADEM VOCE AUDITUR CUM ALIIS CANTANS ET CLAMANS QUAM SOLUS? \textsuperscript{2}AUT QUIA COADUNATE ALIQUID FACERE, AUT REPRIMERE AUT IMPELLERE, NON TANTA PROPE SUNT QUANTUS NUMERUS. \textsuperscript{3}SED SICUT LINEA BIPEDALIS NON DUPLUM SED QUADRUPLUM ALIQUID SCRIBIT, ITA COMPOSITA PLUS POSSUNT, AUT SECUNDUM NUMERUM AUT QUANDO FUERINT DIUERSA. \textsuperscript{4}COADUNATIS IGITUR EXISTENTIBUS VNA FIT VOCIS FORTITUDO ET SIMUL IMPELLIT AEREM. QUARE MULTIPLICITER PREIT, \textsuperscript{5}ETENIM EX OMNIBUS VOX VNIUSCUIUSQUE MULTIPLEX}{\lemma{PROPTER \dots\ MULTIPLEX}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}} | PROPTER QUID LONGIUS \textit{V\textsuperscript{m}V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} vnus et idem numero eadem similiter voce cantans \edtext{et}{\Cfootnote{\textit{om}. \textit{κ}}} vociferans \edtext{remotius auditur quando cum aliis \edtext{vna}{\Cfootnote{una voce \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} cantat et clamat quam si solus tantum cantaret aut vociferaret}{\lemma{remotius \dots\ vociferaret}\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}?} Non tamen adeo \edtext{distantur}{\Cfootnote{distanter \textit{V\textsuperscript{o}V\textsuperscript{z}}}}, sicut vnus post alium ordinatus distantia \edtext{vociferaret}{\Cfootnote{vociferat \textit{π}}} \edtext{vt in}{\Cfootnote{vt \textit{κ} | in \textit{P\textsuperscript{y}}}} proicientibus lapidem successiue.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens \edtext{primo causam esse}{\Cfootnote{causam esse primo \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}}}} \textit{quia coniunctim aliquid agere, puta \edtext{est}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{y}M\textsuperscript{a}}}} reprimere \edtext{coangustante}{\Cfootnote{coangustare \textit{βM\textsuperscript{a}} | coangustando \textit{π}}} \edtext{an}{\Cfootnote{aut \textit{απM\textsuperscript{a}}}} \edtext{impellere}{\Cfootnote{compellere \textit{V\textsuperscript{o}}}} contendendo, vel aliquid \edtext{singulariter}{\Cfootnote{simile \textit{π}}} tale, non \edtext{tamen}{\Cfootnote{tantum \textit{λE\textsuperscript{a}}}} potest \edtext{per se vnumquodque}{\Cfootnote{vnum per se \textit{π}}},} \edtext{sicut quando plura}{\Cfootnote{quantum quando duo simul \textit{π}}} coniunguntur numerum constituentia, \edtext{dummodo}{\Cfootnote{dum tamen \textit{π}}} \edtext{ordinentur}{\Cfootnote{ordinetur \textit{V\textsuperscript{b}V\textsuperscript{m}V\textsuperscript{z}}}} in vnum \edtext{vt}{\Cfootnote{velut \textit{π}}} nauis \edtext{tractus}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} ostendit.
\pend

\pstart
\textbf{4} \edtext{\textit{SED SICUT.}}{\Cfootnote{Deinde. SED SICUT. \textit{V\textsuperscript{a}mv}}} Declarat secundo dicens quod \edtext{istud}{\Cfootnote{illud \textit{π}}} apparet ex tali simili quoniam \textit{sicut linea duorum pedum non solum duplum, sicut est potest \edtext{figurare}{\Cfootnote{firmare \textit{π}}} \edtext{intellecta moueri, sed}{\Cfootnote{intellectum ymmo etiam \textit{π}}} quadruplum et octuplum}, secundum quod diuersimode \edtext{mensurando applicatur}{\Cfootnote{infundo applicantur \textit{M\textsuperscript{a}} | applicatur mensurando \textit{π}}}, \textit{ita vt \edtext{super}{\Cfootnote{semper \textit{V\textsuperscript{o}mv}}} \edtext{plurem numerum habeat}{\Cfootnote{plurem numeri habeat \textit{M\textsuperscript{a}V\textsuperscript{m}} | pluralem numerum habeat \textit{V\textsuperscript{b}mv} | plures vnam habeant \textit{π}}} potentiam, similiter et ad inuicem \edtext{etiam}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{a}}}} in \edtext{quibuscunque}{\Cfootnote{quibusdam \textit{πE\textsuperscript{a}}}} aliis.} Cum autem plura fuerint numero amplius possunt, \edtext{vel}{\Cfootnote{vel quam \textit{π}}} cum diuisa fuerint quocunque alio distinguente dummodo vnum \edtext{proportionaliter compossit}{\Cfootnote{proportionaliter possit \textit{π}}} \edtext{alteri}{\Cfootnote{alteri coadunari \textit{π}}}.
\pend

\pstart
\textbf{5} \edtext{\textit{COADUNATIS.}}{\Cfootnote{\textit{om}. \textit{π} | Deinde. COADUNATIS. \textit{V\textsuperscript{a}mv}}} Concludit tertio \edtext{quod quando}{\Cfootnote{quod \textit{π}}} \textit{aliqua plura numero vel specie aut qualitercunque diuisa \edtext{coniungantur}{\Cfootnote{coniunguntur \textit{V\textsuperscript{o}}}} ad inuicem virtute, vtpote in voce vna ex omnibus \edtext{contingit}{\Cfootnote{contingit simul \textit{V\textsuperscript{o}}}} \edtext{vocum}{\Cfootnote{vocis \textit{π}}} fortitudo simul impellens \edtext{aerem}{\Cfootnote{aera \textit{π}}}. Et ideo \edtext{valde amplius}{\Cfootnote{amplius valde \textit{P\textsuperscript{x}V\textsuperscript{z}} | amplius \textit{M\textsuperscript{a}V\textsuperscript{m}}}} potest preire \edtext{et}{\Cfootnote{et in \textit{απ}}} longinquius audiri quam vox vnius} quod ostenditur. Nam ex voce uniuscuiusque vox quedam vna constituitur multiplicium habens potentiam, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} tractus optime una et diuisim \edtext{denotat nauis}{\Cfootnote{ostendit nauis \textit{π}}}. Et hoc quidem problema partim pertinet \edtext{.xi\textsuperscript{e}.}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} vt \edtext{visum}{\Cfootnote{visum est \textit{λV\textsuperscript{a}V\textsuperscript{o}}}} particule partimque isti.
\pend

\pstart
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\end{center}
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\newpage

\subsection{Tertium problema}\label{part3}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID PER YPATEM CANTANTES MAXIME RUMPIMUS NICHILOMINUS QUAM NITEM ET SUPERIORA, CUM DISTANTIA PLURI? \textsuperscript{2}AUT QUIA DIFFICILLIME HANC CANTANT, ET IPSA EST PRINCIPIUM? \textsuperscript{3}ID AUTEM DIFFICILE PROPTER ELEUATIONEM ET DEPRESSIONEM VOCIS. IN HIIS AUTEM LABOR, LABORANS AUTEM MAGIS CORRUMPITUR}{\lemma{PROPTER \dots\ CORRUMPITUR}\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}} | PROPTER QUID PARYPATEM \textit{V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} \edtext{cum cantamus}{\Cfootnote{cantantes \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} secundum notam quam illa corda representat que vocatur ypotes potissime interrumpimus cantum, non potentes ipsum continuare sicut \edtext{accepimus}{\Cfootnote{accipimus \textit{P\textsuperscript{x}V\textsuperscript{z}} | incepimus \textit{π}}} pre sui difficultate. Et in nullo \edtext{minus}{\Cfootnote{videtur \textit{mv}}} ymmo \edtext{in omnibus}{\Cfootnote{etiam \textit{V\textsuperscript{o}}}} magis frangimus cantum \edtext{propter}{\Cfootnote{per \textit{απ}}} ipsam quam per \edtext{nitem}{\Cfootnote{vitem \textit{V\textsuperscript{o}}}} \edtext{siue}{\Cfootnote{seu \textit{πV\textsuperscript{b}}}} netem. Et vniversaliter magis frangitur cantus et difficilius seruatur quando \edtext{vtitur}{\Cfootnote{vtuntur \textit{π}}} notis per cordas superiores \edtext{denotatis}{\Cfootnote{denotatur \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{x}}}}, et precipue quando distant inter se maxime.} Hoc autem \edtext{fit}{\Cfootnote{fit in \textit{P\textsuperscript{y}V\textsuperscript{z}} | fit instans \textit{γ} | sit \textit{E\textsuperscript{a}V\textsuperscript{m}}}} instrumentis \edtext{cantando aut per notas illas que sunt in instrumentis}{\lemma{cantando \dots\ instrumentis}\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{proportionatas}{\Cfootnote{proportionatis \textit{πE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}}}}. 
\pend

\pstart
\textbf{3} \edtext{Notandum}{\Cfootnote{Notandum quod \textit{P\textsuperscript{x}P\textsuperscript{z}V\textsuperscript{a}}}} ad intellectum eius quod \edtext{hic}{\Cfootnote{hic dicitur \textit{π}}} et multorum in hac particula dicendorum quod antiquitus a \edtext{\hyperref[Pyth]{pictagora}}{\Cfootnote{pitagoricis \textit{γ}}} inuentum fuit instrumentum dictum monocordum, eo quod vnam solam cordam haberet \edtext{taliter formatum}{\Cfootnote{talem formatum \textit{V\textsuperscript{a}V\textsuperscript{z}} | formatum taliter \textit{πE\textsuperscript{a}}}} in quo proportiones musice 5 cum tonis simplices prius cognosci \edtext{iubebat}{\Cfootnote{videbantur \textit{π}}} secundum diuersitatem \edtext{loci}{\Cfootnote{loci et \textit{β}}} instrumenti tacti \edtext{representante.}{\Cfootnote{representantem. \textit{M\textsuperscript{a}P\textsuperscript{x}}}} Deinceps \edtext{inuenta sunt instrumenta}{\Cfootnote{inuenta fuerunt instrumenta \textit{κ}}} cum pluribus cordis: vt duabus \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{q}P\textsuperscript{u}}}} rubeba que similiter organum \edtext{introductum}{\Cfootnote{introductorium \textit{βV\textsuperscript{o}}}} existit. \edtext{Denique}{\Cfootnote{Deindeque \textit{V\textsuperscript{b}mv} | Demumque \textit{βV\textsuperscript{a}} | Deinde quia \textit{V\textsuperscript{o}}}} paulatim sunt cetere \edtext{adinuente}{\Cfootnote{inuente \textit{P\textsuperscript{x}V\textsuperscript{b}}}}. Vnde primo \textit{musice} ait boecius ``\hyperref[Ni]{nichomacum} dicere musicam in principio fuisse simplicem in tantum quod tota constabat ex quatuor cordis. Quod durauit vsque ad \hyperref[Or]{orphei} tempus \edtext{puta, quod}{\Cfootnote{ex \textit{πE\textsuperscript{a}}}} \edtext{prima}{\Cfootnote{prima quidem \textit{π}}} et quarta corda \edtext{constitueret}{\Cfootnote{constituerent \textit{P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{z}} | constabat \textit{π}}} \edtext{consonantiam duplam}{\Cfootnote{consonantia dupla \textit{V\textsuperscript{o}}}} dyapason \edtext{dictam}{\Cfootnote{dicta \textit{V\textsuperscript{o}}}}. Medie \edtext{quidem}{\Cfootnote{vero \textit{π}}} \edtext{harum ad se inuicem}{\Cfootnote{harum ad inuicem \textit{πE\textsuperscript{a}}}} tonum constituebant, et ad \edtext{extremas}{\Cfootnote{extremas partes \textit{V\textsuperscript{b}mv}}} comparate dyapente et dyatesseron, vt apparebit .13°., ita vt in hiis quatuor cordis nichil dissonum existeret ad \edtext{imitationem}{\Cfootnote{mutationem \textit{P\textsuperscript{p}P\textsuperscript{q}P\textsuperscript{x}} | immutationem \textit{E\textsuperscript{a}V\textsuperscript{b}}}} mundane musice antedicte que ex quatuor constat \edtext{vt tactum elementis}{\Cfootnote{vt tactum est elementis \textit{V\textsuperscript{a}V\textsuperscript{m}} | elementis vt tactum est \textit{π}}}. Quarum quatuor cordarum dicitur \edtext{\hyperref[Mer]{mercurium} inuentorem}{\Cfootnote{\hyperref[Mer]{mercurius} inuentor \textit{π}}} fuisse. Vnde adhuc \edtext{aliqua}{\Cfootnote{aliquod \textit{π} | alia \textit{M\textsuperscript{a}V\textsuperscript{m}}}} instrumentorum \edtext{videmus tantum}{\Cfootnote{vidimus \textit{π}}} quatuor cordas habere vt citaram et zigam. quintam vero cordam chorebus \edtext{achis filius}{\Cfootnote{ac hiffilius \textit{π}}} lidorum rex adiunxit, hiagris quidem phirs sextam \edtext{apposuit}{\Cfootnote{apponit \textit{M\textsuperscript{a}V\textsuperscript{m}}}}. \edtext{Septimam}{\Cfootnote{Aliam \textit{E\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{o}}}} vero \hyperref[Ter]{terpanterius lesbius} inuenit planetarum \edtext{7}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{similitudinem}{\Cfootnote{similitudine \textit{γN\textsuperscript{u}V\textsuperscript{o}}}}. Inter quas grauissima erat, ypate dicta quasi maior et \edtext{honorabilior}{\Cfootnote{honor \textit{M\textsuperscript{a}P\textsuperscript{x}}}} vnde et \hyperref[Ju]{iouem} ypaton vocant id est supremum et consulem eodem nomine nuncupantes propter dignitatis excellentiam. Septem vero fuisse cordas antiquitus \edtext{testatur}{\Cfootnote{testabatur \textit{γP\textsuperscript{y}V\textsuperscript{o}}}} .25. \edtext{et .32.}{\Cfootnote{et .33. \textit{γ} | et .34. \textit{V\textsuperscript{o}}}} \edtext{Et}{\Cfootnote{\textit{om}. \textit{π}}} attributa fuit \edtext{is}{\Cfootnote{hec \textit{βπE\textsuperscript{a}}}} saturno propter tarditatem motus et grauitatem soni. Parypate vero secunda quasi iuxta ypatem collocata. Lychanos tertia ideo, quia lychanos \edtext{digitus dicitur}{\Cfootnote{dicitur digitus \textit{β}}} quem latini indicem vocant; greci vero a \edtext{lingendo}{\Cfootnote{ligando \textit{π}}} lychanos. Et quia in \edtext{cantando}{\Cfootnote{cantando ad \textit{απ}}}, hanc cordam tertiam ab ypate, digitus hic pertingit, quare ab \edtext{ipso}{\Cfootnote{illo \textit{λ}}} lychanos appellatur. Quarta vero dicitur mese \edtext{quasi}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{media quia}{\Cfootnote{quia media \textit{πE\textsuperscript{a}}}} inter primam et septimam vere media est quarta. Quinta paramese quasi iuxta mediam collocata. Septima quidem dicitur nete quasi neate id est inferior. Inter quam et paramese est sexta dicta paranete quasi iuxta netem sita. \edtext{Paranese}{\Cfootnote{Paramese \textit{N\textsuperscript{u}V\textsuperscript{a}}}} autem quia \edtext{tertia}{\Cfootnote{\textit{om}. \textit{π}}} est a nete eodem vocabulo trite id est tertia dicitur.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth \textit{De inst. mus.} I.20 ((Friedlein pp. 205–6, ll. 28–9))}} Fiat \edtext{igitur}{\Cfootnote{ergo \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{x}}}} \edtext{talis cordarum}{\Cfootnote{talis \textit{λ}}} parygrafia secundum \hyperref[Ty]{thimotheum} ``precipuum in hac arte virum:'' \edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Met.} II.1, 993b.15–17}}vt \edtext{apparet}{\Cfootnote{apparet in \textit{V\textsuperscript{o}}}} 2° \textit{methaphysice}, incipiendo a superiori cordarum vocem grauissimam reddenti \edtext{ad inferius}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} descendendo \edtext{vt}{\Cfootnote{usque \textit{P\textsuperscript{y}V\textsuperscript{o}}}} ad ultimam acutissimam representantem: 
\pend

\pstart
\begin{edtabularl}
Hypate & prima.\\
Parypate & secunda.\\
Lychanos & tertia.\\
Mese & quarta.\\
\edtext{Paramese}{\Cfootnote{Paramese vel trite \textit{γE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{o}}}} & quinta.\\
Paranete & sexta.\\
Nete & septima.
\end{edtabularl}
\pend

\pstart
\textbf{4} \edtext{Et}{\Cfootnote{\textit{om}. \textit{π}}} iste quidem 7 videntur esse ceteris essentialiores et necessariores, ita vt \hyperref[Ar]{aristoteles} \edtext{supra ipsas}{\Cfootnote{ipsas solas \textit{π}}} \edtext{non tangat}{\Cfootnote{non tangit \textit{γM\textsuperscript{b}P\textsuperscript{y}} | tangat non \textit{π}}} \edtext{vt apparebit ullam}{\Cfootnote{nulli tamen \textit{π}}}, \edtext{fueruntque}{\Cfootnote{fueritque \textit{P\textsuperscript{x}P\textsuperscript{y}} | fuerunt \textit{E\textsuperscript{a}M\textsuperscript{a}} | fuerunt qui \textit{π}}} consequenter .8. 9. \edtext{et 10.}{\Cfootnote{10. \textit{β} | et 10. dixerunt \textit{π}}} iuxta illud psalmiste ``in psalterio \edtext{decacordo}{\Cfootnote{decacordo psallam tibi \textit{P\textsuperscript{y}}}}.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{\textit{Psalm} 143.9 ((Psalm 144:9 in English translations))}} Et \edtext{deinceps}{\Cfootnote{deinde \textit{G\textsuperscript{u}P\textsuperscript{p}V\textsuperscript{o}}}} \edtext{in tantum}{\Cfootnote{iterum \textit{π}}} \edtext{creuit numerus}{\Cfootnote{excreuit numerus \textit{βV\textsuperscript{b}V\textsuperscript{o}} | exercuit numerus \textit{E\textsuperscript{a}V\textsuperscript{a}}}} vt \hyperref[Bo]{boecius} \edtext{18 cum}{\Cfootnote{cum \textit{πE\textsuperscript{a}V\textsuperscript{b}}}} reiteratione ipsarum enumeret sub dyatonico melo et totidem sub cromatico ac 18 sub enarmonico cum variatione \edtext{quadam}{\Cfootnote{quedam \textit{E\textsuperscript{a}V\textsuperscript{o}}}} communi sine zeugnum et diateuzeugnum. Non enim plura \edtext{tribus sunt}{\Cfootnote{sunt tribus \textit{π}}} genera melorum quibus omnis contexitur cantilena. \edtext{Cuius hec est descriptio}{\Cfootnote{Cuius est huius descriptio \textit{E\textsuperscript{a}V\textsuperscript{m}} | Quibus hec est descriptio \textit{β}}} 
\begin{itemize}
\item dyatonici
\item proslabanomenos
\item ypate ypaton
\item \edtext{parypate ypaton}{\Cfootnote{\textit{om}. \textit{γ}}} 
\item lychanos ypato dyatonici
\item ypate meson
\item parypate meson
\item lychanos meson dyatonici
\item mese
\item trite \edtext{sinezeugium}{\Cfootnote{sinezeugnum dyatonici \textit{κ}}} 
\item \edtext{paranete sinezeugium}{\Cfootnote{paranete sinezeumen diatonici \textit{λE\textsuperscript{a}V\textsuperscript{b}} | \textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{a}}}} 
\item \edtext{nete sunezeugium}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}V\textsuperscript{a}}}} 
\item paramese
\item trite diezeugium
\item paranete diezeugium dyatonici
\item nete diezeugium
\item \edtext{trite yperboleon dyatonici}{\Cfootnote{trite yperboleon \textit{βV\textsuperscript{a}}}} 
\item paranete yperboleon dyatonici
\item nete yperboleon.
\end{itemize}
Similis autem per omnia \edtext{figuratio extat}{\Cfootnote{est figuratio \textit{π}}} in cromatico melo \edtext{ac}{\Cfootnote{et \textit{πM\textsuperscript{a}}}} \edtext{consimilis}{\Cfootnote{similis \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} in enarmonico, excepto \edtext{quod vt}{\Cfootnote{quod \textit{γE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}}}} in prima \edtext{subscribitur}{\Cfootnote{scribitur \textit{βπ}}} \edtext{dyatonici}{\Cfootnote{dyatonicum \textit{G\textsuperscript{u}P\textsuperscript{q}} | dyatonicis \textit{P\textsuperscript{a}P\textsuperscript{p}} | dyatonicus \textit{P\textsuperscript{z}V\textsuperscript{o}}}} \edtext{ita}{\Cfootnote{ita vt \textit{γ}}} in secunda cromaticon et in tertia enarmonicum, ita vt omnes coniuncte .54. \edtext{sunt}{\Cfootnote{sint \textit{αV\textsuperscript{o}}}} corde. Moderni autem in aliquibus ultra .64. apposuere vt medius \edtext{canon}{\Cfootnote{canon dictus \textit{απ}}} ostendit. 
\begin{itemize}
\item dyatonici
\item proslabanomenos
\item ypate ypaton
\item parypate ypaton
\item lychanos ypato dyatonici
\item ypate meson
\item lychanos meson
\item Mese
\item Trite sinezeugium
\item Paranete sinezeugium dyatonici
\item Nete sinezeugium
\item Paramese
\item Trite dyazeugium
\item paranete dyezeugium dyatonici
\item zete dyezeugium
\item trite yperboleon dyatonici
\item Nete yperboleon
\end{itemize}

\pend

\pstart
\textbf{5} \edtext{\textit{AUT.}}{\Cfootnote{AUT QUIA. \textit{γ} | Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quia \edtext{cum}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{z}}}} difficultate \edtext{ampliori}{\Cfootnote{\textit{om}. \textit{γ}}} cantus fit secundum ypatem. Ipsa \edtext{enim}{\Cfootnote{\textit{om}. \textit{γ}}} est principium} \edtext{incipiendo}{\Cfootnote{\textit{om}. \textit{κ}}} superius, vt \edtext{prescriptum}{\Cfootnote{prescriptum est \textit{γλV\textsuperscript{o}}}} ad quam. Si perueniri debet vocem oportet eleuare et ab ipsa versus netem \edtext{deuenire}{\Cfootnote{venire \textit{P\textsuperscript{y}V\textsuperscript{z}}}}, cum cantus perficiatur \edtext{sonorus}{\Cfootnote{sonorum \textit{G\textsuperscript{u}P\textsuperscript{x}mv}}} ex contrariis eleuando et deprimendo vocibus.
\pend

\pstart
\textbf{6} \edtext{\textit{ID AUTEM.}}{\Cfootnote{Deinde. ID AUTEM. \textit{V\textsuperscript{a}mv}}} \edtext{Contingit}{\Cfootnote{Concludit \textit{V\textsuperscript{a}mv} | Dicitur \textit{πP\textsuperscript{x}}}} propter hoc \edtext{quod}{\Cfootnote{quia \textit{E\textsuperscript{a}V\textsuperscript{o}}}} contraria sunt \edtext{subitam}{\Cfootnote{subicita \textit{V\textsuperscript{m}V\textsuperscript{z}}}} inducentia alterationem. \textit{In hoc vero \edtext{intercidit}{\Cfootnote{interdicit \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} labor et difficultas, ubi autem \edtext{huius}{\Cfootnote{hoc \textit{M\textsuperscript{a}V\textsuperscript{z}} | huiusmodi \textit{γN\textsuperscript{u}}}} magis accidit corrumpi} sicut etiam in aliis apparet potentiis. \textit{Et ideo agonizantes cantant leges et non choros} 15°. Per alias vero cordas et proprie medias per oppositum cum difficultate cantatur \edtext{minime}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}.
\pend

\pstart
\textbf{7} \edtext{Circa predicta dubitatur}{\Cfootnote{Dubitatur circa dicta \textit{π}}} quoniam videmus \edtext{plura}{\Cfootnote{plurima \textit{αV\textsuperscript{o}}}} instrumenta habentia multo plures cordas \edtext{quam}{\Cfootnote{que \textit{γE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{m}}}} pretacte. Rota enim habet .44. vt .22\textsuperscript{as}. in vno latere et in reliquo \edtext{totidem}{\Cfootnote{\textit{om}. \textit{π}}} medius vero canon .64. et de \edtext{aliis}{\Cfootnote{reliquis \textit{π}}} \edtext{similiter}{\Cfootnote{consimiliter \textit{απ}}}. Dicendum quod ille plures reducuntur ad minores prius institutas. \edtext{Quatuor}{\Cfootnote{Quatuor enim \textit{π}}} superadditarum sumuntur pro vna \edtext{priorum}{\Cfootnote{priorum et \textit{E\textsuperscript{a}V\textsuperscript{o}}}}, ita vt \edtext{.xxviii.}{\Cfootnote{24 \textit{π}}} cadant in .7. notiores, \edtext{et}{\Cfootnote{\textit{om}. \textit{γ}}} .44. \edtext{redeant}{\Cfootnote{redeunt \textit{V\textsuperscript{o}}}} in \edtext{xi.}{\Cfootnote{.xii. \textit{π}}} et 64. in 16. Et similiter de aliis taliter variatis. Sicut \edtext{namque}{\Cfootnote{enim \textit{π}}} ``quatuor positi sunt gradus medicinarum''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Avic. \textit{Can. med.} I.2.2.15}} prius in tegni, \edtext{deinde quoque}{\Cfootnote{deindeque \textit{βγ} | deinde \textit{E\textsuperscript{a}π}}} quinto simplicium \edtext{vnusquisque}{\Cfootnote{\textit{om}. \textit{π}}} diuisus \edtext{est}{\Cfootnote{est quilibet \textit{V\textsuperscript{o}}}} in tres mansiones, vt agnoscatur quantum \edtext{possibile}{\Cfootnote{possibile est \textit{π}}} particularius complexio \edtext{medicine}{\Cfootnote{medie \textit{βE\textsuperscript{a}}}}. Ita posteriores musici non solum cordis fuere contenti quibus essentialiores et \edtext{integraliores}{\Cfootnote{irregulariores \textit{γ}}} denotarentur proportiones. Verum vt \edtext{sonum}{\Cfootnote{solum \textit{V\textsuperscript{a}mv}}} amplius melodiosum \edtext{construerent}{\Cfootnote{constituerent \textit{γπE\textsuperscript{a}}}} inter priores principaliores alias \edtext{adiunxere}{\Cfootnote{adiuxerunt \textit{π}}}, ne inter eas magna permanerent \edtext{interstitia}{\Cfootnote{in tristitia \textit{G\textsuperscript{u}P\textsuperscript{q}mv}}}.
\pend

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\subsection{Quartum problema}\label{part4}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID SECUNDUM HANC DIFFICULTER YPATEN, FACILE ETIAM DIUISIO ALTERIUS? \textsuperscript{2}AUT QUIA CUM REMISSIONE YPATE ET SIMUL POST DIMISSIONEM LEUE SURSUM PROICERE? \textsuperscript{3}PROPTER HOC AUTEM VIDETUR ET QUE DICUNTUR AD VNAM AD HANC QUAM CIRCA NITEM. \textsuperscript{4}OPORTET ENIM CUM DIMISSIONE ET REMISSIONE VALDE PROPRIA CONSUETUDINI AD VOLUNTATEM. \textsuperscript{5}HUIUS AUTEM VTIQUE CONSONANTIA QUEDAM CAUSA}{\lemma{PROPTER \dots\ CAUSA}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}} | PROPTER QUID SECUNDUM HANC \textit{V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} secundum \edtext{istam}{\Cfootnote{illam \textit{M\textsuperscript{a}V\textsuperscript{m}} | hanc \textit{π}}} cordam puta ypatem difficulter cantatur?} Et ideo dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{βP\textsuperscript{x}}}} \edtext{antea}{\Cfootnote{ante \textit{π}}} \textit{\edtext{secundum}{\Cfootnote{secundum eam \textit{απ}}} interruptionem cantus \edtext{peruenire}{\Cfootnote{prouenire \textit{M\textsuperscript{a}V\textsuperscript{o}}}}} et \textit{licet cum difficultate} cantetur. \textit{Tamen diuisio siue distinctio istius corde a quacunque alia est facilis et manifesta et proprie ab inferioribus}, vt nete ac \edtext{ei}{\Cfootnote{ea \textit{V\textsuperscript{o}} | eis \textit{P\textsuperscript{x}V\textsuperscript{a}}}} propinquis.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et prius ad id quod \edtext{subdictum}{\Cfootnote{subditum \textit{βE\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}} \edtext{secundo}{\Cfootnote{est secundo \textit{γV\textsuperscript{o}}}} secundo \edtext{dicens}{\Cfootnote{dicit \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{q}}}} \textit{\edtext{illud}{\Cfootnote{id \textit{λN\textsuperscript{u}V\textsuperscript{b}} | ideo \textit{πP\textsuperscript{y}}}} esse quia faciendo remissionem et quasi quandam interpollationem cum cantus fiat cum ypate \edtext{simulque}{\Cfootnote{simul que \textit{π}}} fere statim post durationem \edtext{vel}{\Cfootnote{et \textit{πE\textsuperscript{a}}}} \edtext{diuisionem}{\Cfootnote{dimissionem \textit{βE\textsuperscript{a}}}} intonationis grauem, leue fit vocem sursum emittere,} cum iam aliqualiter cantans \edtext{quieuerit}{\Cfootnote{conuenerit \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} emittendo spiritum et \edtext{attrahendo}{\Cfootnote{emittendo \textit{π}}} sufficienter in vocis organa. Et ideo percipi videtur discretio in cantu \edtext{facto}{\Cfootnote{facta \textit{π}}} per ypatem cum alia sibi opposita.
\pend

\pstart
\textbf{4} \edtext{\textit{PROPTER}}{\Cfootnote{Deinde. PROPTER \textit{V\textsuperscript{a}mv}}} \edtext{HOC}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}}. Assignat causam prioris \edtext{premissi}{\Cfootnote{dicti \textit{π}}} dicens \textit{\edtext{id}{\Cfootnote{illud \textit{γE\textsuperscript{a}}}} \edtext{fore}{\Cfootnote{esse \textit{π}}} quia melodie que referuntur ad unam cordam, puta ad illam que est circa netem \edtext{vt}{\Cfootnote{aut \textit{π}}} paranetem \edtext{ac etiam paramese}{\Cfootnote{\textit{om}. \textit{π}}} facilime \edtext{cantantur}{\Cfootnote{cantatur \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{a}}}},} eo quod non interuenit multa diuersitas vocum secundum acutum et \edtext{grauem}{\Cfootnote{graue \textit{πE\textsuperscript{a}M\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{b}}}}, \textit{sed tamen licet \edtext{per ypatem}{\Cfootnote{parypatem \textit{V\textsuperscript{o}}}} cantetur difficulter propter grandem notarum \edtext{distantiam}{\Cfootnote{differentiam \textit{π}}}. \edtext{Cum}{\Cfootnote{\textit{om}. \textit{π}}} tamen \edtext{fit}{\Cfootnote{sit \textit{E\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{z}}}} vt tactum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}}}} quadam dimissione et \edtext{remissione}{\Cfootnote{reuersione \textit{γ}}} quasi pausando ita \edtext{ea reddatur}{\Cfootnote{ea redditur \textit{π}}} propria valde consuetudini per voluntatem acquisite,} quis enim sic eligit, tunc vtique consuetudine inducta in cantando per ypatem. Secundum talem dimissionem \edtext{et remissionem}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{a}V\textsuperscript{z}}}} licet de se cantetur difficilime \edtext{illo autem modo}{\Cfootnote{illa autem \textit{π}}} minime, quoniam consuetudo difficilima etiam alleuiat \edtext{cuncta iuxta illud ypocratis \textit{afforisticis} ``assueti assuetos ferre labores et reliqua.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Hipp. \textit{Aph.} II.49}}}{\lemma{cuncta \dots\ reliqua}\Cfootnote{\textit{om}. \textit{π}}} Dicit autem voluntatem quia ``\edtext{consuetudo proprie}{\Cfootnote{proprie consuetudo \textit{π}}} in habentibus rationem acquiritur''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Eth. Nic.} II, 1103a.17}} \edtext{apparens}{\Cfootnote{vt apparet \textit{π}}} \edtext{\textit{ethycorum} secundo}{\Cfootnote{secundo \textit{ethicorum} \textit{π}}}.
\pend

\pstart
\textbf{5} \edtext{\textit{HUIUS}}{\Cfootnote{HUIUSMODI \textit{M\textsuperscript{a}V\textsuperscript{b}} | Deinde. HUIUS \textit{V\textsuperscript{a}mv}}} \textit{AUTEM.} Subdit tertio quod \textit{\edtext{etiam}{\Cfootnote{illud \textit{V\textsuperscript{o}}}} consonantia est \edtext{aliqua}{\Cfootnote{alia \textit{λE\textsuperscript{a}}}} causa \edtext{eius}{\Cfootnote{huius \textit{π}}},} quod \edtext{per ypatem}{\Cfootnote{parypatem \textit{κP\textsuperscript{u}V\textsuperscript{o}}}} cantus sepe non reputatur difficilius nam consonantia fessa \edtext{restaurantur}{\Cfootnote{restauratur \textit{γM\textsuperscript{a}}}} \edtext{vt visum et}{\Cfootnote{et \textit{π} | vt visum est \textit{γ} | vt visum est et \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}}}} \edtext{vigorantur}{\Cfootnote{vigoratur \textit{M\textsuperscript{a}V\textsuperscript{b}}}} debilitata.\edtext{ Εt ideo quesitum ``quare coram infirmis'' secundum alexander ``fistule ac tympana.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.-Alex. Aph. \textit{Prob.} I.78 ((Vat. reg.lat. 747, f. 75v))}}}{\lemma{Et ideo \dots\ tympana}\Cfootnote{\textit{om}. \textit{π}}} Sic igitur licet de se difficulter cantetur ypate per accidens, tamen non in tantum, puta \edtext{vel}{\Cfootnote{\textit{om}. \textit{π}}} ratione consuetudinis interuenientis aut \edtext{delectabilis}{\Cfootnote{delectationis \textit{π}}} ipsius armonie, cum fuerit \edtext{amicius}{\Cfootnote{ac amicius \textit{P\textsuperscript{x}} | animi eius \textit{γE\textsuperscript{a}} | cum eius \textit{M\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{o}} | eius \textit{P\textsuperscript{y}}}} \edtext{adiecta}{\Cfootnote{adiuncta \textit{πE\textsuperscript{a}}}} opposita, et \edtext{quod}{\Cfootnote{quia \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}} distinctio maxime apparet in eo qui secundum ipsam \edtext{cantu}{\Cfootnote{causatur \textit{π}}}.
\pend

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\subsection{Quintum problema}\label{part5}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DELECTABILITER AUDIUNT CANTANTES QUOSCUMQUE, VTIQUE PRESCIENTES ADIPISCUNTUR CANTUS, QUAM QUOS NESCIUNT? \textsuperscript{2}UTRUM QUIA MAGIS MANIFESTUS ADIPISCENS SICUT INTENTIONEM QUANDO COGNOSCUNT QUOD CANITUR? HOC AUTEM DELECTABILE CONSIDERARE, QUIA DELECTABILE DISCERE. HUIUS AUTEM CAUSA HOC QUIDEM ACCIPERE SCIENTIAM. HOC VERO VTI ET RECOGNOSCERE IPSAM. \textsuperscript{3}ADHUC AUTEM ET CONSUETUM DELECTABILE MAGIS INCONSUETO}{\lemma{PROPTER \dots\ INCONSUETO}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}} | PROPTER QUID DELECTABILITER \textit{V\textsuperscript{m}V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} cum delectatione maiori \edtext{audiunt}{\Cfootnote{audiuntur \textit{β}}} homines cantatores ac etiam sonatores cantantes omnes illos cantus quorum \edtext{preaudientes}{\Cfootnote{plaudientes \textit{V\textsuperscript{o}}}} \edtext{habuere}{\Cfootnote{habuerant \textit{V\textsuperscript{o}} | habuerunt \textit{G\textsuperscript{u}P\textsuperscript{q}P\textsuperscript{x}}}} prius aliquam cognitionem?} Siue \edtext{sit}{\Cfootnote{sint \textit{M\textsuperscript{a}V\textsuperscript{z}} | \textit{om}. \textit{πE\textsuperscript{a}}}} secundum verba, vt quod \edtext{aliquid}{\Cfootnote{aliqui \textit{π}}} de ystoria \edtext{gesti}{\Cfootnote{gesta \textit{πE\textsuperscript{a}}}} iam \edtext{audiuerint}{\Cfootnote{audierunt \textit{π}}}, vel \edtext{demum}{\Cfootnote{de modo \textit{απ}}} cantandi, eo quod \edtext{audiuerint}{\Cfootnote{audierunt \textit{λ}}} alias, illum \edtext{et}{\Cfootnote{\textit{om}. \textit{γP\textsuperscript{y}}}} consuetudinem \edtext{aliquam inde}{\Cfootnote{inde \textit{πE\textsuperscript{a}} | aliquam \textit{β}}} acceperunt quam illos cantus quorum \edtext{audientes}{\Cfootnote{\textit{om}. \textit{π}}} nullam \edtext{habuerunt}{\Cfootnote{habuerint \textit{P\textsuperscript{y}V\textsuperscript{m}}}} precognitionem omnino.
\pend

\pstart
\textbf{3} \edtext{\textit{UTRUM.}}{\Cfootnote{Deinde. VTRUM. \textit{V\textsuperscript{a}mv}}} Soluit et prius \edtext{probabilius magis}{\Cfootnote{magis probabiliter \textit{π}}} dicens \edtext{illud}{\Cfootnote{id \textit{βλV\textsuperscript{a}} | ideo \textit{V\textsuperscript{o}}}} esse \textit{quoniam quando homo aliquam prescientiam habuerit circa cantum \edtext{quem audiuit}{\Cfootnote{quam audit \textit{V\textsuperscript{o}}}}, adipiscitur \edtext{et attingit }{\Cfootnote{\textit{om}. \textit{π}}} ei quod est sibi magis manifestum.} Et hoc declarat quemadmodum quando aliquis habuerit intentionem gesti quod cantatur et hystoriam eius quodammodo \edtext{sciuit}{\Cfootnote{sciuerit \textit{βγ} | presciuit \textit{π}}}, tunc \textit{\edtext{hoc}{\Cfootnote{huic \textit{β} | hunc \textit{λ}}} intendere est delectabile precognoscentibus, eo quod iocundum est discere aliquid cuius non perfecte habetur cognitio}, quod \edtext{fit}{\Cfootnote{sit \textit{E\textsuperscript{a}P\textsuperscript{x}}}} iterum audiendo vel considerare, vt \edtext{continetur}{\Cfootnote{continetur in \textit{π}}} .40., ubi \edtext{id}{\Cfootnote{hoc \textit{βγπ}}} repetitur. Que duo interueniunt cum quis habet cognitionem gesti aliquam, nam ipse \edtext{discit}{\Cfootnote{dixit \textit{V\textsuperscript{b}V\textsuperscript{m}}}} in eo quod accipit scientiam cantus illius, eo quod gestum illud licet aliqualiter \edtext{esset}{\Cfootnote{est \textit{P\textsuperscript{x}V\textsuperscript{a}}}} scitum ab eo. Non tamen \edtext{per modum melodie}{\Cfootnote{per motionem die \textit{P\textsuperscript{x}} | promotionem die \textit{M\textsuperscript{a}V\textsuperscript{m}}}} vtitur autem et considerat \edtext{in}{\Cfootnote{\textit{om}. \textit{π}}} eo quod audit gestum cuius \edtext{habuit}{\Cfootnote{habet \textit{πP\textsuperscript{x}}}} cognitionem. Et ideo videmus homines vt plurimum \edtext{cum volunt amplius}{\Cfootnote{quando volunt in \textit{π}}} \edtext{cantu}{\Cfootnote{cantum \textit{E\textsuperscript{a}P\textsuperscript{x}}}} delectari facere sibi ystoriam gesti enarrare, tunc enim dupliciter \edtext{delectantur}{\Cfootnote{delectatur \textit{P\textsuperscript{x}V\textsuperscript{m}} | delectantur et melodia et ystoria et siquidem cum habuerint \textit{α} | delectantur et melodia et ystoria et siquidem cum hoc habuerit \textit{π}.}} \edtext{Triplex enim tunc dilectatio concurrit}{\Cfootnote{\textit{transp. post amplius delectantur} \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}} consuetudinem audiendi \edtext{predictum cantum}{\Cfootnote{cantum predictum \textit{π}}} aliquando in amplius \edtext{delectantur}{\Cfootnote{delectatur \textit{P\textsuperscript{x}V\textsuperscript{o}}}}.
\pend

\pstart
\textbf{4} \edtext{Vnde}{\Cfootnote{Deinde. Vnde \textit{V\textsuperscript{a}mv}}} secundo confirmat causam prefatam, \textit{ADHUC AUTEM,} \edtext{dicens}{\Cfootnote{dicit \textit{κ}}} quod amplius ad hoc facit \edtext{si quod}{\Cfootnote{sicut \textit{π}}} \edtext{auditur fuerit consuetum,}{\Cfootnote{auditur fuerit \textit{κE\textsuperscript{a}} | audiui \textit{π}}} \edtext{quia}{\Cfootnote{\textit{om}. \textit{κπE\textsuperscript{a}}}} \textit{consuetum magis delectat quam oppositum,} cum sit \edtext{facta}{\Cfootnote{factum \textit{κ}}} natura \edtext{quedam}{\Cfootnote{quadam \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}}}} et \edtext{nobis}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{conuenientia}{\Cfootnote{conuenientiam \textit{E\textsuperscript{a}V\textsuperscript{m}}}}. Quod autem ``\edtext{et}{\Cfootnote{et quomodo \textit{απ}}} consueta \edtext{delectent}{\Cfootnote{delectant \textit{βπ}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.-Arist. \textit{Prob.} 18.6, 917a.6–17}} ostensum \edtext{est}{\Cfootnote{est in \textit{V\textsuperscript{o}}}} sexto 18\textsuperscript{e}, et per oppositum quid agant opposita, et ideo audientes qui \edtext{vnius}{\Cfootnote{videntur \textit{V\textsuperscript{o}}}} \edtext{lingue sunt}{\Cfootnote{lingue \textit{βπE\textsuperscript{a}}}} cantum alterius minime nisi consueti delectantur, cum neque ystoriam neque \edtext{modum}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} \edtext{cantandi precognouerint}{\Cfootnote{cantandi precognouerit \textit{V\textsuperscript{m}} | precantandi cognouerit \textit{V\textsuperscript{b}} | precanendi cognouerint \textit{β}}}. In opposito \edtext{autem secus}{\Cfootnote{autem cetus \textit{V\textsuperscript{o}}}}, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{q}}}} michi contingit \edtext{per omne}{\Cfootnote{parisius \textit{π}}} cum ad gallos me transtuli. Vt autem id repetitum 40 \edtext{exponatur}{\Cfootnote{exponitur \textit{γ}}} omne, \hyperref[Ar]{aristoteles} talem addit causam, dicens ideo esse quoniam ille qui cantantem compatitur ipsi et ab ipso \edtext{afficitur}{\Cfootnote{efficitur \textit{V\textsuperscript{a}V\textsuperscript{z}}}}, propter voces quas exprimit que \edtext{audienti}{\Cfootnote{auditui \textit{πE\textsuperscript{a}}}} videntur denotare gaudium, et ideo consonat sibi \edtext{id quod}{\Cfootnote{illud quod \textit{γP\textsuperscript{y}} | quod \textit{π}}} ille cantat. Et quadam naturali inclinatione \edtext{ad delectationem mouetur}{\Cfootnote{mouetur ad delectationem \textit{V\textsuperscript{o}}}} in \edtext{seipso ceu}{\Cfootnote{se seu \textit{V\textsuperscript{o}}}} ille mouetur. Quando enim natura vnius percipit naturam alterius congaudere aut tristari vel se purgare, natura etiam illius \edtext{ad hoc mouetur}{\Cfootnote{mouetur ad hoc \textit{γE\textsuperscript{a}} | mouetur ad se hoc \textit{V\textsuperscript{o}}}} agendum, vt declaratum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}}}} septima. Quare subdit in eius \edtext{verificationem sermonis}{\Cfootnote{verificatione sermonis \textit{γ}}} quod omnis cantans consonat sibi et alium reddit consonum et \edtext{affectum}{\Cfootnote{affectantem \textit{V\textsuperscript{b}mv}}}, dummodo illum cantum non exprimat propter necessitatem, vt \edtext{sperans}{\Cfootnote{superans \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} \edtext{ita}{\Cfootnote{inde \textit{π}}} \edtext{victum}{\Cfootnote{vitum \textit{E\textsuperscript{a}N\textsuperscript{u}}}} lucrari \edtext{ceu}{\Cfootnote{vt \textit{π}}} ystriones aut metu pene vel tristitia, et \edtext{huius}{\Cfootnote{huiusmodi \textit{P\textsuperscript{y}V\textsuperscript{b}}}} propter que talia \edtext{cantantes}{\Cfootnote{\textit{om}. \textit{π}}} non \edtext{magne}{\Cfootnote{magis \textit{πV\textsuperscript{b}}}} \edtext{consonant}{\Cfootnote{consonans \textit{V\textsuperscript{m}V\textsuperscript{z}}}} aliis et maxime extraneis ab illo cantu \edtext{et sibimet ipsis}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} minime.
\pend

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\subsection{Sextum problema}\label{part6}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID PAR YPATE SECUNDUM RATIONEM IN CANTIBUS TRAGICUM. \textsuperscript{2}AUT PROPTER INEQUALITATEM? PASSIUUM ENIM INEQUALE, ET IN MAGNITUDINE FORTUNE AUT TRISTITIE. EQUALE AUTEM MINUS PLORABILE VEL VLULABILE}{\lemma{PROPTER \dots\ VLULABILE}\Cfootnote{PROPTER QUID PARYPATE \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βλE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} rationabile \edtext{existit}{\Cfootnote{est \textit{π}}} quod genus \edtext{cantandi}{\Cfootnote{\textit{om}. \textit{π}}} tragicum per ypatem cantetur?} Nosce \edtext{quod}{\Cfootnote{\textit{om}. \textit{πM\textsuperscript{a}}}} circa genus \edtext{armonicum}{\Cfootnote{armoniarum vel armonicum \textit{π}}} quodcumque duplicem maneriem cantandi constare, quedam enim est \edtext{comedia}{\Cfootnote{comedia et \textit{π}}} altera tragedia. Est autem comedia canticum agreste \edtext{que}{\Cfootnote{\textit{om}. \textit{π}}} res priuatarum et \edtext{humilium}{\Cfootnote{habilium \textit{γ}}} personarum \edtext{comprehendit}{\Cfootnote{comprehendens \textit{π}}}, et dicitur a cosmos quod est villa \edtext{vel laus}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} secundum aliquos. Grece tamen mundus et odi cantus quasi mundanus et vulgaris cantus, de rebus \edtext{enim}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{m}}}} vulgaribus tractat et affinibus cotidiane \edtext{locutionis}{\Cfootnote{locutioni \textit{πE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{a}}}}. Vel dicitur \edtext{a commessatione}{\Cfootnote{a commessatione quia \textit{π} | a commiseratione \textit{P\textsuperscript{x}} | a commensuratione \textit{V\textsuperscript{m}} | accessatione \textit{E\textsuperscript{a}N\textsuperscript{u}}}} post cibum, \edtext{enim}{\Cfootnote{\textit{om}. \textit{π}}} homines solebant ad eam audiendam conuenire. Tragedia vero est modus \edtext{carminis}{\Cfootnote{carminum \textit{β}}} quo poete in luctibus regum vtebantur. \edtext{Dicta}{\Cfootnote{Dictus \textit{π}}} a tragos quod \edtext{est}{\Cfootnote{dicitur \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} yrcus et odi \edtext{cantus}{\Cfootnote{quod est cantus \textit{V\textsuperscript{o}}}} \edtext{quasi}{\Cfootnote{inde \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}} hyrcinus cantus. Et quia materia erat \edtext{tristis}{\Cfootnote{tristis et \textit{λ}}} \edtext{feda}{\Cfootnote{seda \textit{βγM\textsuperscript{a}P\textsuperscript{x}}}}, \edtext{huiusmodi}{\Cfootnote{huius \textit{κE\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{o}}}} animal vt yrcum pro \edtext{munere}{\Cfootnote{numero \textit{V\textsuperscript{o}}}} \edtext{poematis}{\Cfootnote{\textit{om}. \textit{π}}} accipiebant additur. \edtext{Et}{\Cfootnote{Etiam \textit{πN\textsuperscript{u}}}} maneries a \edtext{quibus}{\Cfootnote{quibusdam \textit{πM\textsuperscript{a}N\textsuperscript{u}}}} tertia satira dicta que carmine \edtext{construitur}{\Cfootnote{constituitur \textit{γ}}} \edtext{reprehensorio}{\Cfootnote{reprehensoris \textit{V\textsuperscript{o}}}} \edtext{ceu}{\Cfootnote{seu \textit{V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} modus \edtext{\hyperref[Hor]{oratii} fuit}{\Cfootnote{fuit \hyperref[Hor]{oratii} \textit{απ}}} maxime.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse sumptam \textit{ab inequalitate, vtrobique \edtext{enim}{\Cfootnote{videlicet \textit{βγ}}} inest quedam inequalitas}, vt in cantu tragico et \edtext{per ypatem}{\Cfootnote{parypaten \textit{M\textsuperscript{a}P\textsuperscript{u}P\textsuperscript{z}}}} facto, propter \edtext{eius}{\Cfootnote{eam \textit{π}}} esse \edtext{supremum}{\Cfootnote{supremam \textit{π}}}. Nam in vnoquoque istorum est quedam passio et difficultas, quod autem \edtext{tale}{\Cfootnote{tale est \textit{λ}}}, \edtext{siue}{\Cfootnote{seu \textit{V\textsuperscript{o}} | Inequale seu \textit{G\textsuperscript{u}P\textsuperscript{q}}}} sit \textit{in magnitudine \edtext{\edtext{fortune}{\Cfootnote{fortitudine \textit{γP\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} ceu cantus magne letitie alicuius \edtext{magnatis}{\Cfootnote{magnatis et \textit{γN\textsuperscript{u}}}} probitatis, siue in magnitudine}{\lemma{fortune \dots\ in magnitudine}\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} tristitie}, sicut mortis alicuius excellentis \edtext{vel}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}P\textsuperscript{y}}}} \edtext{huiusmodi}{\Cfootnote{huius \textit{κE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{a}V\textsuperscript{o}}}}, est inequale, non eundem seruans tenorem. Quod autem est inequale et dissonum \edtext{rationabile}{\Cfootnote{rationale \textit{V\textsuperscript{m}V\textsuperscript{z}}}} est, vt cantu exprimatur \edtext{huiusmodi}{\Cfootnote{huius \textit{κE\textsuperscript{a}V\textsuperscript{o}}}}, vt pondus meloticum configeretur rei de qua existit. Tunc cantus consurgit ``eurithimus''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Gal. \textit{De diff. puls.} I.8 ((I.12 in Latin trans. by Burgundio of Pisa. Cf. Pietro d'Abano 
 Conciliator diff. 
 83))}} sicut de \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} pulsu scribitur \textit{de differentiis pulsuum} a galeno. \edtext{Ceu}{\Cfootnote{Seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} in bellis tubarum clangor bellantes concitat in furorem, in oppositis autem vt in choreis oppositum \edtext{expetitur}{\Cfootnote{experitur \textit{γV\textsuperscript{o}}}}, ac similiter in mortuorum lamentatione. Et \edtext{id}{\Cfootnote{illud \textit{α} | hoc \textit{π}}} est quod \edtext{subditur}{\Cfootnote{subdit \textit{πE\textsuperscript{a}}}} a contrariis probando pretactum, inquiens quod \textit{illud quod est equale in cantu et vnum conseruans tenorem non est \edtext{plorabile}{\Cfootnote{proportionabile \textit{γ}}} aut \edtext{ululabile}{\Cfootnote{lamentabile \textit{π}}}}, cum non sit quid passiuum, propter quod non est conueniens ipsum \edtext{parypatem}{\Cfootnote{per ypatem \textit{βγV\textsuperscript{z}}}} cuius est inequalitatem tenere sed per alias \edtext{cordarum}{\Cfootnote{cordas \textit{πP\textsuperscript{x}V\textsuperscript{b}}}} \edtext{mitiores}{\Cfootnote{anteriores \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} exponere.
\pend

\pstart
\textbf{4} Nosce quod ``pondus est illud in \edtext{quo}{\Cfootnote{quod \textit{V\textsuperscript{o}}}} proportiones'' secundum auicenna ``\edtext{cadunt musice}{\Cfootnote{musice cadunt \textit{π}}}.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Avic. \textit{Can. med.} I.2.3 ((p. 141 of 1608 ed.))}} Hoc autem pondus \edtext{rebus expedit}{\Cfootnote{expedit conformari rebus \textit{π}}} circa quas extat \edtext{configurari}{\Cfootnote{\textit{om}. \textit{π}}}, cum eo exprimantur passiones que sunt in anima. ``Voces enim sunt note passionum que sunt in \edtext{ea}{\Cfootnote{anima \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} illeque rerum''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Peri herm.} I.1, 16a.5–6}} primo \textit{peryarmenias.} Vnde si res fuerit \edtext{alta}{\Cfootnote{altera \textit{P\textsuperscript{p}P\textsuperscript{q}P\textsuperscript{u}}}} in signis \edtext{contingit}{\Cfootnote{conuenit \textit{M\textsuperscript{a}N\textsuperscript{u}P\textsuperscript{z}}}} tragedia ipsam exprimere. Si autem econtra comedia vt modus coaptetur rei \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} proprietas subiecto.
\pend

\pstart
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\subsection{Septimum problema}\label{part7}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID ANTIQUI .VII CORDARUM FACIENTES ARMONIAM YPATEM, SED NON NITEM RELIQUERUNT? \textsuperscript{2}UTRUM HOC MENDACIUM, VTRAMQUE ENIM RELIQUERUNT, TERTIAM VERO ELEUAUERUNT? \textsuperscript{3}AUT NON, SED QUIA GRAUIOR FORTIFICAT ACUTIORIS SONUM, QUARE MAGIS YPATE REDDIT ANTIFONUM QUAM NITE? \textsuperscript{4}QUIA ACUTUM POTENTIA MAGIS, GRAUE AUTEM FACILIUS CANTATUR}{\lemma{PROPTER \dots\ CANTATUR}\Cfootnote{PROPTER QUID ANTIQUI \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} est quod antiquitus \edtext{constituentes}{\Cfootnote{constitutiones \textit{P\textsuperscript{x}V\textsuperscript{m}}}} armoniam musicalem ex .7. cordis \edtext{diuiserunt}{\Cfootnote{dimiserunt \textit{λE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{b}V\textsuperscript{o}}}} ypatem \edtext{non}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} auferentes \edtext{ipsam}{\Cfootnote{eam \textit{π}}} ab armonia, non autem relinquerunt \edtext{nete}{\Cfootnote{netem \textit{γE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{o}}}},} \edtext{verum}{\Cfootnote{velut \textit{π}}} \edtext{inde abstulerunt}{\Cfootnote{inabstulerunt \textit{V\textsuperscript{o}}}}?
\pend

\pstart
\textbf{3} \edtext{\textit{UTRUM.}}{\Cfootnote{Deinde. VTRUM. \textit{V\textsuperscript{a}mv}}} Soluit et \edtext{prius}{\Cfootnote{primo \textit{πV\textsuperscript{a}V\textsuperscript{z}}}} \edtext{problema}{\Cfootnote{probatur \textit{β} | probabitur \textit{V\textsuperscript{o}} | probabiliter \textit{γE\textsuperscript{a}}}} dicens \textit{\edtext{id}{\Cfootnote{illud \textit{E\textsuperscript{a}V\textsuperscript{a}}}} \edtext{non esse}{\Cfootnote{non \textit{γ} | esse \textit{κ}}} verum quod premissum \edtext{est}{\Cfootnote{\textit{om}. \textit{βV\textsuperscript{b}V\textsuperscript{o}}}}, quoniam non \edtext{seruauerunt solum}{\Cfootnote{solum seruauerunt \textit{π}}} ypatem, ymmo \edtext{vtramque}{\Cfootnote{vtraque \textit{β}}} puta ypatem et netem. Sed quam abstulerunt fuit tertia, vt paramese vel lychanos, vtraque enim est tertia \edtext{eius}{\Cfootnote{huius \textit{π}}} quod est \edtext{7. esse}{\Cfootnote{esse 7. \textit{λ}}} cordas.} Vnde 32°. dicetur quod \textit{\hyperref[Ter]{tripandius} extrahens tertiam cordam a septicordo netem \edtext{apposuit}{\Cfootnote{apponunt \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}}.} Sic \edtext{ergo}{\Cfootnote{igitur \textit{απ}}} per \edtext{interemptionem}{\Cfootnote{interpretationem \textit{β}}} soluit.
\pend

\pstart
\textbf{4} \edtext{\textit{AUT}}{\Cfootnote{Deinde. AUT \textit{V\textsuperscript{a}mv}}} \textit{NON.} Soluit magis ad \edtext{orationem}{\Cfootnote{rationem \textit{P\textsuperscript{y}V\textsuperscript{a}}}} \edtext{secundo}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}} \edtext{dicens}{\Cfootnote{dicit \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} \edtext{nichil}{\Cfootnote{hoc \textit{γ}}} esse quod dicitur vt quod abstulissent netem et dimisissent stare ypatem. Sed hoc dictum fortassis ex hoc pullulauit, quia \textit{sonus \edtext{cordis}{\Cfootnote{corde \textit{βγπM\textsuperscript{a}V\textsuperscript{z}}}} grauis \edtext{vigorat}{\Cfootnote{fortificat \textit{G\textsuperscript{u}} | figurat \textit{V\textsuperscript{o}}}} sonum acutioris.} Suo enim multo tempore et duratione graue substentat acutum ne cito pereat cum breui tempore feriat. \textit{Propter quod ypate longo tempore \edtext{aere}{\Cfootnote{aerem \textit{γM\textsuperscript{a}V\textsuperscript{o}}}} percutiente \edtext{quam}{\Cfootnote{quem \textit{γ}}} nete reddat, ob eius acuitatem, que \edtext{potentia}{\Cfootnote{ponitur \textit{V\textsuperscript{o}}}} vel fortitudine indiget in cantando, difficile namque acutum cantare sed graue facilius,} sicut pluries dicetur. Sic itaque quia grauis corda sonum fortificat acute et ad se conuertit. Et quia \edtext{difficile}{\Cfootnote{difficilime est \textit{γ} | \textit{om}. \textit{β}}} acutum cantare, quis credidit ex hoc cum \edtext{nete}{\Cfootnote{netem \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} sonum non \edtext{perciperet}{\Cfootnote{percipet \textit{P\textsuperscript{x}V\textsuperscript{a}} | percipiet \textit{M\textsuperscript{a}V\textsuperscript{z}}}} proprium quod esset sublata? Et fortassis aliquis leget \edtext{illud}{\Cfootnote{istud \textit{E\textsuperscript{a}V\textsuperscript{z}} | illud vt tactum multe \textit{γ}}} \textit{\edtext{QUIA ACUTUM}{\Cfootnote{\textit{om}. \textit{π}}}}, quod aliam solutionem subiungat.
\pend

\pstart
\textbf{5} Nosce quod ``armonia que est vna trium partium \edtext{vt tactum est musice}{\Cfootnote{vt tactum musice \textit{E\textsuperscript{a}N\textsuperscript{u}} | musice vt tactum est \textit{π} | \textit{om}. \textit{γ}}}, proprie pertinet ad comedos et tragedos vel choros, et omnino ad eos qui precipue voce cantant. Ex ea namque \edtext{causatur}{\Cfootnote{causatur talis \textit{βπE\textsuperscript{a}}}} armonia que est \edtext{vocis}{\Cfootnote{vocum \textit{βπE\textsuperscript{a}}}} modulatio et concordia plurium sonorum vel coaptatio. Simphonia est \edtext{vocis}{\Cfootnote{vocum \textit{βπ}}} modulatio temperata ex graui et \edtext{acuto accordantibus}{\Cfootnote{acuto a cordantibus \textit{γπE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} sonis, cuius quidem \edtext{contraria}{\Cfootnote{contraria est \textit{π}}} dyaphonia. Euphonia est suauitas \edtext{vocis}{\Cfootnote{vocum \textit{βM\textsuperscript{a}V\textsuperscript{m}}}}, hec autem melos a melle et suauitate dicta. Dyastema est \edtext{vocis}{\Cfootnote{vocum \textit{βE\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} spatium ex duobus vel pluribus sonis \edtext{coaptatum}{\Cfootnote{coadaptatum \textit{M\textsuperscript{a}V\textsuperscript{m}}}}. Dyesis est \edtext{spatia}{\Cfootnote{spatium \textit{πE\textsuperscript{a}}}} \edtext{quedam et}{\Cfootnote{quoddam \textit{π}}} \edtext{deducens}{\Cfootnote{deductiones \textit{βπE\textsuperscript{a}V\textsuperscript{a}}}} modulandi et vergendi de vno in alterum sonum.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Isidore of Seville \textit{Etym.} III.20.1–6}} Boecius autem ``\edtext{dyesim dicit}{\Cfootnote{dicit Dyesim \textit{π}}} \edtext{vnius}{\Cfootnote{minus \textit{πE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{m}}}} fore semitonii.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth \textit{De inst. mus.} II.28 ((Friedlein p. 260, ll. 21–24))}} ``Tonus \edtext{est}{\Cfootnote{autem \textit{V\textsuperscript{o}}}} \edtext{acuta vocis}{\Cfootnote{vocis acuta \textit{P\textsuperscript{x}V\textsuperscript{z}} | acuta arta vocis \textit{π}}} enunciatio. Αrsis quidem vocis eleuatio, thesis depressio.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Isidore of Seville \textit{Etym.} III.20.7, 9}} Sic igitur ex symphonia potest haberi quod antiphonum vel antiphona sit quedam modulatio \edtext{vel}{\Cfootnote{et \textit{απ}}} intonatio contra simphoniam, vt sonus qui post eam refrangitur auditum demultiens.
\pend

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\subsection{Octauum problema}\label{part8}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID GRAUIS ACUTE FORTIFICAT SONUM? \textsuperscript{2}AUT QUIA MAIUS HOC GRAUE? OBTUSO ENIM ASSIMULATUR, HOC VERO ACUTO ANGULO}{\lemma{PROPTER \dots\ ANGULO}\Cfootnote{PROPTER QUID GRAUIS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \edtext{Quia}{\Cfootnote{Quare \textit{E\textsuperscript{a}P\textsuperscript{x}}}} dixerat \textit{grauem cordam sonum fortificare acute,} \edtext{causam siquidem querit}{\Cfootnote{querit causam \textit{π}}} huius dicens \textit{\edtext{quare}{\Cfootnote{quia \textit{β}}} grauis corde sonus \edtext{vigorat}{\Cfootnote{vigilat \textit{β}}} eum qui acute?}
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse quoniam \textit{illud quod est graue maius est et fortius. Quod ostenditur quoniam \textit{angulo \edtext{simulatur}{\Cfootnote{si multiplicatur \textit{γ}}} \edtext{obtuso}{\Cfootnote{acuto \textit{π}}} trianguli, qui quidem maior est et fortior \edtext{etiam}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} vigore acuto \edtext{acuto vt res ostendunt nature et artis, acutum vero angulo proportionatur acuto}{\Cfootnote{\textit{om}. \textit{κ}}},} \edtext{ceu}{\Cfootnote{seu \textit{πE\textsuperscript{a}}}} minori et debiliori. Quod autem \edtext{maius}{\Cfootnote{maius est \textit{π}}} et fortius vigorat quod minus et debilius,} \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}V\textsuperscript{o}}}} quidem graue acuti respectu, quare ipsum vigorat, \edtext{vt 12°. etiam apparebit}{\Cfootnote{\textit{om}. \textit{π}}}.
\pend

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\subsection{Nonum problem}\label{part9}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DELECTABILITER VNITATEM CANTUS AUDIMUS SI QUIS AD FISTULAM AUT LYRAM CANTAT ET AD CORDAS, ET EUNDEM CANTUM CANTANT VTROBIQUE? \textsuperscript{2}SI ENIM ADHUC MAGIS IDEM, POTIUS OPORTEBAT AD MULTOS FISTULATORES ET ADHUC DELECTABILITER ESSE. \textsuperscript{3}AUT QUIA ADIPISCENS MANIFESTUS INTENTIONEM MAGIS QUANDO AD FISTULAM AUT LYRAM? QUOD VERO AD MULTOS FISTULATORES AUT LYRAS MULTAS NON EST DELECTABILE PROPTER ID QUOD DESTRUIT MELODIAM}{\lemma{PROPTER \dots\ MELODIAM}\Cfootnote{PROPTER QUID DELECTABILITER \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} \edtext{cum delectatione}{\Cfootnote{delectabiliter \textit{π}}} audimus cantum unitum seu in vnum \edtext{coniectum}{\Cfootnote{coniunctum \textit{απM\textsuperscript{a}V\textsuperscript{m}}}}, vt si quis cantat in fistula vel lyra et vniuersaliter in cordis \edtext{ceu}{\Cfootnote{seu \textit{πE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{m}}}} \edtext{in}{\Cfootnote{\textit{om}. \textit{βπ}}} psalterio canone et \edtext{huius}{\Cfootnote{huiusmodi \textit{βV\textsuperscript{b}}}} instrumentis ex cordis multis compositis, dummodo vtrobique \edtext{cantet}{\Cfootnote{causet \textit{P\textsuperscript{y}V\textsuperscript{o}}}} eundem cantum?} Quoniam si vnus cantus seu dantia fieret \edtext{in}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{a}} | in vno \textit{πP\textsuperscript{x}}}} instrumento et \edtext{alius}{\Cfootnote{alius in \textit{λ}}} voce \edtext{cantantis}{\Cfootnote{causantis \textit{P\textsuperscript{x}V\textsuperscript{a}V\textsuperscript{m}} | canentis \textit{E\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}, \edtext{in}{\Cfootnote{etiam \textit{γ}}} illo non \edtext{consurgeret}{\Cfootnote{consurget \textit{M\textsuperscript{a}P\textsuperscript{y}}}} propter dissonantiam delectatio.
\pend

\pstart
\textbf{3} \edtext{\textit{SI}}{\Cfootnote{Deinde. SI \textit{V\textsuperscript{a}mv}}} \textit{ADHUC.} Quasi arguit in contrarium dicens quod \textit{si \edtext{unius}{\Cfootnote{huius \textit{P\textsuperscript{y}V\textsuperscript{a}} | huiusmodi \textit{V\textsuperscript{b}N\textsuperscript{u}} | talis \textit{π}}} delectatio contingat propter vnitatem cantus et ydemptitatem, tunc oportebit magis delectari cum quis cantabit cum multis fistulantibus vel lyrantibus,} cum magis \edtext{videantur}{\Cfootnote{videatur \textit{N\textsuperscript{u}P\textsuperscript{x}}}} super vnum opus \edtext{unitari}{\Cfootnote{imitari \textit{V\textsuperscript{a}V\textsuperscript{o}}}}, vt plures trahentes nauem \edtext{imitantur}{\Cfootnote{vnitantur \textit{M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}, vt melius trahant.
\pend

\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \edtext{quia}{\Cfootnote{quoniam \textit{λ}}} \textit{cum aliquis \edtext{cantat}{\Cfootnote{cantet \textit{E\textsuperscript{a}V\textsuperscript{o}}}} \edtext{singulariter}{\Cfootnote{signanter \textit{γ} | singulariter cantat \textit{π}}} intentionem et maneriem cantus magis vnitatur vox cum sono instrumenti quam \edtext{quando}{\Cfootnote{cum \textit{πE\textsuperscript{a}}}} cantaret in diuersis instrumentis}, \edtext{etiam}{\Cfootnote{et \textit{P\textsuperscript{y}V\textsuperscript{z}}}} eodem cantu vtrobique existente. Et hoc quia in cantu \edtext{quedam}{\Cfootnote{quid \textit{V\textsuperscript{o}}}} reseruantur proportiones vt dicetur commensurate. Quibus obseruatis iocunda \edtext{consurgit}{\Cfootnote{consistit \textit{V\textsuperscript{o}}}} armonia, que quidem corrumpitur et indelectabilis redditur cum cantans ad multa cantauerit instrumenta.
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\textbf{5} Quod autem \edtext{inductum}{\Cfootnote{inductum est \textit{πM\textsuperscript{a}V\textsuperscript{m}}}} \edtext{in}{\Cfootnote{ad \textit{π}}} contrarium prius non valet quia licet multi possint idem cantare, non tamen eodem modo. \edtext{Non solum enim requiritur}{\Cfootnote{Et non requitur solum \textit{π}}} ydemptitas ex parte illius quod cantatur et cuiuslibet cantantis ad se ipsum ymo \edtext{uniuscuiusque}{\Cfootnote{uniuscuiusque et \textit{γλE\textsuperscript{a}}}} ad alium et ad omnes, ita quod ex omnibus resultent proportiones debite in cantu. Quod autem \edtext{subditur}{\Cfootnote{subdit \textit{V\textsuperscript{a}} | subicitur \textit{V\textsuperscript{o}} | subiecti \textit{E\textsuperscript{a}P\textsuperscript{x}} | subieci \textit{λV\textsuperscript{b}}}} de \edtext{nauis tractu}{\Cfootnote{tractu nauis \textit{α} | tractu nauis vel ligni \textit{π}}} non \edtext{huic}{\Cfootnote{hoc \textit{V\textsuperscript{m}V\textsuperscript{z}}}} \edtext{similatur}{\Cfootnote{assimilatur \textit{π}}} dicto, quoniam plurimi possunt \edtext{simul idem}{\Cfootnote{idem simul \textit{π}}} vt nauem eodem modo et tempore trahere vt audientes vocem rectoris nauis. \edtext{Non}{\Cfootnote{Vt \textit{P\textsuperscript{x}V\textsuperscript{m}}}} autem talis vnitas potest in cantu plurium \edtext{obseruari}{\Cfootnote{reseruari \textit{π}}} armonico, cum \edtext{eum}{\Cfootnote{enim \textit{V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} \edtext{ex}{\Cfootnote{\textit{om}. \textit{π}}} pluribus et contrariis oporteat \edtext{constituere}{\Cfootnote{construere \textit{V\textsuperscript{o}P\textsuperscript{y}V\textsuperscript{z}}}} in \edtext{unum}{\Cfootnote{unam \textit{V\textsuperscript{o}}}}. Tandem \edtext{redditur}{\Cfootnote{redditis \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{m}} | reductis \textit{V\textsuperscript{o}}}} quod \edtext{fit}{\Cfootnote{sit \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{a}}}} impossibile pluribus entibus vocibus \edtext{et}{\Cfootnote{in \textit{M\textsuperscript{a}V\textsuperscript{m}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{z}}}} instrumentis.
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\subsection{Decimum problema}\label{part10}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID SI DELECTABILIS HOMINIS VOX QUE SINE SERMONE CANTANTIS NON DELECTABILE EST VT VERNANTIUM, SED FISTULA AUT LYRA? \textsuperscript{2}AUT NON ILLIC, SI NON IMMUTATUR, SIMILITER DELECTABILE? \textsuperscript{3}AT VERO SED PROPTER OPUS IPSUM. VOX ENIM DELECTABILIS QUE EST HOMINIS, PERCUSSIUA AUTEM MAGIS INSTRUMENTA ORIS. PROPTER QUOD DELECTABILIUS AUDIRE QUAM VERNARE}{\lemma{PROPTER \dots\ VERNARE}\Cfootnote{PROPTER QUID NON DELECTABILIS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} si vox humana est delectabilis \edtext{auditu}{\Cfootnote{auditui \textit{V\textsuperscript{o}}}}}, vnde ``\edtext{\hyperref[Mu]{museus}}{\Cfootnote{musicus \textit{γ} | dicitur \textit{π}}}'' \edtext{\textit{politicorum} .8°.}{\Cfootnote{octauo \textit{politicorum} \textit{π}}} ``\edtext{delectabilissimum}{\Cfootnote{delectabili fine \textit{κ} | quod delectabilissimum \textit{π}}} est hominibus cantare,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII, 1339b.22}} \textit{cum \edtext{quis}{\Cfootnote{aliquis \textit{πE\textsuperscript{a}}}} cantat \edtext{etiam}{\Cfootnote{\textit{om}. \textit{π}}} non proferendo sermonem omnino} aut \edtext{insignificatium}{\Cfootnote{insignium \textit{V\textsuperscript{o}}}}, \edtext{cur}{\Cfootnote{\textit{om}. \textit{π}}} non \edtext{est delectabilis}{\Cfootnote{delectabilis est \textit{απ}}} \edtext{vox alicuius vernantium}{\Cfootnote{Videtur tamen \textit{π}}}, cum sit hominis vox, \edtext{et}{\Cfootnote{eti \textit{P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{m}} | etiam \textit{γ}}} lyre et fistule sonus est delectabilis? \edtext{Qui}{\Cfootnote{Que \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{o}}}} minorem communicationem \edtext{videtur cum homine}{\Cfootnote{cum homine videtur \textit{π}}} habere quam vox ipsa humana. In hiis itaque verbis \hyperref[Ar]{aristoteles} succincte tetigit motiuum quesiti et confirmationem eius.
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\textbf{3} Notandum quod \textit{vernare} \edtext{\textit{vocis}}{\Cfootnote{vocem \textit{π}}} est \textit{expressionem facere in qua non \edtext{manifestatur}{\Cfootnote{manifestetur \textit{λE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}} aliqua \edtext{sensibilis}{\Cfootnote{subtilitas \textit{π}}} diuersitas extensione et depressione vocis, ita quod eius remaneat equalitas,} ad modum quo ver \edtext{equale ponitur}{\Cfootnote{ponitur equale \textit{π}}} a medicis. Est enim quasi \edtext{illud quod dicitur}{\Cfootnote{\textit{om}. \textit{π}}} bordonizare \edtext{quod}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} per se \edtext{minime}{\Cfootnote{minimam \textit{π}}} \edtext{videtur}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} delectationem inducere.
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\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et \edtext{prius}{\Cfootnote{primo \textit{P\textsuperscript{y}V\textsuperscript{a}}}} respondet ad confirmationem, dicens quod \textit{illic scilicet in \edtext{fistula}{\Cfootnote{fistulis \textit{π}}} et lyra non est delectari, quoniam similiter \edtext{in eis ita se habet}{\Cfootnote{etiam se habet eis \textit{π}}} vt \edtext{in}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}P\textsuperscript{x}}}} vernante}, \edtext{quia}{\Cfootnote{quoniam \textit{λπ}}} si non fiat immutatio soni secundum ascensionem et depressionem, vt inde \edtext{consurgant}{\Cfootnote{desurgant \textit{V\textsuperscript{m}V\textsuperscript{z}}}} debite proportiones armonice, non contingit delectari ex \edtext{huius}{\Cfootnote{huiusmodi \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{m}V\textsuperscript{o}}}} instrumenti sono. \edtext{Sic}{\Cfootnote{Sicut \textit{λ}}} igitur vernare non est delectabile, eo quod proportiones \edtext{non cadunt}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} \edtext{in ipso}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{melite}{\Cfootnote{multipliciter \textit{π}}} auditum \edtext{demulcentes}{\Cfootnote{demulceantes \textit{P\textsuperscript{x}V\textsuperscript{m}}}}.
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\pstart
\textbf{5} \edtext{\textit{AUT}}{\Cfootnote{AT \textit{V\textsuperscript{b}V\textsuperscript{o}V\textsuperscript{z}} | Deinde. AUT \textit{V\textsuperscript{a}mv}}} \textit{UERO.} Respondet motiuo ipsius \edtext{causam dando}{\Cfootnote{dando causam \textit{π}}} quare vox humana cantantis sit delectabilis. \textit{Hoc enim est propter operationem ex qua consequitur. In ipsa namque magis percutiuntur instrumenta oris ordinata ad formationem cantus delectabilis quam \edtext{in}{\Cfootnote{ex \textit{π}}} vernare. Ex \edtext{huius}{\Cfootnote{huiusmodi \textit{V\textsuperscript{b}V\textsuperscript{m}V\textsuperscript{o}}}} \edtext{namque}{\Cfootnote{autem \textit{π}}} \edtext{maiori}{\Cfootnote{maiore \textit{V\textsuperscript{o}}}} percussione instrumentorum oris que fit in cantu, nunc in vocem extollendo \edtext{nunc}{\Cfootnote{tunc \textit{αV\textsuperscript{m}}}} in deprimendo,} modo vero medie conseruando resultat \edtext{cantus}{\Cfootnote{modus \textit{π}}} armonicus suauis valde. Et ideo inquit \textit{delectabilius \edtext{fore}{\Cfootnote{esse \textit{π}}} vocem hominis cantantis audire, cum hic \edtext{huiusmodi}{\Cfootnote{huius \textit{M\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{z}}}} contingat quam vocem vernantis, ubi minime.} Vox etiam hominis delectabilis quia consueta et quoniam potest mutari \edtext{componi et}{\Cfootnote{et potest \textit{E\textsuperscript{a}P\textsuperscript{z}V\textsuperscript{o}}}} contraponi facillime.
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\subsection{Vndecimum problema}\label{part11}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID SONANS ACUTIOR? \textsuperscript{2}AUT QUIA MINUS IMBECILLIOR FACTA EST}{\lemma{PROPTER \dots\ EST}\Cfootnote{PROPTER QUID SONANS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} aliquis sonans sonitu vel voce acutior redditur et per consequens alius grauior?}
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse quia \edtext{hic}{\Cfootnote{\textit{om}. \textit{γ}}} sonans est minus sonans quam illud quod grauiter sonat. \textit{Quod autem sit minus ostenditur quia fit a virtute \edtext{que}{\Cfootnote{quia \textit{P\textsuperscript{x}P\textsuperscript{y}}}} facta est imbecillior.} Grauiter autem sonans a virtute \edtext{causatur fortiori}{\Cfootnote{fortiori causatur \textit{π}}}, propter quod dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}P\textsuperscript{y}}}} \edtext{.xi\textsuperscript{a}.}{\Cfootnote{\textit{om}. \textit{π}}} ``quod pueri femine \edtext{eunuchi}{\Cfootnote{eunuci et \textit{γ}}} infirmi habent voces acutas,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XI.16, 900b.15–17}} oppositi \edtext{autem}{\Cfootnote{vero \textit{π}}} graues \edtext{propter}{\Cfootnote{per \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}} oppositum.
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\subsection[Duodecimum problema … ]{Duodecimum problema}\label{part12}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID CORDARUM GRAUIOR SEMPER SONUM ACCIPIT? SI ENIM OPORTET DIMITTERE QUE EST CIRCA MEDIAM PARAMESEM CUM MEDIA ALTERA, FIT MEDIUM NIHILOMINUS. SI AUTEM MEDIA VTRAQUE, ALTA NON FUERIT. \textsuperscript{2}AUT QUIA GRAUE MAGNUM EST, QUARE FORTE? ET INEST IN MAGNO PARUUM, ET ASSUMPTIONE DUE NITE IN YPATE FIUNT}{\lemma{PROPTER \dots\ FIUNT}\Cfootnote{PROPTER QUID CORDARUM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} de cordis illa que grauioris \edtext{extat}{\Cfootnote{est \textit{π}}} toni semper accipit \edtext{sonum}{\Cfootnote{tonum \textit{πP\textsuperscript{x}}}}} et representat pre aliis ipsum vt subiectum proprietatem, dum tamen equaliter fiat percussio et \edtext{ad hoc}{\Cfootnote{adhuc \textit{πP\textsuperscript{y}V\textsuperscript{z}}}}, \textit{si oportet propter aliquam causam \edtext{vt}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} defectum ipsius \edtext{aut aliter}{\Cfootnote{aliquando \textit{V\textsuperscript{o}}}} dimittere paramese que est iuxta mediam,} scilicet iuxta mese \edtext{siue mediam}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{uere}{\Cfootnote{ipsa \textit{M\textsuperscript{a}V\textsuperscript{z}} | hec \textit{πE\textsuperscript{a}}}}. Namque .7. cordarum vt visum est media \textit{\edtext{cum alia quodammodo media}{\Cfootnote{quodammodo cum aliquo \textit{π}}} existente} vt lychanos, ita quod ea \edtext{sonum}{\Cfootnote{sonus \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}} \edtext{non}{\Cfootnote{nisi \textit{P\textsuperscript{x}mv}}} \edtext{constituatur}{\Cfootnote{constituat \textit{π}}} in armonia. Nihilominus \edtext{ex}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} extremis puta ypate nete parypate \edtext{etiam et}{\Cfootnote{et etiam \textit{V\textsuperscript{o}} | etiam \textit{P\textsuperscript{x}V\textsuperscript{a}}}} paranete, \edtext{sicut ex illis tribus mediis}{\Cfootnote{\textit{om}. \textit{π}}} resultat quoddam medium constitutiuum armonie, in quo grauior cordarum sonum preaccipit aliis acutis. Communis quidem littera circa mediam\edtext{partem}{\Cfootnote{partem habet \textit{πP\textsuperscript{y}} | \textit{om}. \textit{mv}}} \edtext{hoc}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}} tamen licet possit tollerari series tamen iam exposita continens paramese \edtext{planior et sauior}{\Cfootnote{sauior et planior \textit{P\textsuperscript{y}V\textsuperscript{a}} | planior et senior \textit{P\textsuperscript{x}V\textsuperscript{m}}}}. \textit{\edtext{Sed}{\Cfootnote{Et \textit{V\textsuperscript{a}V\textsuperscript{o}}}} si \edtext{vtraque media}{\Cfootnote{vtreque medie \textit{V\textsuperscript{a}}}} \edtext{coniungantur}{\Cfootnote{coniungatur \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} ad inuicem vtpote dicatur intermediarum, sicut \edtext{sunt}{\Cfootnote{est \textit{E\textsuperscript{a}V\textsuperscript{o}}}} lychanos mese paramese siue trite, non constituitur sonus eleuatus, ita vt inde resultet grauis,} tanquam \edtext{corde}{\Cfootnote{cordis \textit{E\textsuperscript{a}V\textsuperscript{b}}}} grauioris reddentis sonum grauiorem, cum sonus \edtext{harum}{\Cfootnote{earum \textit{π}}} \edtext{existat}{\Cfootnote{extat \textit{V\textsuperscript{a}V\textsuperscript{z}}}} in medio inter \edtext{graue}{\Cfootnote{grauem \textit{λP\textsuperscript{y}V\textsuperscript{a}}}} et acutum. Sic itaque videtur duo querere puta \edtext{quia}{\Cfootnote{quare \textit{απM\textsuperscript{a}}}} grauior cordarum sonum preaccipiat et \edtext{cur}{\Cfootnote{cum \textit{πE\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{z}}}} \edtext{ex}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{z}}}} extremis concursus fiat in medium reddens grauem sonum non autem ex mediis.
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{AUT QUIA. \textit{α} | Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse quia \textit{\edtext{id}{\Cfootnote{illud \textit{αV\textsuperscript{o}}}} \edtext{quod}{\Cfootnote{\textit{om}. \textit{π}}} graue magnum et forte diuque \edtext{perdurans}{\Cfootnote{permanens seu perdurans \textit{π}}} preaccipit vigorem sui oppositi, talis siquidem est grauior cordarum ad alias, \edtext{etiam}{\Cfootnote{et \textit{M\textsuperscript{a}V\textsuperscript{z}}}} in magno \edtext{continetur et}{\Cfootnote{continuo et \textit{M\textsuperscript{a}V\textsuperscript{z}}}} \edtext{et}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}P\textsuperscript{x}}}} occultatur parvuum, sicut in \edtext{superiori}{\Cfootnote{siteriorum \textit{V\textsuperscript{o}}}}, taliter quod sonus duarum nitarum assumptus est in sono vno parypatem causato.} Ex quo habetur quod grauior cordarum sonum preaccipit, et \edtext{quod}{\Cfootnote{\textit{om}. \textit{κP\textsuperscript{z}}}} ex \edtext{ultimis extremis}{\Cfootnote{extremis ultimis \textit{π}}} medium potest resultare, sicut apparebit \edtext{.23°}{\Cfootnote{22° \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{z}}}}.
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\subsection[Tertiumdecimum problema … ]{Tertiumdecimum problema}\label{part13}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID IN DYAPASON ACUTI QUIDEM ANTIFONUM FIT GRAUE, HUIUS AUTEM ACUTUM NON? \textsuperscript{2}AUT QUIA MAXIME QUIDEM IN AMBOBUS EST AMBORUM MELODIA? SI AUTEM NON, NON GRAUE, MAIUS ENIM EST}{\lemma{PROPTER \dots\ EST}\Cfootnote{PROPTER QUID IN DYAPASON \textit{P\textsuperscript{p}P\textsuperscript{u}V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} in proportione illa que dicitur dyapason vel proportio dupla vel \edtext{totius}{\Cfootnote{tonus \textit{π}}}, illud quod habet rationem acuti plerumque \edtext{pertransit}{\Cfootnote{transit \textit{πE\textsuperscript{a}P\textsuperscript{y}}}} in \edtext{grauem}{\Cfootnote{graue \textit{γE\textsuperscript{a}}}}, \edtext{quod autem}{\Cfootnote{illud autem quod \textit{π}}} habet rationem grauis non permutatur in acutum?}
\pend

\pstart
\textbf{3} \edtext{Videndum}{\Cfootnote{Videndum est \textit{πV\textsuperscript{a}}}} nunc gratia \edtext{eius}{\Cfootnote{huius \textit{π}}} quod dicitur dyapason quot et que \edtext{sunt}{\Cfootnote{sint \textit{γE\textsuperscript{a}V\textsuperscript{z}}}} proportiones \edtext{musicales}{\Cfootnote{musicales et \textit{γπP\textsuperscript{y}}}} quomodo sumantur, et \edtext{quia}{\Cfootnote{dico quod \textit{π}}} originantur ex proportionibus arismetricis \edtext{intuendum prius illas.}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} Sunt autem secundum boecium in primis \textit{arismetrice} et \textit{musice} numeri ``5. \edtext{sumpte}{\Cfootnote{sumptis \textit{π}}} penes numerum inequalem. Quarum tres sunt simplices puta \edtext{multiplex et submultiplex}{\Cfootnote{multiplex submultiplex \textit{V\textsuperscript{o}} | multiplex \textit{κ}}}, superparticularis \edtext{et}{\Cfootnote{\textit{om}. \textit{π}}} subsuperparticularis, superpartiens et subsuperpartiens. Due autem composite quarum prima componitur ex multiplici et superparticulari \edtext{et ex}{\Cfootnote{et \textit{π}}} \edtext{multiplici}{\Cfootnote{supermultiplici \textit{γ} | submultiplici \textit{P\textsuperscript{y}V\textsuperscript{o}}}} et subsuperparticulari, secunda ex multiplici superparticulari et \edtext{submultiplici}{\Cfootnote{submultiplici et \textit{E\textsuperscript{a}P\textsuperscript{z}V\textsuperscript{a}}}} \edtext{subsuperparticulari}{\Cfootnote{subsuperpartiente \textit{V\textsuperscript{o}}}}.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} I.4 ((Friedlein p. 191, ll. 6–10)); Boeth. \textit{De inst. arith.} I.22 ((Friedlein p. 46, ll. 6–17))}}
\pend

\pstart
\textbf{4} Multiplex autem \edtext{est quando}{\Cfootnote{est \textit{P\textsuperscript{x}M\textsuperscript{a}} | \textit{om}. \textit{V\textsuperscript{m}V\textsuperscript{z}}}} numerus maior minorem in se \edtext{continet}{\Cfootnote{continens \textit{M\textsuperscript{a}V\textsuperscript{z}}}} pluries, vt bis vel ter \edtext{aut}{\Cfootnote{vel \textit{π}}} quater et sic deinceps, vt nichil desit nichilque exsuperet. \edtext{Huius}{\Cfootnote{Huiusmodi \textit{E\textsuperscript{a}V\textsuperscript{b}}}} autem species sunt duplex triplex quadruplex et sic in infinitum. Duplex quidem \edtext{cum}{\Cfootnote{vt \textit{V\textsuperscript{o}}}} vnus numerus alium bis continet que dicitur proportio dupla vt quatuor ad 2. Triplex cum vnus numerus continet alium \edtext{totum }{\Cfootnote{numerum \textit{V\textsuperscript{o}}}} tribus vicibus sicut trinarius \edtext{se habet ad vnitatem}{\Cfootnote{ad vnitatem se habet \textit{π}}}.
\pend

\pstart
\textbf{5} Proportio autem secunda iuxta aliam inequalitatem, vt superparticularis cum eius \edtext{opposita}{\Cfootnote{opponitur \textit{γV\textsuperscript{o}}}}, est cum maior \edtext{numerus}{\Cfootnote{\textit{om}. \textit{π}}} continet in se totum semel \edtext{et}{\Cfootnote{et adhuc \textit{π}}} vnam eius \edtext{insuper}{\Cfootnote{\textit{om}. \textit{π}}} partem, vt aut \edtext{dimidia}{\Cfootnote{dimidiam \textit{πE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{a}}}} \edtext{vel}{\Cfootnote{aut \textit{π}}} \edtext{tertia}{\Cfootnote{tertiam et \textit{απM\textsuperscript{a}}}} ita deinceps. \edtext{Cuius}{\Cfootnote{Huius \textit{γ}}} quidem proportionis modi sunt sexquialtera, \edtext{sexquitertia vel epitrita}{\Cfootnote{sexquinta vel epitrica \textit{M\textsuperscript{a}V\textsuperscript{z}} | sexquitertia \textit{π}}}, sexquiquarta et \edtext{huius}{\Cfootnote{huiusmodi \textit{P\textsuperscript{y}V\textsuperscript{b}}}}. Sexquialtera seu emiola est cum \edtext{quis}{\Cfootnote{aliquis \textit{π}}} numerus alium semel continet et insuper \edtext{eius mediam partem}{\Cfootnote{partem eius mediam \textit{λ}}}, vt \edtext{ternarius se habet ad binarium}{\Cfootnote{ternarius ad binarium se habet \textit{λ}}}. Sexquitertia \edtext{cum}{\Cfootnote{\textit{om}. \textit{γ}}} semel et eius tertiam partem, vt quaternarius ad ternarium.
\pend

\pstart
\textbf{6} Superpartiens \edtext{est}{\Cfootnote{\textit{om}. \textit{κ}}} quando maior numerus continet minorem et insuper \edtext{eius aliquas}{\Cfootnote{aliquas eius \textit{π}}} partes, vt \edtext{duas}{\Cfootnote{3\textsuperscript{as} \textit{V\textsuperscript{o}}}} tres et cetera. Cuius species sunt superbipartiens supertripartiens et cetera. Superbipartiens est cum numerus continet numerum semel et eius duas partes, vt quinarius ad ternarium se habet. \edtext{Supertripartiens}{\Cfootnote{Supertripartiens est \textit{γ}}} quando semel continet totum \edtext{et eius}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} \edtext{tres partes}{\Cfootnote{tertiam partem \textit{γE\textsuperscript{a}}}}, vt 7. se habet ad 4. Superquadripartiens \edtext{et cetera, vt}{\Cfootnote{etiam, sicut \textit{π}}} .9. \edtext{se habet}{\Cfootnote{\textit{om}. \textit{π}}} ad .5.
\pend

\pstart
\textbf{7} \edtext{Compositarum}{\Cfootnote{Compositorum \textit{π}}} vero multiplex superparticularis est cum numerus \edtext{continet maior}{\Cfootnote{maior continet \textit{πM\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}} minorem totum multotiens et insuper \edtext{aliquam}{\Cfootnote{aliam \textit{πE\textsuperscript{a}}}} partem eius, vt medietatem vel tertiam et huius. \edtext{Huius}{\Cfootnote{Huiusmodi \textit{P\textsuperscript{y}V\textsuperscript{b}}}} autem \edtext{species sunt}{\Cfootnote{sunt species \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}V\textsuperscript{o}}}} duplex sexquialtera, \edtext{duplex sexquitertia}{\Cfootnote{duplex sexquitertia et \textit{λ}}}, duplex sexquiquarta, et cetera, et \edtext{id ratione}{\Cfootnote{id rationem \textit{E\textsuperscript{a}V\textsuperscript{b}}}} \edtext{duplicitatis}{\Cfootnote{duplicis \textit{E\textsuperscript{a}V\textsuperscript{o}}}}; ex parte vero \edtext{triplicitatis}{\Cfootnote{triplicis \textit{V\textsuperscript{o}}}} \edtext{triplex sexquialtera,}{\Cfootnote{\textit{om}. \textit{λ}}} \edtext{triplex sexquitertia}{\Cfootnote{triplex sexquitertia duplex sexquitertia \textit{V\textsuperscript{m}V\textsuperscript{z}}}}, \edtext{et cetera}{\Cfootnote{triplex et cetera \textit{P\textsuperscript{y}} | triplex sexquiquarta \textit{V\textsuperscript{o}}}}. \edtext{Est}{\Cfootnote{\textit{om}. \textit{π}}} duplex sexquialtera quando maior \edtext{numerus}{\Cfootnote{numerus bis \textit{π}}} \edtext{continet}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{z}V\textsuperscript{m}}}} minorem totum et insuper eius partem mediam, vt \edtext{quinarius}{\Cfootnote{.5. \textit{γ}}} se habet ad \edtext{binarium}{\Cfootnote{.2. \textit{V\textsuperscript{a}mv}}}. Duplex sexquitertia \edtext{quando}{\Cfootnote{est quando \textit{γ} | quando ipsum \textit{π}}} continet totum et \edtext{ipsius}{\Cfootnote{eius \textit{π}}} tertiam partem, vt .7. se habet ad tria. Duplex sexquiquarta \edtext{et cetera}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{x}}}}, vt .9. \edtext{se habet}{\Cfootnote{\textit{om}. \textit{π}}} ad quatuor. \edtext{Triplex sexquialtera}{\Cfootnote{Triplex sexquitertia \textit{π}}} \edtext{est}{\Cfootnote{\textit{om}. \textit{π}}} quando alter \edtext{alterum continet totum}{\Cfootnote{continet alterum totum ter \textit{π}}} \edtext{et}{\Cfootnote{vt \textit{P\textsuperscript{x}V\textsuperscript{m}}}} adhuc eius tertiam partem, vt x. ad tria. \edtext{Triplex sexquialtera}{\Cfootnote{Quadruplex sexquialtera \textit{P\textsuperscript{y}V\textsuperscript{b}} | Triplex sexquiquarta \textit{M\textsuperscript{a}V\textsuperscript{o}}}} \edtext{quando}{\Cfootnote{est quando \textit{γ}}} quartam partem continet ultra totum, vt 13. ad quatuor.
\pend

\pstart
\textbf{8} Multiplex superpartiens \edtext{est}{\Cfootnote{\textit{om}. \textit{π}}} cum maior numerus continet minorem plus quam semel vel multotiens et \edtext{pluries}{\Cfootnote{plures \textit{P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}}, cum hoc ipsius \edtext{partem}{\Cfootnote{partes \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}}. Cuius etiam \edtext{infinite sunt}{\Cfootnote{sunt infinite \textit{π}}} species, vt \edtext{duplex superbipartiens}{\Cfootnote[nosep]{\textit{mv} | duplex superpartiens \textit{ακπ}}}, \edtext{duplex supertripartiens}{\Cfootnote{et cetera, duplex supertripartiens \textit{γP\textsuperscript{y}} | triplex superpartiens \textit{V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}}}}, et cetera, et id ratione duplicitatis; ex parte \edtext{quidem}{\Cfootnote{quid \textit{V\textsuperscript{o}}}} triplicitatis erunt triplex superbipartiens, \edtext{triplex supertripartiens}{\Cfootnote{triplex superpartiens \textit{π}}}, et cetera. Duplex superbipartiens est quando numerus maior \edtext{continet}{\Cfootnote{continet bis \textit{π}}} minorem et insuper duas eius partes, vt .8. se \edtext{habet}{\Cfootnote{habent \textit{π}}} ad \edtext{tria}{\Cfootnote{tria \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}}. Duplex supertripartiens \edtext{et cetera}{\Cfootnote{est \textit{V\textsuperscript{o}} | est quando \textit{P\textsuperscript{x}}}}, vt .11. se habet ad tria. \edtext{Quadruplex superbipartiens}{\Cfootnote{Triplex supertripartiens \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}V\textsuperscript{o}}}} \edtext{est quando}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}} et cetera, vt 15. se \edtext{habet}{\Cfootnote{habent \textit{G\textsuperscript{u}P\textsuperscript{u}P\textsuperscript{y}}}} ad quatuor. Hee \edtext{quidem}{\Cfootnote{qua \textit{V\textsuperscript{o}}}} sunt numerorum \edtext{inequalitates}{\Cfootnote{inequales \textit{M\textsuperscript{a}V\textsuperscript{m}}}} ex quibus \edtext{.5.}{\Cfootnote{\textit{om}. \textit{π}}} resultant proportiones arismetrice cum suis oppositis antedictis.
\pend

\pstart
\textbf{9} Restat \edtext{quidem breuiter videre secundo}{\Cfootnote{breuiter quidem videre secundo \textit{P\textsuperscript{y}V\textsuperscript{b}} | breuiter secundo videre \textit{πV\textsuperscript{a}}}}: \edtext{quot et que}{\Cfootnote{que et quot sunt \textit{π}}} ex hiis \edtext{sint}{\Cfootnote{sunt \textit{E\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{m}} | \textit{om}. \textit{π}}} consonantie \edtext{siue simphonie}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} musicales, et a \edtext{quibus}{\Cfootnote{quibus proportionibus \textit{απ}}} \edtext{sumantur}{\Cfootnote{summatur \textit{P\textsuperscript{x}V\textsuperscript{b}V\textsuperscript{m}} | sumuntur \textit{V\textsuperscript{o}}}} \edtext{arismetricis}{\Cfootnote{arismetricum \textit{γ}}}. Secundum \edtext{quidem}{\Cfootnote{quid \textit{V\textsuperscript{o}} | \textit{om}. \textit{λ}}} \hyperref[Mac]{macrobium} et \hyperref[Bo]{boecium} ex omni numerorum varietate pauci \edtext{numeri}{\Cfootnote{\textit{om}. \textit{π}}} reperiuntur qui conueniant sibi ad consonantiam musicalem constituendam. Sunt autem hii .6. \edtext{omnes}{\Cfootnote{omnis \textit{π}}} iuxta quos .5. consonantie constituuntur, et tonus ex quo \edtext{constituunt}{\Cfootnote{constituitur \textit{πV\textsuperscript{z}}}} dyatonum, scilicet epitricus, emiolus, duplaris, triplaris, \edtext{quadruplarus}{\Cfootnote{quadruplum \textit{E\textsuperscript{a}V\textsuperscript{o}}}}, \edtext{epogdous}{\Cfootnote{epogdonium \textit{πE\textsuperscript{a}M\textsuperscript{a}}}}.
\pend

\pstart
\textbf{10} Epitricus vel sexquitertius est vt aliqualiter iam patuit cum maior numerus continet minorem totum et eius tertiam partem, vt quatuor \edtext{se habet}{\Cfootnote{\textit{om}. \textit{π}}} ad tria. Ex quo nascitur simphonia vel consonantia \edtext{dyatesseron dicta}{\Cfootnote{dicta dyatesseron \textit{π}}} id est per quatuor, que in sexquitertia \edtext{proportione consistit}{\Cfootnote{consistit proportione \textit{V\textsuperscript{o}}}}, vt cum una vox in grauitate vel acuitate se \edtext{se habuerit ad aliam sicut quatuor ad tria}{\Cfootnote{hii \textit{κ}}}.
\pend

\pstart
\textbf{11} \edtext{Iuxta vero emiolam vel sexquialteram vt cum maior numerus}{\Cfootnote{\textit{om}. \textit{κ}}} \edtext{minorem continet}{\Cfootnote{continet minorem \textit{πV\textsuperscript{a}}}} totum et \edtext{medietatem ipsius}{\Cfootnote{ipsius medietatem \textit{π}}}, \edtext{ceu}{\Cfootnote{seu \textit{V\textsuperscript{o}} | vt \textit{G\textsuperscript{u}P\textsuperscript{p}}}} \edtext{se habet}{\Cfootnote{\textit{om}. \textit{π}}} tria ad duo. Oritur consonantia dicta dyapente vel per \edtext{quinque}{\Cfootnote{quandoque \textit{P\textsuperscript{x}V\textsuperscript{m}}}} \edtext{in quinto enim}{\Cfootnote{in quinto \textit{π}}} loco terminatur in monocordo, sicut dyatesseron in \edtext{quatuor}{\Cfootnote{quarto \textit{γP\textsuperscript{y}V\textsuperscript{o}}}} \edtext{ceu}{\Cfootnote{seu \textit{πE\textsuperscript{a}}}} cum una vox se habuerit ad aliam in grauitate et leuitate et cetera.
\pend

\pstart
\textbf{12} Secundum \edtext{siquidem}{\Cfootnote{quidem \textit{λ}}} duplarem numerum vel proportionem, vt quatuor se habet ad duo sumitur proportio dyapason siue per totum, terminat enim \edtext{et}{\Cfootnote{uel \textit{π}}} continet totum monacordum et maxime primum, cadit enim super \edtext{.8.}{\Cfootnote{.4. \textit{π}}} incipiens a gama greco.
\pend

\pstart
\textbf{13} \edtext{Scilicet}{\Cfootnote{\textit{om}. \textit{λV\textsuperscript{b}}}} iuxta \edtext{vero triplarem}{\Cfootnote{triplarem vero \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}V\textsuperscript{o}}}} numerum seu proportionem, vt cum numerus se \edtext{habet}{\Cfootnote{habeat \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{a}}}} ad numerum \edtext{ceu}{\Cfootnote{sicut \textit{V\textsuperscript{o}}}} tria ad vnum, componitur enim ex multiplici et superparticulari \edtext{sicut se habet tria ad vnum}{\Cfootnote{\textit{om}. \textit{π}}}, nascitur simphonia composita \edtext{vt}{\Cfootnote{ex \textit{π}}} \edtext{dyapente}{\Cfootnote{dyapente et \textit{πV\textsuperscript{b}}}} dyapason vt quando \edtext{vna}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} vox excedit alteram et cetera. Iuxta vero numerum quadruplum, cum vnus alium quater continet, vt quatuor se habet ad vnum oritur bis diapason.
\pend

\pstart
\textbf{14} Epogdous est numerus qui \edtext{inter}{\Cfootnote{intra \textit{λE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}} se minorem \edtext{concludit}{\Cfootnote{continet \textit{π}}} et insuper \edtext{octauam}{\Cfootnote{8 \textit{E\textsuperscript{a}V\textsuperscript{o}}}} eius partem, \edtext{ceu}{\Cfootnote{seu \textit{V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} 9 se habet ad .8. Ex \edtext{hoc}{\Cfootnote{quo \textit{π}}} quidem numero sonus consurgit quem musici \edtext{tonum vocant}{\Cfootnote{tonum vocarunt \textit{λ} | tonum vocauerunt \textit{V\textsuperscript{o}}}}, qui autem tono minus semitonum dicitur, quod \edtext{non}{\Cfootnote{non est \textit{αM\textsuperscript{a}V\textsuperscript{o}}}} \edtext{extimandum}{\Cfootnote{existimandum \textit{V\textsuperscript{o}}}} tonum \edtext{vere}{\Cfootnote{fere \textit{π}}} fore sicut neque semiuocalem in litteris. \edtext{Non enim}{\Cfootnote{Nam \textit{π}}} tonus naturaliter in duo \edtext{equalia}{\Cfootnote{\textit{om}. \textit{π}}} potest diuidi, cum constet ex numero nouenario, \edtext{quod}{\Cfootnote{qui \textit{V\textsuperscript{a}V\textsuperscript{z}}}} in duo diuidi minime \edtext{potest equaliter}{\Cfootnote{potest \textit{κ} | equaliter potest \textit{πE\textsuperscript{a}}}}. \edtext{Vnde et}{\Cfootnote{Et ideo \textit{π}}} \edtext{semitonium vocarunt}{\Cfootnote{vocauerunt semitonum \textit{π}}} sonum tono minorem. Est autem semitonum duplex \edtext{maius scilicet}{\Cfootnote{scilicet maius \textit{V\textsuperscript{a}V\textsuperscript{z}} | maius \textit{V\textsuperscript{o}}}} et minus. \edtext{Maius}{\Cfootnote{Minus \textit{E\textsuperscript{a}V\textsuperscript{o}}}} \edtext{quidem}{\Cfootnote{videm \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{b}} | quid \textit{V\textsuperscript{o}}}} dicitur apothomi, apud vero \hyperref[Bo]{boecium} decisio, minus \edtext{autem}{\Cfootnote{aut \textit{π}}} dyesis vel limes, quod tam paruo distare a tono deprehensum est quantum \edtext{hii}{\Cfootnote{\textit{om}. \textit{π}}} duo numeri distant: \edtext{vt}{\Cfootnote{\textit{om}. \textit{λ}}} 243. \edtext{et .256.}{\Cfootnote{2256. \textit{γ}}}. Quorum quidem vnusquisque constat ex duabus vocibus \edtext{puta}{\Cfootnote{prout \textit{V\textsuperscript{o}}}} \edtext{vt}{\Cfootnote{vt et \textit{π}}} mi, est \edtext{et}{\Cfootnote{etiam \textit{π}}} ditonus, qui ex duobus consurgit perfectis tonis, et \edtext{semidytonus}{\Cfootnote{semitonus \textit{λπ}}}, qui ex duobus non \edtext{integris}{\Cfootnote{integer \textit{V\textsuperscript{o}}}}. Vt \edtext{id autem}{\Cfootnote{autem hec \textit{π}}} per amplius \edtext{appareat}{\Cfootnote{apparent \textit{V\textsuperscript{o}}}} parigraphia \edtext{conscribatur}{\Cfootnote{scribatur \textit{πP\textsuperscript{y}}}} \edtext{ista}{\Cfootnote{talis \textit{π}}}.
\pend

\pstart
\begin{edtabularl}
Γ. & 1296. & ut. & \\
a. & 1152. & re. & \\
b. & 1034. & mi. & dytonus.\\
c. & 972. & faut. & dyatesseron. \\
d. & 864. & solre. & dyapente. \\
e. & \edtext{768.}{\Cfootnote{786. \textit{M\textsuperscript{a}P\textsuperscript{x}}}}. & lami. & \\
f. & 739. & faut. & \\
g. & 648. & solreut. & dyapason.\\
a. & 576. & lamire. & \\
b. & 554. & mi. & \\
b. & 512. & fa. & \\
c. & \edtext{486.}{\Cfootnote{446. \textit{π}}}. & solfaut. & dyatesseron dyapason.\\
d. & 432. & lasolre. & dyapente dyapason. \\
e. & 384. & lami. & \\
f. & \edtext{366.}{\Cfootnote{369. \textit{πE\textsuperscript{a}V\textsuperscript{a}} | 389. \textit{P\textsuperscript{x}P\textsuperscript{}y}}}. & faut. & \\
g. & 324. & solreut. & bis dyapason.\\
a. & 288. & lamire. & \\
\edlabel{tablacstart}b. & 277. & mi. & \\
b. & 256. & fa. & \\
c. & 243. & solfa. & \\
\edlabel{tablacend}\edtext{d}{\Cfootnote{c \textit{E\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{o}}}}. & 216. & 
\edtext{lasol.}
{\xxref{tablacstart}{tablacend}\lemma{b. 277. mi. \dots\ d. 216. lasol.}\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}P\textsuperscript{x}}}} &
\end{edtabularl}
\pend

\pstart
\textbf{15} Sunt itaque consonantie .5. scilicet: dyatesseron, dyapente, dyapason, \edtext{dyapente dyapason, et}{\Cfootnote{dyapente dyapason \textit{γV\textsuperscript{o}} | dyatesseron et \textit{λ}}} bis dyapason. Que quidem hiis tonis et semitonis constituuntur, \edtext{dyatesseron}{\Cfootnote{diatonici \textit{P\textsuperscript{x}V\textsuperscript{m}}}} enim constat ex duobus tonis et semitono \edtext{non integro, dyapente tribus tonis et minore semitono}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{a}}}}, dyapason 5. tonis et \edtext{semitonis duobus}{\Cfootnote{duobus semitonis \textit{π}}} minoribus, \edtext{neque}{\Cfootnote{nullaque \textit{π}}} \edtext{peruenit}{\Cfootnote{prouenit \textit{G\textsuperscript{u}P\textsuperscript{p}} | prout \textit{V\textsuperscript{o}}}} ad \edtext{6.}{\Cfootnote{5. \textit{π}}} tonos perfecte. Ex quibus apparet quod dyatesseron et \edtext{dyapente}{\Cfootnote{diapentesseron \textit{κ}}} \edtext{consurgunt}{\Cfootnote{consurgit \textit{π}}} in \edtext{dyapason}{\Cfootnote{diapasseron \textit{P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{m}} | \textit{om}. \textit{π}}}. \edtext{Dyapente dyapason}{\Cfootnote{Dyapente dyapason que \textit{π} | Et in pentadyaseron \textit{κ}}} ex 9 tonis \edtext{et ex}{\Cfootnote{et \textit{απ}}} semitono minori constat, \edtext{bis vero}{\Cfootnote{sed ipsa bis \textit{π}}} dyapason ex 12. \edtext{perficitur}{\Cfootnote{perfectis \textit{π}}} tonis. Et hec \edtext{quidem}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{x}}}} amplius \edtext{sunt}{\Cfootnote{sunt exposita \textit{V\textsuperscript{o}}}} in \textit{conciliatoris} \edtext{mei}{\Cfootnote{nostri \textit{λ} | \textit{om}. \textit{π}}} differentia .83\textsuperscript{a}. \edtext{exposita}{\Cfootnote{\textit{om}. \textit{π}}}.
\pend

\pstart
\textbf{16} Nosce quoque \edtext{ex \textit{musice} quinto}{\Cfootnote{ex quinto \textit{musice} \textit{π}}} ``quod \hyperref[Pt]{ptholomeus} \edtext{sextam hiis addit}{\Cfootnote{sextam hiis addidit \textit{P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}} | addidit sextam \textit{π}}} consonantiam compositam ex dyatesseron et dyapason, que constat ex multiplici \edtext{superpartiente}{\Cfootnote{superpartienti \textit{P\textsuperscript{y}V\textsuperscript{a}} | superparticulare \textit{M\textsuperscript{a}V\textsuperscript{b}}}} \edtext{qua proportione}{\Cfootnote{quam proportionem \textit{V\textsuperscript{m}V\textsuperscript{z}} | in qua proportione \textit{π}}} \edtext{se habet}{\Cfootnote{habet \textit{E\textsuperscript{a}V\textsuperscript{z}}}} .8. ad tria. Continet enim .8. bis tria, propter multiplicem proportionem ex qua \edtext{visum}{\Cfootnote{visum est \textit{πE\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{a}}}} consurgere dyapason, et duas \edtext{eius partes}{\Cfootnote{partes eius \textit{π} | eius partes eius \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} quod est \edtext{tertia}{\Cfootnote{3. \textit{π}}} scilicet duo propter superpartientem a qua nascitur dyatesseron.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} V.8–9}} Communiter tamen pictagorici \edtext{musice}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} \edtext{institutores}{\Cfootnote{instructores \textit{λ}}} posuerunt omnes \edtext{proportiones}{\Cfootnote{consonantias \textit{π}}} musicales consurgere ex multiplici et superparticulari, \edtext{nullamque}{\Cfootnote{nullam vero \textit{π}}} ex \edtext{superpartiente}{\Cfootnote{superpartienti \textit{αV\textsuperscript{m}V\textsuperscript{z}}}}, quod videtur \hyperref[Ar]{aristoteles} sentire .41°.
\pend

\pstart
\textbf{17} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{\edtext{quoniam}{\Cfootnote{quia \textit{π}}} in ambobus scilicet \edtext{in}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{o}}}} graui et acuto est melodia vtrorumque, vnum enim alteri \edtext{coniunctum}{\Cfootnote{consitum \textit{λ}}} melodiam reddit, quia graue et acutum sunt dissimilia.} \edtext{Et}{\Cfootnote{\textit{om}. \textit{γ}}} \edtext{\hyperref[Ty]{thimoteus}}{\Cfootnote{entimomus \textit{γ}}} quidem \edtext{extimauit}{\Cfootnote{existimauit \textit{V\textsuperscript{o}}}} musice primo melodiam vel consonantiam fieri ex dissimilibus, non autem ex \edtext{similibus}{\Cfootnote{consimilibus \textit{V\textsuperscript{o}}}}. \textit{Si autem proueniat vtrorunque in vtrisque concordia non fit melodia, cum autem sic fuerit quod graue \edtext{maius sit}{\Cfootnote{sit maius \textit{πP\textsuperscript{y}}}} acuto et fortius,} tunc quod est acutum assumit rationem grauis, ita quod acutum sentitur graue, \edtext{vt}{\Cfootnote{nisi \textit{M\textsuperscript{a}V\textsuperscript{m}}}} cum due nite \edtext{concurrunt}{\Cfootnote{concordant \textit{π}}} in ypatem. Dicitur autem melodia a melle, quoniam est consonantia delectabilissima valde ex acuto et graui debite composita.
\pend

\pstart
\vspace{1em}
\begin{center}
    {\Large \ding{167}}
\end{center}
\pend
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\newpage

\subsection[Quartumdecimum problema … ]{Quartumdecimum problema}\label{part14}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID LATET DYAPASON ET VIDETUR VNIUS SONI ESSE, VT IN FINIKIO ET IN HOMINE? QUE ENIM IN ACUTIS EXISTUNT NON VNIUS SONI, SED PROPORTIONALITER AD INUICEM DYAPASON. \textsuperscript{2}AUT QUIA SICUT IPSE ESSE VIDETUR SONUS PROPTER ANALOGUM, EQUALITAS IN SONIS, EQUALE AUTEM VNIUS? HOC AUTEM IDEM IN FISTULIS DECIPIUNTUR}{\lemma{PROPTER \dots\ DECIPIUNTUR}\Cfootnote{PROPTER QUID LATET DYAPASON \textit{V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} dyapason cum sit proportio satis distans quoniam dupla latet \edtext{in tantum}{\Cfootnote{iterum \textit{M\textsuperscript{a}V\textsuperscript{m}}}} \edtext{quod}{\Cfootnote{quare \textit{κ}}} videtur sonus ex ipsa \edtext{resultatus}{\Cfootnote{resultans \textit{π}}} esse vnus et quasi vnum tonum simpliciter habere?} Et hoc declarat sicut apparet \edtext{et cetera}{\Cfootnote{\textit{om}. \textit{π}}}. Communiter quidem inuenitur littera in finitis, alia infantis \textit{\edtext{aliter finikio}{\Cfootnote{alia finikio \textit{γP\textsuperscript{y}} | \textit{om}. \textit{πE\textsuperscript{a}}}},} sed forte debet \edtext{dicere in fidibus}{\Cfootnote{dici in fidibus \textit{P\textsuperscript{y}V\textsuperscript{z}} | esse in fidibus \textit{π}}} seu cordis. \textit{Et in homine similiter} \edtext{declarat}{\Cfootnote{declaratur \textit{αλ}}} ita quod per hoc tangat duplicem melodiam tactu vel pulsu factam et \edtext{voce}{\Cfootnote{note \textit{E\textsuperscript{a}V\textsuperscript{o}}}}. \textit{Soni siquidem existentes acuti quia non concurrunt adeo in vnum, non sunt \edtext{unius}{\Cfootnote{minus \textit{γ}}} soni vt graues aut \edtext{sic}{\Cfootnote{sunt \textit{απ}}} compositi ex grauibus sed in dyapason. Hoc accidit quoniam toni se habent proportionaliter ad inuicem ex graui \edtext{et}{\Cfootnote{ex \textit{E\textsuperscript{a}V\textsuperscript{m}}}} acuto,} \textit{est enim optima consonantiarum} \edtext{.35°.}{\Cfootnote{.3. quinto \textit{κV\textsuperscript{b}V\textsuperscript{o}}}}
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{\edtext{quare}{\Cfootnote{quia \textit{P\textsuperscript{x}V\textsuperscript{z}}}} sonus apparet vnicus esse propter anologiam et proportionem sonorum ad inuicem in dyapason, ex qua resultat equalitas grandis in sonis.} Quod autem \edtext{equalitatem habet}{\Cfootnote{habet equalitatem \textit{λ}}} \edtext{vnius}{\Cfootnote{unus \textit{P\textsuperscript{y}V\textsuperscript{o}}}} \edtext{soni existit}{\Cfootnote{existit soni \textit{απ}}}, et non solum \edtext{hoc}{\Cfootnote{\textit{om}. \textit{π}}} accidit in \edtext{sono}{\Cfootnote{\textit{om}. \textit{π}}} voce facto \edtext{aut}{\Cfootnote{\textit{om}. \textit{π}}} pulsu, sed \edtext{etiam}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}}}} \edtext{in sono}{\Cfootnote{\textit{om}. \textit{π}}} flatu facto, \edtext{vel}{\Cfootnote{vt \textit{αV\textsuperscript{o}}}} in fistulis. In illis enim decipiuntur etiam vt in vocis sono, propter \edtext{debitam enim}{\Cfootnote{enim debitam \textit{π}}} tonorum consonantiam accidit plures tonos non discerni sed vnum apparere cum diuersitates illic non \edtext{extent}{\Cfootnote{existant \textit{πP\textsuperscript{x}}}} sensibiles, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} videmus \edtext{ex}{\Cfootnote{in \textit{πE\textsuperscript{a}V\textsuperscript{z}}}} saporibus \edtext{composita ex}{\Cfootnote{compositis ex \textit{π} | composita \textit{γλE\textsuperscript{a}}}} diuersis in quibus vnus consurgit delectabilis et nullius excessus, \edtext{vt}{\Cfootnote{vt qui \textit{πE\textsuperscript{a}}}} conficitur decoctione \edtext{ex}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{cucurbitis}{\Cfootnote{cucurbite \textit{π}}}, cepis, vel \edtext{alleis}{\Cfootnote{aliis \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}}, cum tamen vnum sit saporis insipidi et alterum acuti.
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\subsection[Quintumdecimum problema … ]{Quintumdecimum problema}\label{part15}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID LEGES QUIDEM NON IN CONUERSIONIBUS FACIEBANT, ALII VERO CANTUS CHORICI? \textsuperscript{2}AUT QUIA LEGES QUIDEM AGONIZANTIUM FUERUNT? QUIBUS VTIQUE MUTARI POTENTIBUS ET DISTENDI, CANTUS FIEBAT LONGUS ET MULTIFORMIS. QUEMADMODUM IGITUR ET VERBA, ET CANTUS MUTATIONEM SEQUEBANTUR, SEMPER ALTERA FACTA. \textsuperscript{3}MAGIS ENIM CANTU NECESSE IMITARI QUAM VERBIS. PROPTER QUOD ET DITIRAMBI, QUONIAM AUTEM IMITATIUI, FIEBANT. NON ADHUC HABENT CONUERSIUOS, PRIUS AUTEM HABEBANT. CAUSA AUTEM QUONIAM ANTIQUITUS LIBERI ROTABANT IPSI, MULTOS IGITUR AGONISTICE CANTARE DIFFICILE ERAT, QUARE IN ARMONIA CANTUM DELECTABILITER. MUTARE ENIM MULTIS MUTATIONIBUS VNO FACILE QUAM MULTIS ET AGONISTE ID EST CONSUETUDINEM SERUANT. PROPTER QUOD SIMPLICIORES FACIEBANT SIBI MELODIAS. CONUERSIUUS AUTEM SIMPLEX, NUMERUS ENIM EST ET VNO MENSURATUR. \textsuperscript{4}EADEM AUTEM EST CAUSA ET PROPTER QUOD QUE QUIDEM A SCENA NON CONUERSIUA, CHORI ENIM CONUERSIUA. QUI ENIM CANTAT IN SCENA AGONISTA ET MUTATOR, CHORUS AUTEM MINUS MUTATUR}{\lemma{PROPTER \dots\ MUTATUR}\Cfootnote{PROPTER QUID LEGES \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} antiquitus cantatores constituerunt quosdam cantus dictos leges non habentes conuersiones, sed alii cantus chorici et conuersiui \edtext{fuere}{\Cfootnote{fide \textit{V\textsuperscript{o}}}}?} 
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\textbf{3} Nosce quod per leges non in conuersionibus factas audio cantus simplices expansos in quibus potius fit commutatio verborum quam cantus, ita quod in eis non constituantur circuli conuersiui, sicut est cantus alicuius insignis gesti. \edtext{Choricos autem cantus intelligo conuersiuos primis vtcunque \edtext{oppositos}{\Cfootnote{oppositis \textit{γ}}}, vt apparebit in causa. Est autem chorus curator inoperabilis, beniuolentia enim solum prebet quibus adest .48.}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} 
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\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et primo assignat causam quesiti. Secundo \edtext{ex hac}{\Cfootnote{\textit{om}. \textit{π}}} infert \edtext{causam}{\Cfootnote{causam etiam \textit{πE\textsuperscript{a}}}} cuiusdam alterius cantus \edtext{secunda \textit{EADEM AUTEM}}{\Cfootnote{EADEM \textit{V\textsuperscript{o}}}}. \edtext{Etiam}{\Cfootnote{Et \textit{M\textsuperscript{a}V\textsuperscript{z}} | \textit{om}. \textit{V\textsuperscript{o}}}} primo respondet ad primam partem dicens \textit{causam esse quoniam leges fuerunt cantus constituti ab agonizantibus et bellatoribus, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} sunt cantus gestorum insignium. Quas \edtext{quidem}{\Cfootnote{quam \textit{P\textsuperscript{x}V\textsuperscript{m}}}} leges vt possent consequi recitando quod in eis continebatur \edtext{distendebant}{\Cfootnote{discebant \textit{π}}} cantando, nunquam faciendo conuersionem vel circulationem ad id vnde \edtext{inceperant}{\Cfootnote{ceperant \textit{λ}}} redeuntes. Propter quod \edtext{eorum}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} cantus fiebat longus et multarum manerierum, secundum verborum \edtext{prelatorum}{\Cfootnote{prolatorum \textit{M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}} diuersitatem. Nam secundum quod verba gestorum fiebant continue \edtext{altera}{\Cfootnote{alta \textit{γP\textsuperscript{y}V\textsuperscript{m}V\textsuperscript{o}}}} vel melius altera et diuersa mutatione similiter. \edtext{Et}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} in cantu similem obseruabant mutationem, \edtext{vt}{\Cfootnote{vt ad \textit{π}}} commutationem verborum cantus comitaretur \edtext{mutatio}{\Cfootnote{commutatio \textit{V\textsuperscript{o}}}}, ad verba enim amplius quam ad cantum attendebant.} Propter quod dicte sunt leges cantus \edtext{huius}{\Cfootnote{huiusmodi \textit{P\textsuperscript{y}V\textsuperscript{b}} | \textit{om}. \textit{π}}} a legendo, \edtext{velut}{\Cfootnote{visum \textit{G\textsuperscript{u}P\textsuperscript{p}}}} in lectionibus et proprie diuinis, vt apparebit \edtext{.48.}{\Cfootnote{.44. \textit{π}}}, vna \edtext{dictio}{\Cfootnote{dicto \textit{P\textsuperscript{x}V\textsuperscript{m}}}} post aliam profertur non multum vocem diuersificando.
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\textbf{5} \edtext{\textit{MAGIS}}{\Cfootnote{Deinde. MAGIS \textit{V\textsuperscript{a}mv}}} \textit{ENIM.} \edtext{Tangit}{\Cfootnote{Tangit enim \textit{M\textsuperscript{a}V\textsuperscript{m}}}} causam \edtext{alterius}{\Cfootnote{plus alterius \textit{πE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{a}}}} partis, dicens quod in cantu \edtext{armonico}{\Cfootnote{armonisato \textit{V\textsuperscript{o}}}} sufficienter non debet \edtext{per oppositum accidere}{\Cfootnote{hoc accidere per oppositum \textit{π}}}, nam in eo quod dictum est, \textit{in legibus amplius \edtext{oportet}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{imitari}{\Cfootnote{immutari \textit{λE\textsuperscript{a}}}} cantum reddendo ipsum melodiosum quam verba,} cum cantus sit magis finis musici actus verbis, ampliorem inducens delectationem, propter quod \edtext{dictum}{\Cfootnote{dictum est \textit{λV\textsuperscript{a}}}} 10° quod \textit{vox delectabilis \edtext{est cantantis}{\Cfootnote{cantantis est \textit{π} | est cantus \textit{E\textsuperscript{a}V\textsuperscript{a}}}} sine \edtext{sermone}{\Cfootnote{sermonem \textit{κ} | prouisione \textit{P\textsuperscript{p}P\textsuperscript{q}}}} \edtext{etiam. Et}{\Cfootnote{et cetera. Et \textit{M\textsuperscript{a}V\textsuperscript{m}} | etiam. Et etiam \textit{π}}}} \textit{\edtext{ideo}{\Cfootnote{\textit{om}. \textit{π}}} quidam modi cantandi dicti ditirambi fiunt, cum quis magis cantum imitatur verbis.} Est autem ditirambus cantus qui concidit in frigistum, vnde .8°. \textit{politicorum} ``\hyperref[Phil]{filoxenus} conatus in dorista facere ditirambum \edtext{fabulas}{\Cfootnote{fabulans \textit{γE\textsuperscript{a}}}} non fuit potens, sed ab \edtext{ipsa natura}{\Cfootnote{ipso \textit{π}}} decidit in frigistum.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII, 1342b9}} Hec autem .48°. erunt manifesta adhuc. Siquidem \edtext{rediens}{\Cfootnote{rediens ad \textit{π}}} leges inquit quia \edtext{cantus huius dicti}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{leges fiebant vt dictum}{\Cfootnote{vt dictum est leges fiebant \textit{π} | leges fiebant vt dictum est \textit{γλ}}} continue faciendo \edtext{per mutationem}{\Cfootnote{mutationem \textit{π}}}. \textit{Secundum verba ideo non sunt amplius conuersiui, sed \edtext{tamen}{\Cfootnote{cum \textit{P\textsuperscript{y}V\textsuperscript{a}}}} antiquitus fuerunt. Cuius quidem causa quoniam ab antiquo ipsimet liberi nobilesque faciebant cantus \edtext{rotationes}{\Cfootnote{rotatos \textit{γ} | rotatiuos \textit{πE\textsuperscript{a}}}} vel conuersiuos quoniam plures erant, et ideo difficile fiebat eis cantare agonistice vel \edtext{leges}{\Cfootnote{legere \textit{E\textsuperscript{a}V\textsuperscript{o}}}} facere.} Choros autem faciebant siue cantus conuersiuos qui fiunt in choris, ubi sufficientes \edtext{et quieti}{\Cfootnote{etiam qui \textit{π}}} amplius \edtext{imitari conantur}{\Cfootnote{volunt imitari \textit{π}}} melodiam quam verba. \textit{Propter \edtext{quod}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{delectabile}{\Cfootnote{delectabile est \textit{λ}}} cantare secundum armoniam illam \edtext{commutatam}{\Cfootnote{commutationem \textit{V\textsuperscript{o}}}} ubi pauci conueniebant. Nam hii de facili possunt mutare cantum pluribus mutationibus, \edtext{et id est quod dicitur: quod vni mutare cantum pluribus mutationibus}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}} facilius est quam multis, quare \edtext{agoniste}{\Cfootnote{agonistice \textit{π}}} cantantes vel leges consuetudinem huius vt tactum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} \edtext{obseruant}{\Cfootnote{obseruabant \textit{π}}}.} \edtext{Qui}{\Cfootnote{Que \textit{V\textsuperscript{o}}}} \edtext{mutatione}{\Cfootnote{mutationem \textit{λ}}} verborum \edtext{mutantur}{\Cfootnote{imitantur \textit{M\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{o}V\textsuperscript{z}} | immutantur \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}}, \textit{et ideo melodias valde simplices faciebant et incompositas, quoniam \edtext{conuersiuus}{\Cfootnote{conuersus \textit{π}}} cantus quo vtebantur est numerus \edtext{quidem}{\Cfootnote{quidam \textit{γλE\textsuperscript{a}}}} simplex, qui vnitate tantum \edtext{mensurabatur}{\Cfootnote{mensurabis \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}},} ita quod conuersio si \edtext{aliqua}{\Cfootnote{alia \textit{γE\textsuperscript{a}V\textsuperscript{z}}}} erat ibi potius in simplicem quam in \edtext{compositum}{\Cfootnote{compositam \textit{V\textsuperscript{o}}}}, referebatur, vt in numeris \edtext{dicimus}{\Cfootnote{dicto \textit{P\textsuperscript{x}V\textsuperscript{m}}}} illum numerum simplicem qui continue crescit mensuratus per vnitatem. Similiter et cantus potest simplex dici \edtext{qui}{\Cfootnote{que \textit{P\textsuperscript{x}P\textsuperscript{z}}}} \edtext{contineatur}{\Cfootnote{continuatur \textit{πP\textsuperscript{y}}}} \edtext{per additionem solam}{\Cfootnote{solum per additionem \textit{π}}} verbi ad \edtext{ista}{\Cfootnote{verba \textit{πE\textsuperscript{a}P\textsuperscript{y}}}}.
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\pstart
\textbf{6} \edtext{\textit{EADEM.}}{\Cfootnote{EANDEM. \textit{πE\textsuperscript{a}} | Deinde. EADEM. \textit{V\textsuperscript{a}mv}}} Applicat \edtext{istam}{\Cfootnote{illam \textit{P\textsuperscript{x}V\textsuperscript{z}}}} causam ad quesitum \edtext{adhuc}{\Cfootnote{ad hoc \textit{E\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}} | aliud huic \textit{γπ}}}, \edtext{simile}{\Cfootnote{similem \textit{λ}}} dicens quod \textit{\edtext{ista}{\Cfootnote{illa \textit{V\textsuperscript{b}V\textsuperscript{m}}}} eadem \edtext{causa est}{\Cfootnote{est causa \textit{γE\textsuperscript{a}}}}, \edtext{quia}{\Cfootnote{quare \textit{απ}}} \edtext{illi}{\Cfootnote{isti \textit{γP\textsuperscript{y}V\textsuperscript{o}}}} cantus qui \edtext{a scena}{\Cfootnote{scena \textit{E\textsuperscript{a}V\textsuperscript{o}}}} dicuntur non sunt conuersiui, chori autem conuertuntur, quoniam ille qui \edtext{cantat}{\Cfootnote{cantant \textit{P\textsuperscript{x}V\textsuperscript{a}}}} a scena est sicut \edtext{agonistice}{\Cfootnote{agonista \textit{M\textsuperscript{a}V\textsuperscript{z}} | agoniste \textit{P\textsuperscript{y}V\textsuperscript{a}}}} cantans aut \edtext{leges}{\Cfootnote{legens \textit{πE\textsuperscript{a}}}},} cantat enim agonizantium gesta commutans cantum secundum verba vt agoniste \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} visum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}}, una enim dictio indiget \edtext{aspiratione}{\Cfootnote{inspiratione \textit{π}}}, \edtext{alia vero}{\Cfootnote{alia \textit{V\textsuperscript{o}}}} lenitate et huius consimili alteratione.
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\textbf{7} \textit{Sed \edtext{choricus}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} cantus \edtext{minus}{\Cfootnote{vnus \textit{πE\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{a}}}} \edtext{commutatur}{\Cfootnote{commutatus \textit{π} | commitatur \textit{P\textsuperscript{x}V\textsuperscript{a}V\textsuperscript{m}}}},} amplius enim vnum tenorem et circulum retinet conseruatum. Notandum quod a scena multas habet significationes, aliquando namque idem est quod \edtext{umbra}{\Cfootnote{nichilominus \textit{G\textsuperscript{u}P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{o}}}} vel locus umbratus et coopertus in teatro \edtext{dicitur}{\Cfootnote{diuiditur \textit{γ}}}, \edtext{interdum}{\Cfootnote{etiam interdum \textit{π} | in circulum \textit{γ}}} totum teatrum, et quandoque recitatio operationis. Vnde olim poete libros suos deferebant a scene vt \edtext{appropriarentur}{\Cfootnote{approbarentur \textit{πP\textsuperscript{y}}}}. Dicitur \edtext{etiam}{\Cfootnote{et \textit{λE\textsuperscript{a}V\textsuperscript{b}}}} tabernaculum mercenariorum coopertum \edtext{asseribus}{\Cfootnote{aceribus \textit{V\textsuperscript{o}}}}. Hic autem per ascena potest intelligi: locus ubi colliguntur homines vt gesta \edtext{librorum}{\Cfootnote{liberorum \textit{V\textsuperscript{o}}}} cantare \edtext{audiant}{\Cfootnote{\textit{redit}ant \textit{N\textsuperscript{u}}}} et insignium. \edtext{Et id etiam .29. et 48°. apparebit}{\Cfootnote{\textit{om}. \textit{π}}}.
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\subsection[Sextumdecimum problema … ]{Sextumdecimum problema}\label{part16}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DELECTABILIUS ANTIFONUM CONSONO? \textsuperscript{2}AUT QUIA MAGIS MANIFESTUM FIT CONSONARE? \textsuperscript{3}AUT QUANDO AD CONSONANTIAM CANTANT, NECESSE ENIM ALTERAM CONSONARE? QUARE DUE AD VNAM VOCEM FACTE DESTRUUNT ALTERAM}{\lemma{PROPTER \dots\ ALTERAM}\Cfootnote{PROPTER QUID DELECTABILIUS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} cantus antiphonus in quo quedam \edtext{extat resonantia}{\Cfootnote{restat consonantia \textit{κ}}} vel \edtext{contra}{\Cfootnote{etiam \textit{π}}} \edtext{consonantiam}{\Cfootnote{sonantia \textit{P\textsuperscript{y}V\textsuperscript{b}}}} tonorum super tonos est \edtext{delectabilior}{\Cfootnote{delectabilis magis \textit{π}}} \edtext{auditu}{\Cfootnote{auditui \textit{π}}} quam sonus qui est consonus et vniformis in quo non sentitur \edtext{contra}{\Cfootnote{aliqua \textit{π}}} \edtext{resonantia}{\Cfootnote{resonantiam \textit{κ}}}.} \edtext{Apparentur}{\Cfootnote{Apparent \textit{E\textsuperscript{a}N\textsuperscript{u}} | Apparatur \textit{V\textsuperscript{o}}}} verbi gratia sicut \edtext{si}{\Cfootnote{\textit{om}. \textit{κ}}} duo succinant equaliter et tertius precinat \edtext{hic}{\Cfootnote{hoc \textit{E\textsuperscript{a}V\textsuperscript{z}}}} precinens \edtext{contrasonat}{\Cfootnote{consonat \textit{P\textsuperscript{x}V\textsuperscript{b}}}} et antiphonizat super illos duos.
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit \edtext{assignans}{\Cfootnote{et assignat \textit{πP\textsuperscript{y}}}} duplicem causam, dicens \textit{\edtext{id}{\Cfootnote{illud \textit{γ} | ideo \textit{πE\textsuperscript{a}}}} esse quoniam sonus \edtext{antiphonus}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{redditur}{\Cfootnote{redditur et \textit{β}}} manifestior et sensibilior sono consonante et vniformi.} Quod autem \edtext{sensibilius est sufficienter}{\Cfootnote{est sensibilius sufficienter \textit{π}}} magis delectat, eo quod non est tale sicut apparet in tactu relatione aliorum sensuum.
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\textbf{4} \edtext{\textit{AUT}}{\Cfootnote{Deinde. AUT \textit{V\textsuperscript{a}mv}}} \textit{\edtext{QUANDO.}{\Cfootnote{QUESITI. \textit{P\textsuperscript{x}V\textsuperscript{m}}}}} Assignat aliam causam, dicens quod \textit{quando aliqui cantant consonantiam facientes in qua acutum non excedat graue \edtext{aut}{\Cfootnote{vel \textit{π}}} econuerso, necesse est quod vox \edtext{vnius}{\Cfootnote{huius \textit{πE\textsuperscript{a}}}} consonet omnino voci alterius,} \edtext{sed}{\Cfootnote{si \textit{αM\textsuperscript{a}V\textsuperscript{z}V\textsuperscript{o}}}} debet fieri consonantia. Si autem tertia adueniens non antiphonet, ymmo consonizare velit, oportet eam destrui, \textit{\edtext{quoniam}{\Cfootnote{quando \textit{π}}} due voces \edtext{equalis vigoris ad tertiam relate}{\Cfootnote{ad tertiam relate equalis vigoris \textit{γ}}} \edtext{destruunt}{\Cfootnote{destruant \textit{E\textsuperscript{a}P\textsuperscript{x}} | et destruunt \textit{π}}} et occultant \edtext{aliam}{\Cfootnote{alteram \textit{π}}} penitus tertiam.}
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\textbf{5} \edtext{Notandum}{\Cfootnote{Notandum quod \textit{V\textsuperscript{o}}}} secundum boecium primo \textit{musice} \edtext{quod}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} ``consonantia est acuti soni grauisque mixtura \edtext{suauiter}{\Cfootnote{suauiter et \textit{πE\textsuperscript{a}M\textsuperscript{a}}}} \edtext{vniformiterque}{\Cfootnote{vniformiter \textit{πE\textsuperscript{a}}}} auribus \edtext{accidens}{\Cfootnote{accedens \textit{π}}}. Sed dissonantia est duorum sonorum sibimet impermixtorum ad \edtext{aurem}{\Cfootnote{aures \textit{P\textsuperscript{y}V\textsuperscript{b}}}} \edtext{veniens}{\Cfootnote{perueniens \textit{λ}}} aspera \edtext{atque}{\Cfootnote{et \textit{λ}}} iniocunda percussio. Nam dum sibimet \edtext{misceri}{\Cfootnote{conmisceri \textit{π}}} volunt et quodammodo \edtext{integri}{\Cfootnote{integer \textit{M\textsuperscript{a}V\textsuperscript{o}}}} \edtext{vterque nititur}{\Cfootnote{nititur vterque \textit{π}}} peruenire, cunque alter alteri officit ad sensum insuauiter, vterque transmittitur.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} I.8 ((Friedlein 195, ll. 6–13))}} Videtur siquidem quantum facit ad sermonem \hyperref[Ar]{aristoteles} quod antiphonus sit delectabilior ex contrariis valde distantibus consurgens vt ex ypate et \edtext{nete}{\Cfootnote{nete vel vite \textit{π}}}, consonans autem quando \edtext{ex non}{\Cfootnote{non ex \textit{P\textsuperscript{y}V\textsuperscript{a}} | non \textit{π}}} adeo \edtext{differentibus}{\Cfootnote{distantibus siue differentibus \textit{π}}}, sicut ex cordis mediis, vt apparebit consequenter.
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\subsection[Decimumseptimum problema … ]{Decimumseptimum problema}\label{part17}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID QUINQUE NON CANTANT ANTIFONAM? \textsuperscript{2}AUT QUIA NON EADEM CONSONA CONSONANTIE, QUEMADMODUM IN DYAPASON? ILLA ENIM IN GRAUI PROPORTIONALE, SICUT ACUTA IN ACUTO, QUEMADMODUM IGITUR EADEM EST SIMUL ET ALIA. QUE AUTEM IN DYAPENTE ET DYATESSERON NON HABENT, ITA QUARE NON APPARET QUANDO ANTIFONE SONUS, NON ENIM EST IDEM}{\lemma{PROPTER \dots\ IDEM}\Cfootnote{PROPTER QUID QUANDOQUE \textit{V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} .5. cantantes, vel etiam quatuor, vnam non cantant \edtext{antiphonam}{\Cfootnote{antiphona \textit{E\textsuperscript{a}N\textsuperscript{u}}}},} tres autem vel sex possunt cantare?
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quoniam \edtext{ipsa}{\Cfootnote{ista \textit{απ}}} proportio \edtext{que}{\Cfootnote{quam \textit{απ}}} \edtext{habet}{\Cfootnote{habent \textit{βγπV\textsuperscript{m}V\textsuperscript{z}}}} quinque ad inuicem non est proportio debita que consonet et correspondeat tali consonantie in qua \edtext{sit}{\Cfootnote{\textit{om}. \textit{κ}}} antiphonizare, quemadmodum \edtext{antiphonizare contingit}{\Cfootnote{contingit \textit{πE\textsuperscript{a}}}} in dyapason, ubi est proportio dupla. Ipsa enim habet proportionem in graui, graue enim proportionatum est graui in ipsa, sicut acutum acuto et graue cum acuto \edtext{etiam}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{y}}}}.} Hoc autem non est in aliis. \textit{Ymmo sicut ipsa est eadem inter se et proportionalis vno modo ita, secundum alium modum \edtext{habet}{\Cfootnote{\textit{om}. \textit{κ}}} \edtext{quandam}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}V\textsuperscript{o}}}} alietatem et diuersitatem inter se, eo quod in ipsa est excessus \edtext{manifestus}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}, etsi sit ad equalitatem reductus. Sed consonantie vel proportiones que sunt circa dyatesseron vel dyapente non sic \edtext{se habent}{\Cfootnote{se habet \textit{V\textsuperscript{m}V\textsuperscript{z}}}}. Neque enim in hiis \edtext{tantus}{\Cfootnote{cantus \textit{γE\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{z}}}} \edtext{est}{\Cfootnote{\textit{om}. \textit{π}}} excessus, neque ad equalitatem \edtext{sit}{\Cfootnote{sic \textit{βM\textsuperscript{a}V\textsuperscript{o}}}} reductus. Et ideo sonus licet fiat antiphonus, secundum \edtext{istas}{\Cfootnote{illas \textit{γ}}} \edtext{duas}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} proportiones non redditur manifestus. Non enim est in ipso idemptitas cum quodam excessu sicut dictum est in dyapason euenire.} Id autem potest declarari ex naturis, quia si debet cantari antiphona, oportet debitam proportionem \edtext{interuenire}{\Cfootnote{ante venire \textit{π}}} \edtext{superantium ad superatos}{\Cfootnote{sirantium ad frantos \textit{γ}}} et econtra secundum \edtext{graue et acutum}{\Cfootnote{acutum et graue \textit{β}}}. Sed id non accidit in 5. vocibus, nam \edtext{vel}{\Cfootnote{universaliter \textit{κ}}} quatuor antiphonizabunt \edtext{vno}{\Cfootnote{\textit{om}. \textit{β}}} aut tres duobus, \edtext{non quidem}{\Cfootnote{numquam \textit{P\textsuperscript{x}V\textsuperscript{m}}}} quatuor contra vnum, quia vox eius oppositis obtunderetur omnino, ita vt non \edtext{possit}{\Cfootnote{possint \textit{π}}} sufficienter percipi. Neque tres contra duos, quoniam propter paruum excessum non poterit excedentia manifestari vnius ad alterum, quare non contraphonizare contingit cum \edtext{id}{\Cfootnote{illud \textit{E\textsuperscript{a}V\textsuperscript{b}}}} requiratur omnifariam. Vnde relinquitur \edtext{quod antiphonizare, vt apparebit}{\Cfootnote{vt apparebit quod antiphonizare \textit{π}}} extet secundum dyapason proprie.
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\subsection[Decimumoctauum problema … ]{Decimumoctauum problema}\label{part18}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DYAPASON CONSONANTIA CANTATUR SOLA? ORGANIZANT ENIM SECUNDUM HANC, ALIAM NULLAM. \textsuperscript{2}AUT QUIA SOLA EX ANTIPHONIS EST CORDIS, IN ANTIPHONIS AUTEM ET ALTERAM SI CANTAT, IDEM FACIT. VNA ENIM QUODAMMODO VTRORUNQUE HABET CORDAS, QUARE ET VNA CANTATA IN HAC CONSONANTIA CANTATUR SYMPHONIA ET AMBO CANTANTES. \textsuperscript{3}AUT HAC QUIDEM CANTATA. HAC VERO FISTULATA, QUEMADMODUM VNAM VTRANQUE CANTANT. PROPTER QUOD SOLA CANTATUR, QUIA VNAM HABENT CORDE VOCEM ANTIPHONAM}{\lemma{PROPTER \dots\ ANTIPHONAM}\Cfootnote{PROPTER QUID DYAPASON \textit{V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} dyapason sola consonantiarum cantatur maxime consonans inter ceteras? Quod ostenditur \edtext{quia}{\Cfootnote{quoniam \textit{P\textsuperscript{y}V\textsuperscript{b}}}} secundum ipsam cantantes organizant voces in altum \edtext{leuantes}{\Cfootnote{eleuantes \textit{απ}}}, quod secundum nullam aliarum ita manifeste agere contingit.}
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse primam \textit{quoniam ipsa \edtext{sola}{\Cfootnote{\textit{om}. \textit{π}}} constituitur ex cordis antiphonis seu consonantibus vt \edtext{graui}{\Cfootnote{graui et \textit{π}}} acuto \edtext{per}{\Cfootnote{propter \textit{G\textsuperscript{u}P\textsuperscript{p}}}} proportionem duplam. \edtext{In antiphonis enim cordis ita se habet quod si altera cantetur}{\Cfootnote{\textit{om}. \textit{γ}}} siue tangatur, vt \edtext{media in quam}{\Cfootnote{medio in qua \textit{M\textsuperscript{a}V\textsuperscript{z}}}} ambe declinant extreme vt apparebit, idem fit,} \edtext{ac si}{\Cfootnote{si \textit{κ}}} ambe cantarentur, \edtext{eo}{\Cfootnote{ita \textit{P\textsuperscript{y}V\textsuperscript{o}}}} quod illa vna habet in se aliqualiter \edtext{vt in}{\Cfootnote{\textit{om}. \textit{π}}} virtute cordas vtrorumque sonorum puta grauis et acuti, quare accidit in \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} consonantia quod vna cantata vel pulsata cantetur et altera, ac si ambe cantarentur, propter conuenientiam quam \edtext{habent}{\Cfootnote{habet \textit{M\textsuperscript{a}V\textsuperscript{z}}}} in medio, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} ostendunt pigmenta ex diuersis optime constructa in saporem \edtext{consurgentia}{\Cfootnote{concurrentia \textit{π}}} \edtext{delectamentum}{\Cfootnote{delectatiuum \textit{P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}} | delectatium \textit{E\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{a}}}}.
\pend

\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Assignat aliam causam dicens ideo esse \textit{quia si vna \edtext{cordarum}{\Cfootnote{corda \textit{β}}} cantetur vt ypate et altera fistuletur vt nete cantando et fistulando vtrobique cantatur \edtext{vna}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} quia media. Et ideo sola symphonia cantatur, \edtext{quoniam}{\Cfootnote{et propter hoc \textit{G\textsuperscript{u}P\textsuperscript{p}}}} corde cantate seu fistulate vnam habent vocem scilicet antiphonam,} de quo \edtext{planius}{\Cfootnote{plenius \textit{P\textsuperscript{y}V\textsuperscript{a}} | amplius \textit{π}}} apparebit 19°. et 20°. Omnia enim \edtext{hec}{\Cfootnote{hinc \textit{α}}} problemata \edtext{concidunt, super eodem enim \edtext{fundantur}{\Cfootnote{fundatur \textit{V\textsuperscript{o}}}} medio}{\Cfootnote{concidunt \textit{β}}}. Sic \edtext{ergo}{\Cfootnote{igitur \textit{γλπE\textsuperscript{a}}}} \edtext{ille}{\Cfootnote{iste \textit{απM\textsuperscript{a}V\textsuperscript{z}}}} due \edtext{solutiones}{\Cfootnote{questiones \textit{V\textsuperscript{o}}}} magis differunt ratione modi quam rei, cum vna sit vocalis et \edtext{altera vocalis et}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}V\textsuperscript{o}}}} instrumentalis.
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\subsection[Decimumnonum problema … ]{Decimumnonum problema}\label{part19}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID ANTIPHONIS HOC SOLIS INEST? \textsuperscript{2}AUT QUIA SOLE EQUALITER DISTANT MEDIO? MEDIETAS IGITUR SIMILITUDINEM QUANDAM FACIT SONORUM. ET VIDETUR AUDITUS DICERE QUOD IPSA ET QUOD VTREQUE VLTIME}{\lemma{PROPTER \dots\ VLTIME}\Cfootnote{PROPTER QUID ANTIPHONIS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \edtext{Consequenter}{\Cfootnote{\textit{om}. \textit{π}}} querit causam \edtext{causae}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{x}}}} precedentis problematis, dicens \textit{quare in antiphonis cordis \edtext{inest}{\Cfootnote{ita est \textit{π}}} quod vna cantata \edtext{cantetur et}{\Cfootnote{cantatur \textit{π}}} altera, et quod idem \edtext{consonat}{\Cfootnote{sonat \textit{P\textsuperscript{y}V\textsuperscript{m}}}} vtrobique \edtext{sicut si ambo cantarent}{\Cfootnote{sicut si ambo cantarentur \textit{α} | \textit{om}. \textit{π}}}?}
\pend

\pstart
\textbf{3} \textit{AUT.} Soluit dicens causam esse \textit{quia sole \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} cordarum antiphone in quibus \edtext{existit}{\Cfootnote{consistit \textit{π}}} dyapason equaliter distant a media \edtext{corda}{\Cfootnote{\textit{om}. \textit{π}}},} vt dicatur a mese \edtext{que}{\Cfootnote{quod \textit{πE\textsuperscript{a}}}} est \edtext{vere}{\Cfootnote{nete \textit{γ}}} media 7. \edtext{cordarum}{\Cfootnote{corde \textit{V\textsuperscript{o}}}}, vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{m}}}}. \textit{Illud autem \edtext{quod}{\Cfootnote{quod vere \textit{π}}} est medium conficit quandam \edtext{similitudinem}{\Cfootnote{consonantiam \textit{π}}} et conuenientiam sonorum cum extremis,} reddit enim \edtext{similitudinem}{\Cfootnote{similem \textit{απ}}} fere sonum media tacta, ac si tangerentur ambe ultime. Et hoc est quod dicit: \textit{auditus videtur diiudicare quod ipsa media cordarum et \edtext{quod}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{b}V\textsuperscript{z}}}} \edtext{vtreque}{\Cfootnote{\textit{om}. \textit{π}}} extreme \edtext{constituant}{\Cfootnote{constituunt \textit{π}}} vnum sonum.} Et ideo in antiphonis \edtext{solis}{\Cfootnote{solum \textit{π}}} accidit quod vna corda tacta vel cantata \edtext{cantetur}{\Cfootnote{cantentur \textit{γ}}} \edtext{reliqua}{\Cfootnote{relique \textit{γ} | altera \textit{λ}}}, cum \edtext{in}{\Cfootnote{\textit{om}. \textit{π}}} medium \edtext{concurratur}{\Cfootnote{concurrat \textit{π}}}, quare dyapason tantum inter alias \edtext{proportiones}{\Cfootnote{concordias \textit{G\textsuperscript{u}P\textsuperscript{q}}}} sola ex antiphonis est apparentius tonis.
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\subsection{Vigesimum problem}\label{part20}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID SI QUIS NOSTRUM MEDIAM MOUET, CONCORDANS ALIAS CORDAS, VTITUR INSTRUMENTO, NON SOLUM QUANDO SECUNDUM MEDIE FIAT SONUM DEFICIT, ET VIDETUR NON CONCORDANS, SED ET SECUNDUM ALIAM MELODIAM? SI AUTEM LICHANUM AUT ALIQUEM ALIUM SONUM, TUNC VIDETUR DIFFERE SOLUM QUANDO ILLA QUIS VTITUR. \textsuperscript{2}AUT RATIONABILITER HOC ACCIDIT, OMNES ENIM VTILES CANTUS MULTOTIENS MEDIA VTUNTUR. \textsuperscript{3}ET OMNES BONI POETE FREQUENTER AD MEDIAM OBUIANT. QUAMUIS DISCEDANT, CITO REUERTUNTUR, AD ALIAM AUTEM ITA NEQUE VNAM. \textsuperscript{4}QUEMADMODUM EX ORATIONIBUS: QUIBUSDAM ABLATIS CONIUNCTIONIBUS, NON EST ORATIO GRECA, VT AUTEM ET VERO. ET QUIDAM AUTEM NON DEFICIUNT, PROPTER ID QUOD HIIS QUIDEM NECESSARIUM EST ESSE VTI MULTOTIENS, SI ERIT ORATIO, HIIS AUTEM NON. \textsuperscript{5}ITA AUTEM ET SONORUM MEDIA SICUT CONIUNCTIO EST ET MAXIME BONORUM, PROPTER ID QUOD MULTOTIENS INEST IPSIS SONIS}{\lemma{PROPTER \dots\ SONIS}\Cfootnote{PROPTER QUOD SI QUIS NOSTRUM \textit{V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} si aliquis nostrum scientium \edtext{pulsare}{\Cfootnote{pulsaret \textit{P\textsuperscript{x}V\textsuperscript{m}}}} instrumentum \edtext{tangat}{\Cfootnote{tangit \textit{P\textsuperscript{x}P\textsuperscript{y}}}} cordam mediam vt mese, vtitur instrumento tali modo ac si omnes \edtext{alias}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{y}}}} cordas extremas ad inuicem concordantia reduceret, quod declarat? Non enim quando media corda tangitur sonus deficit et videtur non fieri \edtext{concordantia}{\Cfootnote{concordia \textit{V\textsuperscript{o}}}} in instrumento, sed quod plus est, apparet quod \edtext{tactus}{\Cfootnote{cantus \textit{πP\textsuperscript{x}}}} istius corde resonet in quandam melodiam. Id autem non contingit in aliis cordis extremis, sicut \edtext{est}{\Cfootnote{est in \textit{γ}}} lychanos, que est tertia post ypatem. Si quis enim sonum \edtext{causaret}{\Cfootnote{cantet \textit{V\textsuperscript{o}}}} per cordam hanc vel per aliam vt paramese vel paranete, tunc \edtext{solum}{\Cfootnote{sonum \textit{V\textsuperscript{o}}}} videtur differentia in sono et \edtext{dissimphonia causari}{\Cfootnote{dissymphoniam causare \textit{γ}}} quando quis vtitur tangendo illam cordam que \edtext{non}{\Cfootnote{si \textit{V\textsuperscript{o}}}} est recte media.} Vel aliter quod tunc \edtext{solum}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} est \edtext{differentia sonorum}{\Cfootnote{sonorum differentia \textit{π}}} \edtext{ad}{\Cfootnote{ad se \textit{π}}} inuicem visa, qua discernitur quod vnus est delectabilior \edtext{altero}{\Cfootnote{alio \textit{π}}}, quando quis \edtext{vtitur tali modo}{\Cfootnote{vtitur modo \textit{π}}} media corda recte taliter quod in \edtext{eam}{\Cfootnote{ea \textit{P\textsuperscript{x}V\textsuperscript{a}}}} consonent due ultime, quod est difficile mihi.
\pend

\pstart
\textbf{3} Notandum quod \hyperref[Ar]{aristoteles} \edtext{multotiens dicit}{\Cfootnote{dicit multotiens \textit{λπV\textsuperscript{a}}}} \textit{videtur} eo quod \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} scientia consistit in sensu \edtext{et}{\Cfootnote{vt \textit{γπE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{m}}}} auditu et ratione, \edtext{ceu}{\Cfootnote{seu \textit{V\textsuperscript{o}} | sicut \textit{γ}}} alie. \hyperref[Pyth]{Pictagoras} tamen omne quod ipsius \edtext{nitebatur}{\Cfootnote{videbatur \textit{πE\textsuperscript{a}}}} rationi magis \edtext{inherere}{\Cfootnote{adherere \textit{π}}}, \hyperref[Ax]{aristoxenus} vero sensui, quos ambos \hyperref[Pt]{ptholomeus} in \edtext{eius}{\Cfootnote{sua \textit{π}}} \edtext{\textit{armonica}}{\Cfootnote{rethorica \textit{π}}} detestatur medium obseruans cum declinatione in rationem amplius. 
\pend

\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit per simile in aliis repertum dicens primo \textit{\edtext{rationabiliter id}{\Cfootnote{id rationabiliter \textit{πM\textsuperscript{a}}}} fieri} quod in \edtext{quesito}{\Cfootnote{quesito est \textit{V\textsuperscript{o}}}} \edtext{proposito}{\Cfootnote{propositum \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}}. \textit{Nam omnes cantus vtiles et \edtext{laudati}{\Cfootnote{laudatiui \textit{π}}} attingentes finem musice cuiuscunque \edtext{multotiens}{\Cfootnote{melodie \textit{π}}} \edtext{vtuntur}{\Cfootnote{vtitur \textit{γ}}} corda media} reducendo ceteras et maxime supremas proportionaliter ad ipsam.
\pend

\pstart
\textbf{5} \edtext{\textit{ET}}{\Cfootnote{Deinde. ET \textit{V\textsuperscript{a}mv}}} \textit{OMNES.} Declarat id secundo \edtext{et}{\Cfootnote{\textit{om}. \textit{λE\textsuperscript{a}V\textsuperscript{o}}}} primo in metrica, que est musica quedam vt \edtext{apparuit}{\Cfootnote{apparebit \textit{γπ}}}, dicens quod \textit{omnes boni poete vtentes hac parte musice metrica dicta, \edtext{quam}{\Cfootnote{qua \textit{γ}}} frequenter \edtext{conuertunt se}{\Cfootnote{se conuertunt \textit{π}}} ad mediam cordam siue ad id quod proportionatur illi, quod si \edtext{accidat}{\Cfootnote{accidit \textit{P\textsuperscript{x}V\textsuperscript{z}}}} \edtext{recedere quandoque ab ea}{\Cfootnote{quandoque recedere ab ea \textit{P\textsuperscript{y}V\textsuperscript{b}} | quandoque ab ipsa recedere \textit{π}}}, \edtext{cito}{\Cfootnote{statim \textit{π}}} \edtext{ad ipsam redeunt}{\Cfootnote{redeunt ad ipsam \textit{π}}} tanquam ad perfectum, sed ad nullam \edtext{aliarum}{\Cfootnote{aliam \textit{π}}} \edtext{ita}{\Cfootnote{\textit{om}. \textit{πM\textsuperscript{a}}}} reuertuntur, cum non vt \edtext{hic tactum}{\Cfootnote{hic tactum est \textit{γπ} | hic cantum \textit{λE\textsuperscript{a}N\textsuperscript{u}}}} \edtext{perficiat}{\Cfootnote{proficiat \textit{V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{6} Nosce quod eximii poete \edtext{tunc}{\Cfootnote{cum \textit{π}}} obuiant ad mediam cordam et ad sibi proportionale, quando apud nos vsitate \edtext{construuntur}{\Cfootnote{construunt \textit{E\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}} carmina in quibus sunt tempora sillabarum ad inuicem \edtext{conproportionata}{\Cfootnote{composita \textit{π} | proportionata \textit{P\textsuperscript{y}V\textsuperscript{b}}}} ex breui et \edtext{longo}{\Cfootnote{longa \textit{α}}}. Sillaba namque breuis vnum habet tempus, longa vero duo, quale quod \edtext{constructum}{\Cfootnote{constructum est \textit{γπ}}} ex dactilo vicissim et spondeo et non ex altero. Tantum neque in exametro trocheo, nisi in fine dactilo, \edtext{quidem}{\Cfootnote{quodam \textit{V\textsuperscript{o}}}} \edtext{nunquam}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{neque}{\Cfootnote{neque ex \textit{πE\textsuperscript{a}}}} spondeo in quinto \edtext{pede}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} aut \edtext{iambo}{\Cfootnote{in ambo \textit{E\textsuperscript{a}P\textsuperscript{x}}}} ei opposito. Constat namque ex prima breui et \edtext{ultima}{\Cfootnote{altera \textit{π}}} longa, quo \edtext{vtimur}{\Cfootnote{vtuntur \textit{V\textsuperscript{o}}}} licentia \edtext{quandoque}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{z}}}} poetica in principio \edtext{tertii pedis}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}. Tunc namque \edtext{fit aliter}{\Cfootnote{aliter fit \textit{απ}}} vt cum seceditur a media corda vel ab eo quod sibi \edtext{consimile}{\Cfootnote{simile \textit{πE\textsuperscript{a}V\textsuperscript{z}}}}. Debent namque carmina fore bene \edtext{cantatiua}{\Cfootnote{cantantia \textit{P\textsuperscript{x}P\textsuperscript{z}V\textsuperscript{m}}}} vt que \edtext{\hyperref[Ov]{ouidii}}{\Cfootnote{toni \textit{G\textsuperscript{u}P\textsuperscript{p}} | eum \textit{V\textsuperscript{o}}}} \edtext{non}{\Cfootnote{\textit{om}. \textit{λ}}} autem \edtext{que sunt}{\Cfootnote{sic que \textit{απ}}} \hyperref[Hor]{oracii} et ideo \edtext{aliqui}{\Cfootnote{aliquos \textit{πM\textsuperscript{a}V\textsuperscript{m}}}} versus construunt \edtext{dictos leoninos}{\Cfootnote{leoninos dictos \textit{πP\textsuperscript{y}} | leoninos \textit{M\textsuperscript{a}V\textsuperscript{m}}}} quasi \edtext{rithimicam}{\Cfootnote{rithimos \textit{V\textsuperscript{o}}}} commiscentes metrice.
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\textbf{7} \edtext{\textit{QUEMADMODUM.}}{\Cfootnote{Deinde. QUEMADMODUM. \textit{V\textsuperscript{a}mv}}} Declarat secundo in \edtext{coniunctione}{\Cfootnote{coniectione \textit{P\textsuperscript{x}V\textsuperscript{z}}}} orationis prosaice dicens quod \textit{sicut in aliquibus orationibus \edtext{in quibus}{\Cfootnote{\textit{om}. \textit{κ}}} videlicet multa coadunantur, non constituitur oratio \edtext{greca}{\Cfootnote{gramatica \textit{γ}}}, que est orationum congruissima et perfectissima, remotis quibusdam \edtext{coniunctionibus}{\Cfootnote{coniectionibus \textit{P\textsuperscript{x}V\textsuperscript{z}}}} tanquam vinculis conuenientibus, puta vt \edtext{autem}{\Cfootnote{aurem \textit{P\textsuperscript{x}V\textsuperscript{m}}}} vero quidem et \edtext{huius}{\Cfootnote{huiusmodi \textit{βV\textsuperscript{b}}}}. Vnde et quidam \edtext{construere volentes}{\Cfootnote{volentes construere \textit{π}}} orationem \edtext{grecam}{\Cfootnote{gramaticam \textit{γ}}} congruissimam et perfectam bene sonantem, et ne sequatur vocalis ad vocalem, \edtext{quod}{\Cfootnote{quidam \textit{λ}}} \edtext{grecus}{\Cfootnote{gramaticus \textit{γ}}} \edtext{horret}{\Cfootnote{oret \textit{κ}}} etiam in scriptura, non deficiunt in \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} \edtext{coniunctionalibus}{\Cfootnote{coniunctionibus \textit{P\textsuperscript{x}V\textsuperscript{a}}}} partium ligaturis, cum oporteat eis multotiens vti si \edtext{debet}{\Cfootnote{debeat \textit{πP\textsuperscript{x}V\textsuperscript{z}}}} oratio perfici conuenienter; in hiis autem \edtext{deficiunt}{\Cfootnote{deficiunt, quandoque \textit{απ}}}} \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} et cantus, interdum recedunt a media corda. Quia quandoque non \edtext{sit}{\Cfootnote{fit \textit{P\textsuperscript{y}V\textsuperscript{m}}}} necessarium sepe vti, sicut \edtext{in}{\Cfootnote{\textit{om}. \textit{π}}} orationibus \edtext{minus}{\Cfootnote{viris \textit{G\textsuperscript{u}P\textsuperscript{p}}}} \edtext{grecis}{\Cfootnote{gramaticis \textit{γ}}} aut alio modo, \edtext{velut}{\Cfootnote{valde \textit{β}}} sunt alie partes orationis preter est verbum substantiuum expressum aut intellectum in \edtext{omnibus}{\Cfootnote{orationibus \textit{πV\textsuperscript{z}}}}. Vel aliter, quia coniunctiones sunt duplices, quedam \edtext{namque}{\Cfootnote{enim sunt \textit{π}}} maxime necessarie in \edtext{oratione}{\Cfootnote{orationibus \textit{π}}}, cum protenditur puta \edtext{et proprie}{\Cfootnote{\textit{om}. \textit{π}}} si quia \edtext{aut}{\Cfootnote{autem \textit{αV\textsuperscript{o}}}} et \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}} | similia \textit{π}}}. Alie vero quasi minime, vt quidem siquidem nempe quippe et \edtext{huius}{\Cfootnote{huiusmodi \textit{γE\textsuperscript{a}N\textsuperscript{u}}}}. Modo oratio \edtext{perfecte}{\Cfootnote{perfecta \textit{P\textsuperscript{y}V\textsuperscript{a}}}} \edtext{greca}{\Cfootnote{gramatica \textit{γ}}} constituitur, quando vtitur \edtext{hoc}{\Cfootnote{hac \textit{E\textsuperscript{a}N\textsuperscript{u}P\textsuperscript{x}V\textsuperscript{b}}}} modo coniunctionis duplici, puta priori \edtext{significatiua}{\Cfootnote{significatio \textit{P\textsuperscript{x}V\textsuperscript{m}}}} et non significatiua. Dico autem perfecte \edtext{greca}{\Cfootnote{gramatica \textit{γ}}} quia \edtext{greci}{\Cfootnote{gramatici \textit{γ}}} et potissime \edtext{\hyperref[Ga]{galienus}}{\Cfootnote{gallici \textit{γV\textsuperscript{z}}}} maxime \edtext{vtitur}{\Cfootnote{vtuntur \textit{γλπ}}} \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}P\textsuperscript{q}V\textsuperscript{b}}}} \edtext{ligamentis}{\Cfootnote{ligamentalibus \textit{βE\textsuperscript{a}V\textsuperscript{b}}}} sinchathegorematis \edtext{etiam}{\Cfootnote{et \textit{πV\textsuperscript{m}}}} secundis, vt et plura quandoque adiungat vna puta: ac et, nimirum, quidem, demum. Tales \edtext{autem}{\Cfootnote{\textit{om}. \textit{π}}} coniunctiones apud \edtext{grecos}{\Cfootnote{gramaticos \textit{γ}}} dicuntur parapliromatice, \edtext{id est}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} completiue sermonis, vtpote di. ta. ra. ni. ge. et cetera.
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\pstart
\textbf{8} \edtext{\textit{ITA}}{\Cfootnote{Deinde. ITA \textit{V\textsuperscript{a}mv}}} \textit{AUTEM.} Applicat ad propositum tertio, dicens quod \textit{simili modo se habet cordarum media vel ei proportionale \edtext{in}{\Cfootnote{et \textit{βV\textsuperscript{m}V\textsuperscript{z}}}} sonis, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}V\textsuperscript{o}}}} coniunctio in orationibus \edtext{gramaticis}{\Cfootnote{gramatice \textit{M\textsuperscript{a}V\textsuperscript{a}}}} et proprie bonis \edtext{seu grecis}{\Cfootnote{ceu gramaticis \textit{γ}}}.} Nam \edtext{cordarum}{\Cfootnote{corda \textit{π}}} media est proprie de numero bonarum \edtext{reddentium}{\Cfootnote{cordarum reddentium \textit{π}}} mellifluum sonum et consonum extremis. \textit{Et quia \edtext{hiis}{\Cfootnote{is \textit{β} | id \textit{π}}} multotiens debet \edtext{in esse}{\Cfootnote{esse in \textit{π}}} sonis} ideo \edtext{vt}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{propositum est}{\Cfootnote{propositum \textit{απ}}} multotiens, \edtext{cum possunt}{\Cfootnote{compositum \textit{πE\textsuperscript{a}}}} vtuntur corda media secundum quod ei consonant extreme, et de hoc itidem \edtext{36° apparebit consequenter}{\Cfootnote{apparebit consequenter 36°. \textit{π}}} problemate.
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\subsection[Vigesimumprimum problema … ]{Vigesimumprimum problema}\label{part21}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID CANTANTIUM GRAUIUS CANTANTES ACUTE CANTANTIBUS, SI DISSONANT, MAGIS MANIFESTI FIUNT? SIMILITER AUTEM ET RITHIMO: QUI IN GRAUIORI PECCANTES MANIFESTI MAGIS. \textsuperscript{2}UTRUM QUOD PLUS TEMPUS GRAUI, HIC AUTEM MAGIS SENSIBILIS? \textsuperscript{3}AUT QUIA IN PLURI TEMPORE MULTUM EXHIBET, VELOX AUTEM ET ACUTUM LATET PROPTER VELOCITATEM}{\lemma{PROPTER \dots\ VELOCITATEM}\Cfootnote{PROPTER QUID CANTANTIUM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}?}
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\textbf{2} \textit{\edtext{Quare cantantes}{\Cfootnote{Querit quare cantantes \textit{λ}}} de numero grauius cantantium, si dissonent et \edtext{errent}{\Cfootnote{errarent \textit{γ}}} in cantu suo ab ordine debito, \edtext{magis redduntur}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} \edtext{manifesti}{\Cfootnote{manifesto \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} in suo errore quam illi qui cantant acutius?} Et addit \edtext{quod}{\Cfootnote{quoniam \textit{π}}} \textit{non solum \edtext{accidit hoc}{\Cfootnote{hoc accidit \textit{π}}} in musica armonica, que ex vocibus constat, sed etiam in rithimica}, que coherentiam et consonantiam verborum contemplatur: vtrum bene vel male consonet. \textit{Qui enim \edtext{peccant}{\Cfootnote{potest causat \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} in ipsa secundum grauius \edtext{magis manifestantur}{\Cfootnote{magis manifestatur \textit{πV\textsuperscript{z}}}} quam si \edtext{peccarent}{\Cfootnote{peccaret \textit{V\textsuperscript{o}V\textsuperscript{z}}}} secundum acutius.}
\pend

\pstart
\textbf{3} \edtext{\textit{UTRUM.}}{\Cfootnote{Deinde. VTRUM. \textit{V\textsuperscript{a}mv}}} Soluit et assignat \edtext{duplicem causam quasi}{\Cfootnote{quasi duplicem causam \textit{π}}} \edtext{consimilem}{\Cfootnote{\textit{om}. \textit{π}}}, dicens causam esse \textit{quoniam graue plus habet temporis quam acutum. Vbi autem plus est \edtext{temporis}{\Cfootnote{temporis et \textit{βγ}}}, amplius potest peccari, \edtext{et}{\Cfootnote{vt \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} peccatum similiter magis percipi.}
\pend

\pstart
\textbf{4} \edtext{\textit{AUT}}{\Cfootnote{Deinde. AUT \textit{V\textsuperscript{a}mv}}} \textit{QUIA.} \edtext{Assignat causam dicens}{\Cfootnote{Dicens \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}} | Assignat aliam causam dicens \textit{γπ}}} \edtext{id}{\Cfootnote{illud \textit{E\textsuperscript{a}P\textsuperscript{x}}}} esse \textit{quia \edtext{in}{\Cfootnote{\textit{om}. \textit{π}}} tempore maiori exhibet graue tardum et multum sonum}, \edtext{quare fit esse ipsius manifestum propter sui tarditatem. Sed velox et acutum, quia in minori tempore dat velocem sonum}{\Cfootnote{\textit{om}. \textit{π}}}, ideo latet propter eius velocitatem, ita quod peccatum non perpenditur etiam inexistens notabiliter.
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\subsection[Vigesimumsecundum problema … ]{Vigesimumsecundum problema}\label{part22}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID MULTI MAGIS CANTANTES RITHIMUM SALUANT QUAM PAUCI? \textsuperscript{2}AUT QUIA MAGIS APUD VNUM ET RECTOREM INTENDUNT ET GRAUIUS INCIPIUNT, QUARE FACILE IDEM ATTINGUNT? IN VELOCITATE ENIM PECCATUM EST PLUS}{\lemma{PROPTER \dots\ PLUS}\Cfootnote{PROPTER QUID MULTI \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}P\textsuperscript{q}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} multi cantantes melius \edtext{conseruant}{\Cfootnote{seruant \textit{E\textsuperscript{a}V\textsuperscript{o}}}} rithimum et vniuersaliter omnem \edtext{armoniam}{\Cfootnote{armonicam \textit{M\textsuperscript{a}V\textsuperscript{m}}}} quam acceperunt cantare quam pauci?}
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} \edtext{Soluit dicens}{\Cfootnote{dicit \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{z}}}} causam esse \textit{quoniam ipsi amplius intendunt et \edtext{ordinantur}{\Cfootnote{ordinant \textit{π}}} ad vnum, tanquam ad \edtext{rectorem}{\Cfootnote{rethorem \textit{V\textsuperscript{o}}}} \edtext{ouium}{\Cfootnote{omnium \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}}, et grauius possunt incipere cum fuerint multi quoniam vox vnius fortificat et substentat vocem alterius. Propter quod de facili possunt in vnum deuenire} illudque illesum conseruare in velocitate. \edtext{Autem}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} \edtext{non contingit hoc}{\Cfootnote{hoc non contingit \textit{γE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{o}}}} cum difficilius sit seruare acutum quam graue, vt dicetur \edtext{.26°.}{\Cfootnote{.36°. \textit{π}}}, quare \textit{in velocitate \edtext{magis contingit}{\Cfootnote{contingit magis \textit{γ}}} peccare,} etsi \edtext{peccatum}{\Cfootnote{peccant \textit{β}}} \edtext{plerumque sit}{\Cfootnote{plerumque fit \textit{V\textsuperscript{m}V\textsuperscript{z}} | plurimumque sit \textit{E\textsuperscript{a}P\textsuperscript{y}}}} magis latens vt \edtext{etiam}{\Cfootnote{et \textit{β}}} alii ostendunt actus. Et id quidem \edtext{.45.}{\Cfootnote{.35. \textit{π}}} repetitur.
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\subsection[Vigesimumtertium problema … ]{Vigesimumtertium problema}\label{part23}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DUPLEX NITE YPATE? \textsuperscript{2}AUT PRIMUM QUIDEM QUIA EX MEDIO CORDA CANTATA ET TOTA CONSONANT DYAPASON? \textsuperscript{3}SIMILITER AUTEM SE HABET IN FISTULIS, QUE ENIM PER MEDIUM FORAMEN FISTULE CONSONANT PER TOTAM FISTULAM DYAPASON. ADHUC ET IN ALIIS IN DUPLICI DISTANTIA ACCIPITUR DYAPASON, ET ILLI QUI PERFORANT FISTULAS ITA ACCIPIUNT. SIMILITER ET CUM DYAPENTE EMIOLIUM, QUIA QUI FISTULAS CONCORDANT IN YPATEM CERAM PONUNT, NITEM AUTEM VSQUE AD MEDIUM REPLENT. SIMILITER AUTEM ET PER DYAPENTE EMIOLIO, ET PER QUATUOR EPITRICA DISTANTIA ACCIPIUNT. \textsuperscript{4}AMPLIUS IN TRIANGULIS PSALTERIIS EQUALI DISTENSIONE FACTA CONSONANT DYAPASON. HEC QUIDEM DUPLEX EXISTENS, ALIA VERO MEDIA LONGITUDINE}{\lemma{PROPTER \dots\ LONGITUDINE}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \edtext{Quare}{\Cfootnote{Quia \textit{λE\textsuperscript{a}V\textsuperscript{o}}}} in \edtext{12°.}{\Cfootnote{2°. \textit{π}}} dixerat \textit{duplicem \edtext{nitem}{\Cfootnote{vicem \textit{π}}} constituere ypatem}? Ideo causam huius querit, dicens: \textit{quare duplex \edtext{nite}{\Cfootnote{vite \textit{V\textsuperscript{o}}}} constituit ypatem?}
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{AUT PRIMUM. \textit{γE\textsuperscript{a}V\textsuperscript{o}}} | Deinde. AUT. \textit{V\textsuperscript{a}mv}} Soluit et primo ostendit quomodo ex duplici \edtext{nite}{\Cfootnote{vite \textit{V\textsuperscript{o}}}} causetur ypate in cantu, secundo \edtext{in instrumentis}{\Cfootnote{instrumentis in \textit{π}}} flatu, tertio \edtext{ductu}{\Cfootnote{in ductu \textit{κ}}}, \edtext{secunda}{\Cfootnote{secunda ibi \textit{γ}}} \textit{SIMILITER}, \edtext{tertia}{\Cfootnote{tertia ibi \textit{γ}}} \textit{AMPLIUS}, \edtext{dicens}{\Cfootnote{dicit \textit{E\textsuperscript{a}V\textsuperscript{z}} | dicens primo \textit{π}}} causam esse \edtext{ex hoc primitus sumendam}{\Cfootnote{sumendam ex hoc \textit{π}}}, \textit{quoniam quando media cordarum cuiuslibet instrumenti \edtext{cantatur}{\Cfootnote{incantatur \textit{P\textsuperscript{x}P\textsuperscript{y}}}}, tunc et omnes \edtext{alie}{\Cfootnote{aut \textit{M\textsuperscript{a}V\textsuperscript{z}}}} coordinantur ad \edtext{ipsam}{\Cfootnote{illam \textit{απ}}} mediam et \edtext{consonant}{\Cfootnote{consonat \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} per eam, cum ex ea \edtext{constituatur}{\Cfootnote{constituitur \textit{κV\textsuperscript{a}}}} dyapason}, vt \edtext{apparuit}{\Cfootnote{apparebit \textit{πE\textsuperscript{a}}}} \edtext{20.}{\Cfootnote{2°. \textit{M\textsuperscript{a}P\textsuperscript{x}} | 3°. \textit{π}}}
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\pstart
\textbf{4} \edtext{\textit{SIMILITER.}}{\Cfootnote{Deinde. SIMILITER. \textit{V\textsuperscript{a}mv}}} Ostendit secundo similiter \edtext{fore}{\Cfootnote{\textit{om}. \textit{λ}}} in instrumentis \edtext{et}{\Cfootnote{in \textit{απ}}} \edtext{flatis}{\Cfootnote{sufflatis \textit{P\textsuperscript{y}V\textsuperscript{o}} | sufflatiuis \textit{γE\textsuperscript{a}N\textsuperscript{u}}}}, dicens \textit{\edtext{simili modo}{\Cfootnote{similiter \textit{π}}} se habere in fistulis,} in quibus sunt quedam foramina habentia similitudinem cum cordis, in eis enim \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{a}}}} magnum foramen, cuius sonus grauis, \edtext{proportionatum}{\Cfootnote{proportionatur \textit{πV\textsuperscript{z}}}} ypate et \edtext{secundum}{\Cfootnote{secundus \textit{V\textsuperscript{o}}}} vt medium, \edtext{assimilatum}{\Cfootnote{assimilatur \textit{π}}} mese et plura parua. Super que \edtext{fistulantur}{\Cfootnote{fistulatur \textit{βV\textsuperscript{o}}}} attributa \edtext{nete}{\Cfootnote{vite \textit{P\textsuperscript{p}P\textsuperscript{z}V\textsuperscript{o}}}}, \textit{in quibus siquidem soni qui fiunt per medium foramen fistule consonant et inducunt per totam fistulam secundum aliqua eius foramina omnia dyapason, cum in ipsum, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} in medium, \edtext{reducantur}{\Cfootnote{reducuntur \textit{P\textsuperscript{y}mv}}} reliqua. Similiter et in aliis \edtext{id}{\Cfootnote{illud \textit{P\textsuperscript{x}V\textsuperscript{b}} | \textit{om}. \textit{π}}} accidit instrumentis, in quibus predicta potest \edtext{aptari}{\Cfootnote{adaptari \textit{απV\textsuperscript{z}}}} foraminum similitudo, \edtext{nam}{\Cfootnote{vero \textit{P\textsuperscript{x}V\textsuperscript{m}}}} in eis etiam dyapason constituitur \edtext{in esse}{\Cfootnote{inest \textit{V\textsuperscript{o}}}} secundum duplicem distantiam:} puta vnius extremi ad medium vna, et alterius ad ipsum reliqua, vel quia ex numero consurgit duplari. Quod ostenditur quia \textit{illi qui perforant fistulas tali modo suscipiunt \edtext{distantiam}{\Cfootnote{distantia \textit{βV\textsuperscript{b}V\textsuperscript{o}V\textsuperscript{z}}}} vt inde consurgat dyapason cum reliquis proportionibus. Nam taliter \edtext{operantur}{\Cfootnote{opponatur \textit{γ}}}: perforando et coaptando vt secundum proportionem sexquialteram \edtext{reddatur}{\Cfootnote{\textit{om}. \textit{κ}}} dyapente, quod declarat. Nam qui fistulas taliter \edtext{consonas}{\Cfootnote{consonans \textit{P\textsuperscript{x}V\textsuperscript{a}}}} faciunt vt consurgat dyapason apponunt \edtext{ceram}{\Cfootnote{seram \textit{V\textsuperscript{o}}}} vel aliquod \edtext{viscosum glutinosum}{\Cfootnote{glutinosum viscosum \textit{β} | viscosum siue glutinosum \textit{πE\textsuperscript{a}}}} in foramine maiori proportionato ypate, vt sonus eius subgrauetur, et foramina per nitem representata replent \edtext{cera}{\Cfootnote{ceram \textit{P\textsuperscript{x}V\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}} | sera \textit{V\textsuperscript{o}}}} usque ad medium,} vt eis angustatis sonus reddatur acutior, \edtext{ita}{\Cfootnote{\textit{om}. \textit{γ}}} vt possit duplex \edtext{nite}{\Cfootnote{vite \textit{V\textsuperscript{o}}}} in \edtext{ypatem}{\Cfootnote{ypate \textit{αV\textsuperscript{o}}}} \edtext{consurgere vel crescere}{\Cfootnote{concurrere \textit{αV\textsuperscript{o}}}}, cum vna exacuatur et \edtext{reliqua}{\Cfootnote{altera \textit{V\textsuperscript{o}}}} subgrauetur \edtext{sibi}{\Cfootnote{sibique \textit{π}}} \edtext{appropinquantes}{\Cfootnote{appropinquent \textit{π}}} \edtext{vtcunque}{\Cfootnote{vtrunque \textit{M\textsuperscript{a}V\textsuperscript{o}}}} eodem modo secundum \edtext{debitam distantiam foraminum}{\Cfootnote{debita distantia foraminum \textit{E\textsuperscript{a}N\textsuperscript{u}} | debita foramina \textit{κ}}} cum cere vel alterius appositione, ita quod etiam \edtext{lignum}{\Cfootnote{ligamen \textit{γ}}} \edtext{submittunt}{\Cfootnote{subimmittunt \textit{βV\textsuperscript{b}}}} \edtext{subtile}{\Cfootnote{fistule \textit{γ}}} in foramen medium secundum \textit{sexquialteram proportionem, \edtext{seu}{\Cfootnote{ceu \textit{P\textsuperscript{x}V\textsuperscript{z}}}} emiolam faciunt dyapente. Similiter per epitricam \edtext{distantiam}{\Cfootnote{distantia \textit{E\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{m}}}} que est cum numerus continet \edtext{alium totum}{\Cfootnote{totum alium \textit{E\textsuperscript{a}V\textsuperscript{z}} | alterum totum \textit{π}}} vt visum est et eius tertiam partem, vt quatuor ad tria se habet. Accipiunt per quatuor, id est dyatesseron grece}.
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\pstart
\textbf{5} \edtext{\textit{AMPLIUS.}}{\Cfootnote{Deinde. AMPLIUS. \textit{V\textsuperscript{a}mv}}} Declarat tertio \edtext{in}{\Cfootnote{\textit{om}. \textit{λE\textsuperscript{a}V\textsuperscript{a}}}} instrumento \edtext{tactibili}{\Cfootnote{tangibili \textit{πP\textsuperscript{y}}}}, dicens quod insuper etiam \textit{simili modo in psalteriis, que habent figuram triangularem, et in canonibus \edtext{facta}{\Cfootnote{factam \textit{V\textsuperscript{o}}}} proportionali \edtext{distensione}{\Cfootnote{distantia \textit{π}}} cordarum \edtext{ipsarum}{\Cfootnote{ipsorum \textit{γE\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{o}}}} ex acuto et graui consonant et adducunt dyapason. In ipsa enim duplex corda, vt ypate et \edtext{nite}{\Cfootnote{vite \textit{V\textsuperscript{o}}}}, et alia in quam \edtext{concidunt}{\Cfootnote{concidat \textit{E\textsuperscript{a}V\textsuperscript{b}} | conferat \textit{V\textsuperscript{o}}}} que est media \edtext{distantia}{\Cfootnote{distante \textit{π}}} vere, \edtext{vt}{\Cfootnote{et \textit{V\textsuperscript{o}V\textsuperscript{z}} | a \textit{G\textsuperscript{u}P\textsuperscript{p}}}} mese.} Vel aliter duplex nete concurrit in ypate: et constituitur tunc sonus \edtext{corde medie}{\Cfootnote{cordis medie \textit{E\textsuperscript{a}V\textsuperscript{o}} | totidem medie \textit{γ}}} secundum distantiam vt \edtext{apparebit}{\Cfootnote{apparuit \textit{βγ}}} \edtext{.20.}{\Cfootnote{.22. \textit{V\textsuperscript{o}}}}
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\subsection[Vigesimumquartum problema … ]{Vigesimumquartum problema}\label{part24}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID IN TENUIBUS MENTEM APPONERE, YPATE SOLA VIDETUR CONTRASONARE? \textsuperscript{2}AUT QUIA CONNATURALIS MAXIME FIT SONO SONUS QUI EST AB IPSA PROPTER ID QUOD CONSONANS EST? SIMUL AUGERI AUTEM SIMILI VIDETUR SOLUS. ALII VERO PROPTER PARUITATEM INMANIFESTI}{\lemma{PROPTER \dots\ INMANIFESTI}\Cfootnote{PROPTER QUID IN TENUIBUS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} \edtext{cum}{\Cfootnote{cum in \textit{βγV\textsuperscript{o}}}} subtilibus cordis, vt nete et paranete, sonum acutum et paruum reddentibus \edtext{quis}{\Cfootnote{siquis \textit{β}}} mentem id est rationem et sensum seu \edtext{auditum}{\Cfootnote{aditum \textit{β}}} apponit?} Hec enim scientia, vt pretactum \edtext{est}{\Cfootnote{\textit{om}. \textit{α}}}, constat in ratione et sensu. Tunc sola ypate sine aliis \edtext{videtur}{\Cfootnote{vt \textit{β}}} \edtext{antisonare}{\Cfootnote{antiphonare \textit{π}}} sonitu \edtext{reliquas}{\Cfootnote{reliqua \textit{V\textsuperscript{o}}}} superando, quod prius dictum est antiphonare.
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quia sonus qui ab ipsa causatur principalissime efficitur connaturalis et proportionalis alii sono, propter hoc quod cum sua \edtext{consonantia}{\Cfootnote{contrasonantia \textit{βγV\textsuperscript{z}}}} est etiam \edtext{simul}{\Cfootnote{similis \textit{P\textsuperscript{x}V\textsuperscript{z}}}} consonans}. Quando \edtext{enim}{\Cfootnote{est \textit{P\textsuperscript{x}V\textsuperscript{z}}}} sonus ipsius cum sono alterius corde \edtext{inferiori}{\Cfootnote{inferioris \textit{αV\textsuperscript{o}V\textsuperscript{z}}}} sub medio existentis \edtext{construitur}{\Cfootnote{constituitur \textit{βγπ}}}, consonantia vt iam patuit dyapason. \textit{Et similiter \edtext{solus sonus huius corde}{\Cfootnote{sonus huius corde solus \textit{π}}} inter alios sonos videtur augeri et \edtext{crementum}{\Cfootnote{incrementum \textit{P\textsuperscript{y}V\textsuperscript{z}}}} in tonando accipere}, quia ipsa reddit sonum sola grauiorem aliis, alie vero acutum \edtext{aut}{\Cfootnote{et \textit{λ}}} minus grauem. Sonus autem \edtext{grauis}{\Cfootnote{grauis sonum \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}} ebetat acute, \textit{sed soni aliarum cordarum subtilium}, quia sunt \edtext{parui}{\Cfootnote{praui \textit{V\textsuperscript{o}mv}}}, non percipiuntur manifeste, propter quod \edtext{non apparent}{\Cfootnote{non apparet \textit{P\textsuperscript{y}V\textsuperscript{m}}}} \edtext{consonare}{\Cfootnote{contrasonare \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}V\textsuperscript{z}}}}. Stat \edtext{igitur}{\Cfootnote{enim \textit{πE\textsuperscript{a}}}} in hoc, \edtext{quia}{\Cfootnote{quod \textit{P\textsuperscript{y}V\textsuperscript{m}}}} sonus ypate fortis est, aliarum vero debilis. Qui autem \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}}}} \edtext{consonat}{\Cfootnote{contrasonat \textit{γλE\textsuperscript{a}N\textsuperscript{u}} | contra consonat \textit{V\textsuperscript{o}}}} debili cum eo quod recipit, \edtext{magnus}{\Cfootnote{maior \textit{πP\textsuperscript{y}}}} enim motus minorem debilitat et obtundit. \edtext{Huius}{\Cfootnote{Huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} autem sonus \edtext{cordis}{\Cfootnote{corde \textit{βγV\textsuperscript{z}}}} licet sit fortior ceteris, quia tamen grauior et dimensionatior sonum aliorum de facili \edtext{recipit}{\Cfootnote{suscipit \textit{απ}}} et exaltat, ita vt dyapason inducat.
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\subsection[Vigesimumquintum problema … ]{Vigesimumquintum problema}\label{part25}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID MEDIA VOCATUR IN ARMONIIS, OCTO VERO NON EST MEDIUM? \textsuperscript{2}AUT QUIA VII. CORDARUM ERANT ARMONIE ANTIQUITUS? SEPTEM AUTEM HABENT MEDIUM}{\lemma{PROPTER \dots\ MEDIUM}\Cfootnote{PROPTER QUID MEDIA VOCATUR \textit{V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} in armoniis quedam cordarum vt \edtext{mese}{\Cfootnote{nete \textit{G\textsuperscript{u}P\textsuperscript{q}}}} appellatur media \edtext{recte}{\Cfootnote{ratione \textit{P\textsuperscript{y}V\textsuperscript{a}}}} .7. cordarum?} Equaliter enim distat ab eis, \textit{sed octo cordarum non est medium} directe vt arismetricum, \edtext{sed}{\Cfootnote{\textit{om}. \textit{κ}}} potius geometricum. Et ideo .8. cordarum non vocatur aliqua media.
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quoniam antiquitus, vt \edtext{visum est}{\Cfootnote{visum \textit{βE\textsuperscript{a}}}}, non fuerunt nisi septem corde in instrumentis armonicis. Septem autem \edtext{medium habet}{\Cfootnote{habet medium \textit{βG\textsuperscript{u}M\textsuperscript{a}}}} equidistans ab extremis,} puta .4., vt mese. Id autem \edtext{factum}{\Cfootnote{factum est \textit{π}}} quia dyapason, que est optima consonantiarum, ordinatur amplius cum fuerit medium \edtext{verum}{\Cfootnote{\textit{om}. \textit{π}}} in cordis inuenire. Et ideo \textit{musice} primo ``cum \hyperref[Co]{estiacus colophonius} \edtext{decimam}{\Cfootnote{quartam \textit{κπ}}} cordam \edtext{.9.}{\Cfootnote{cum \textit{π}}} preexistentibus addidisset, \hyperref[Ty]{thimoteus} \edtext{superveniens}{\Cfootnote{superveniens et \textit{E\textsuperscript{a}V\textsuperscript{o}}}} \hyperref[Ty]{milexius} \edtext{xi\textsuperscript{am}.}{\Cfootnote{nullam \textit{G\textsuperscript{u}P\textsuperscript{a}}}} adiunxit,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} I.20 ((Friedlein, pp. 208–209))}} ita vt signato \edtext{vero}{\Cfootnote{vt \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} medio in hiis puta sexta consonet dyapason.
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\pstart
\textbf{4} Nosce, vt \edtext{indicatum}{\Cfootnote{dictum \textit{π}}} \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{b}V\textsuperscript{o}}}} superius, \textit{.7. \edtext{antiquitus fuisse cordas}{\Cfootnote{fuisse cordas antiquitus \textit{π}}}} ``7. planetarum,'' secundum boecium, ``imitatione, ita vt suprema earum ypate reddatur saturno propter tarditatem motus''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} I.20 ((Friedlein 206, ll. 10–1, 15–6))}} vtrobique. Infima lune \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{P\textsuperscript{x}V\textsuperscript{a}}}} nete ob eius velocitatem. Que autem \edtext{intermedio}{\Cfootnote{in intermedio \textit{N\textsuperscript{u}V\textsuperscript{m}V\textsuperscript{o}}}} ordine planetarum \edtext{attribuantur}{\Cfootnote{atribuatur \textit{V\textsuperscript{m}V\textsuperscript{z}} | attribuebantur \textit{πP\textsuperscript{y}}}} mediis iuxta eorum mediocrem cursum. Et id quidem 32°. apparebit et magis 44°.
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\subsection[Vigesimumsextum problema … ]{Vigesimumsextum problema}\label{part26}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID IN ACUTO DISCORDANT PLURIMI? \textsuperscript{2}UTRUM QUIA FACILIUS ACUTUM DIMITTERE QUAM GRAUE? \textsuperscript{3}AUT QUIA PEIUS GRAUI? PECCATUM PEIORIS OPERATIO}{\lemma{PROPTER \dots\ OPERATIO}\Cfootnote{PROPTER QUID IN ACUTO \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} plurimi cantantes vna discordant amplius in acuto quam in graui?}
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\pstart
\textbf{3} \edtext{\textit{UTRUM.}}{\Cfootnote{Deinde. VTRUM. \textit{V\textsuperscript{a}mv}}} \edtext{Soluit}{\Cfootnote{Soluit et \textit{π}}} \edtext{assignans}{\Cfootnote{assignat \textit{π}}} duplicem causam probabilem prius, dicens \edtext{id}{\Cfootnote{illud \textit{P\textsuperscript{x}V\textsuperscript{b}}}} esse \textit{quoniam leuius est dimittere acutum?} Et recessus ab eo sepius contingit quam a graui propter breuitatem temporis \edtext{existentie}{\Cfootnote{ex natura \textit{P\textsuperscript{y}V\textsuperscript{a}}}} ipsius, propter quod dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}}}} 22°.: quod \textit{in velocitate peccatum \edtext{plus}{\Cfootnote{magis \textit{π}}} existit.} Non tamen vt \edtext{expositum}{\Cfootnote{expositum est \textit{π}}} adeo est manifestum \edtext{vt}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} peccatum in graui, propter quod apparet hoc in nullo illi aduersari.
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\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} \edtext{Soluit assignans}{\Cfootnote{Soluit assignando \textit{M\textsuperscript{a}V\textsuperscript{m}} | Assignat \textit{απ}}} aliam causam necessariam magis, dicens \edtext{id}{\Cfootnote{hoc \textit{γ}}} esse \textit{\edtext{quoniam}{\Cfootnote{quia \textit{λ}}} peccatum quod committitur in acutum cantando \edtext{deterius est seu difficilius quam peccatum quod committitur in \edtext{graue}{\Cfootnote{grauem \textit{λ}}} cantando}{\lemma{deterius \dots\ cantando}\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}}. Cum \edtext{molestius}{\Cfootnote{molestius sit \textit{απM\textsuperscript{a}V\textsuperscript{z}}}} cantare acuta quam grauia 37. \textit{Peioris autem seu difficilioris actus peccatum \edtext{maius}{\Cfootnote{maius est \textit{π}}}.} Et \edtext{ideo}{\Cfootnote{\textit{om}. \textit{γ}}} in ipso amplius accidit discordari. Quod enim \edtext{difficilius}{\Cfootnote{difficilius est \textit{π}}} agere magis contingit peccato corrumpere, vt aurum arte construere. \edtext{Et id etiam .46°. rescribetur}{\Cfootnote{\textit{om}. \textit{π}}}.
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\subsection[Vigesimumseptimum problema … ]{Vigesimumseptimum problema}\label{part27}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID AUDIBILE SOLUM HABET SENSIBILIUM? ET ENIM, IN QUANTUM ABSQUE RATIONE, CANTUS TAMEN HABET CONSUETUDINEM, SED NON COLOR, NEQUE ODOR, NEQUE HUMOR HABET. \textsuperscript{2}AUT QUIA MOTUM HABET SOLUM VNIFORMEM, QUO SONUS NOS MOUET? HUIUS ENIM ET ALIIS INEST, MOUET ENIM ET COLOR VISUM, SED CONSEQUENTEM HUIUS SONO SENTIMUS MOTUM. HEC ENIM HABET SIMILLIMA IN RITHIMIS ET IN SONORUM ORDINATIONE ACUTORUM ET GRAUIUM, NON IN MIXTIONE. SED CONSONANTIA NON HABET CONSUETUDINEM, IN ALIIS AUTEM SENSIBILIBUS HOC NON EST. \textsuperscript{3}MOTUS AUTEM IPSI OPERATIUI SUNT, OPERATIONES ENIM CONSUETUDINIS SIGNIFICATIO EST}{\lemma{PROPTER \dots\ EST}\Cfootnote{PROPTER QUID AUDIBILE \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} audibili tantummodo \edtext{de numero sensibilium hoc inest}{\Cfootnote{hoc inest de numero sensibilium \textit{π} | sensibilium hoc inest \textit{E\textsuperscript{a}P\textsuperscript{y}}}}? Quia etiam in quantum est cantus sine ratione \edtext{solo}{\Cfootnote{sono \textit{κ}}}, \edtext{tamen}{\Cfootnote{cum \textit{V\textsuperscript{o}}}} auditus \edtext{sensibili}{\Cfootnote{sensu \textit{απ}}}}, quia hec scientia, vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{z}}}}, perficitur ratione et sensu. Quis \edtext{eum}{\Cfootnote{enim \textit{κ}}} reddit \edtext{sibi}{\Cfootnote{\textit{om}. \textit{π}}} consuetum, non autem \edtext{color odori aut sapori}{\Cfootnote{colorem aut odorem aut saporem \textit{V\textsuperscript{o}}}} id adest quod ea fiant consueta in tantum vt que audibilia.
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et primo tangit partem cause ratione \edtext{audibilium}{\Cfootnote{audibilis \textit{π}}}, secundo ex parte eorum quibus \edtext{acquiritur}{\Cfootnote{adequatur \textit{β}}} \edtext{consuetudo}{\Cfootnote{consuetudo secunda\textsuperscript{a} \textit{M\textsuperscript{a}V\textsuperscript{m}} | consuetudo secunda ibi \textit{γV\textsuperscript{z}}}} \textit{MOTUS AUTEM}. Dicit igitur primo id esse \textit{quoniam audibile solum habet motum vniformem, propter \edtext{quod}{\Cfootnote{quem \textit{βγM\textsuperscript{a}V\textsuperscript{z}}}} sonus habet nos mouere.} ``Mouet enim potentiam audiendi, cum sit virtus passiua, et eam deducit in \edtext{actum}{\Cfootnote{acutum \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{x}}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{De an.} II.12, 424a19 ((Cf. 416b33 and 419b5))}} vt \edtext{apparet}{\Cfootnote{patet in \textit{π}}} secundo \textit{de anima.} \textit{Sed hic modus mouendi aliis \edtext{etiam inest}{\Cfootnote{inest etiam \textit{π}}} obiectis sensuum, nam color mouet \edtext{sensum}{\Cfootnote{visum \textit{απ}}}} et in \edtext{actum}{\Cfootnote{acutum \textit{βγP\textsuperscript{x}}}} deducit; et similiter odor et sapor. \textit{\edtext{Sed}{\Cfootnote{Hoc \textit{π}}} in auditu \edtext{ultra hoc}{\Cfootnote{universaliter \textit{G\textsuperscript{u}P\textsuperscript{p}}}} existit \edtext{quod}{\Cfootnote{quia \textit{πV\textsuperscript{m}}}} percipiendo sonum sentimus etiam motum quo mediante defertur ad aures quod etiam non solum accidit in \edtext{sono}{\Cfootnote{sono in \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}} | sono vel \textit{G\textsuperscript{u}P\textsuperscript{p}}}} communi, verum \edtext{etiam}{\Cfootnote{\textit{om}. \textit{π}}} similiter se habet in rithimis et in coordinatione seu \edtext{coniunctione}{\Cfootnote{coniectione \textit{P\textsuperscript{x}V\textsuperscript{z}}}} grauium sonorum et acutorum ad inuicem vnius quasi iuxta alium, vt non reddatur \edtext{vna}{\Cfootnote{vera \textit{λ} | recta \textit{π}}} commixtio.} Quia in ea non \edtext{id accidit}{\Cfootnote{accidit id \textit{π}}}, cum sonus adeo vniformiter aerem et suauiter \edtext{concitet}{\Cfootnote{continet \textit{β}}} vt aures \edtext{universaliter}{\Cfootnote{vel \textit{γE\textsuperscript{a}} | nichil \textit{βπ}}} percipiant sensibile ex aeris motu. Et \edtext{ideo}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} videmus infirmos \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}} | \textit{om}. \textit{π}}} adeo melodioso \edtext{delectari}{\Cfootnote{delectari sono \textit{γπE\textsuperscript{a}N\textsuperscript{u}}}}, non autem priori eum fugientes et maxime galleatam habentes passionem seu ouum. \textit{Et ideo consonantia sic armonica non recipit consuetudinem,} quemadmodum neque alii pretacti sensuum, \edtext{cum nichil}{\Cfootnote{cum nichil aliud \textit{π} | vel naturaliter \textit{κ}}} sensibilius preter perceptionem speciei \edtext{sonabilis}{\Cfootnote{sensibilis \textit{π}}} percipiatur ad auditum \edtext{peruenire}{\Cfootnote{perueniente \textit{λ}}}, quod \edtext{et}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}} in reliquis dictum est sensibus accidere. \textit{Sed in aliis \edtext{sensibus}{\Cfootnote{sensibilibus \textit{αV\textsuperscript{o}}}}, vt \edtext{color odor et sapor}{\Cfootnote{colori odori et sapori \textit{γP\textsuperscript{z}} | coloribus odoribus et saporibus \textit{β}}}, non inest,} vt eorum \edtext{sumantur}{\Cfootnote{seruatur \textit{π}}} species cum aeris motus perceptione.
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\textbf{4} Nota quod sicut duplex est mixtio, \edtext{vt}{\Cfootnote{vna \textit{π}}} ad sensum \edtext{quedam}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{et}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} alia ad intellectum, id est vera et non vera, ita \edtext{et duplex grauis et acuti}{\Cfootnote{duplex acuti et grauis \textit{βV\textsuperscript{a}} | et duplex est grauis et acuti \textit{π}}} permixtio, taliter vt ex vna reddatur \edtext{vera}{\Cfootnote{vna \textit{β}}} consonantia, ex \edtext{altera}{\Cfootnote{alia \textit{π}}} vero apparens magis.
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\textbf{5} \edtext{\textit{MOTUS.}}{\Cfootnote{Deinde. \textit{V\textsuperscript{a}mv}}} Τangit causam sumptam \edtext{ex secunda parte}{\Cfootnote{ex alia parte \textit{π}}} dicens cum igitur \edtext{visibilia}{\Cfootnote{audibilia \textit{π}}} vt ad multum recurrant ad organum cum motu aeris \edtext{sensibili}{\Cfootnote{sensibilis \textit{M\textsuperscript{a}V\textsuperscript{z}} | sensibiliter \textit{γP\textsuperscript{x}}}}, \textit{et motus \edtext{sint}{\Cfootnote{sicut \textit{βE\textsuperscript{a}V\textsuperscript{z}}}} operatiui}. Operationes enim fiunt per eos \edtext{actiones proprie dicte, cum ``\edtext{proprium actionis}{\Cfootnote{proprius actionis \textit{β} | proprium \textit{κ}}} sit esse in motu,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.-Herm. Tris. \textit{De sex re. princ.} II.1}} auctor \textit{.6. principiorum.}}{\Cfootnote{\textit{om}. \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{o}}}} Et \textit{signum consuetudinis acquirende vel acquisite sit operatio, } nam ``consuetudo \edtext{est}{\Cfootnote{est in \textit{β}}} multotiens aliquid facere,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.-Arist. \textit{Prob.} XXI.14, 928b25–26}} .13°. 21\textsuperscript{a}. \textit{Consuetudo \edtext{siquidem}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{z}}}} \edtext{adipiscetur magis}{\Cfootnote{magis adipiscetur \textit{πV\textsuperscript{z}}}} in audibilibus quam in ceteris \edtext{sensibilibus}{\Cfootnote{sensibus \textit{P\textsuperscript{x}V\textsuperscript{b}V\textsuperscript{o}}}}}, \edtext{et id tangetur etiam 29°.}{\Cfootnote{\textit{om}. \textit{π}}}, nosce quod causa \edtext{fundatur \hyperref[Ar]{aristoteles}}{\Cfootnote{\hyperref[Ar]{aristoteles} fundatur \textit{βπV\textsuperscript{z}}}} super hoc: quod generatio \edtext{soni et vocis}{\Cfootnote{vocis et soni \textit{P\textsuperscript{y}V\textsuperscript{a}}}} consistit in quodam \edtext{fieri et}{\Cfootnote{motu et \textit{λ}}} transmutari. Sunt namque de genere successiuorum entium, cum \edtext{sunt}{\Cfootnote{sint \textit{βπ}}} in motu, et fiunt tantum \edtext{color}{\Cfootnote{color vero \textit{π}}} odor et sapor de \edtext{manerie}{\Cfootnote{numero \textit{π}}} permanentium, que post factum \edtext{habent esse}{\Cfootnote{esse remanent \textit{π}}} et non prius. Quod \edtext{indicat}{\Cfootnote{denotat \textit{π}}} aristoteles, \edtext{quia singulariter \textit{de anima} secundo}{\Cfootnote{quia secundo \textit{de anima} \textit{π}}} ``determinauit de \edtext{modo generationis}{\Cfootnote{generatione \textit{π}}} soni et vocis,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{De an.} II.8, 419b9}} \edtext{et}{\Cfootnote{Et modo inmutandi et \textit{π}}} non \edtext{aliorum, sed inde sensu et sensato}{\Cfootnote{sic de aliis \textit{π}}}. Cum igitur modus generationis soni et perceptionis \edtext{vt ad}{\Cfootnote{ad \textit{V\textsuperscript{o}}}} multum conueniat cum operatione \edtext{siue}{\Cfootnote{seu \textit{V\textsuperscript{o}}}} actione, que fit mediante motu \edtext{qua}{\Cfootnote{quam \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} consuetudo vt \edtext{dictum}{\Cfootnote{dictum est \textit{γπE\textsuperscript{a}V\textsuperscript{z}}}} acquiritur \edtext{visibilia}{\Cfootnote{audibilia \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{y}}}} pre aliis \edtext{sensibilibus}{\Cfootnote{sensibus \textit{πM\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{z}}}} \edtext{reddentur}{\Cfootnote{redduntur \textit{V\textsuperscript{o}} | reddetur \textit{E\textsuperscript{a}P\textsuperscript{y}}}} consueta, vt \edtext{ostendunt}{\Cfootnote{ostendit \textit{P\textsuperscript{x}V\textsuperscript{b}}}} \edtext{eris}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}}}} et nauium malleatores non audiendo, quod in illis \edtext{non}{\Cfootnote{non vt \textit{β}}} \edtext{acquiratur}{\Cfootnote{acquiratur etiam \textit{α}}} consuetudo \edtext{vtcunque}{\Cfootnote{vtrumque \textit{λP\textsuperscript{y}}}}.
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\subsection[Vigesimumoctauum problema … ]{Vigesimumoctauum problema}\label{part28}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID LEGES APPELLANTUR QUAS CANTANT? \textsuperscript{2}AUT QUIA PRIUSQUAM SCIANT LITTERAS, VIDERUNT LEGES NON LATEANT, QUEMADMODUM IN AGATHUSIS AMPLIUS CONSUEUERUNT? ET POSTERIORIBUS IGITUR CANTIBUS PRIMOS IDEM APPELLAUERUNT QUIDAM PRIMAS}{\lemma{PROPTER \dots\ PRIMAS}\Cfootnote{PROPTER QUID SECUNDUM LEGES \textit{V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} illi cantus quos primitus \edtext{aliqui cantant}{\Cfootnote{cantant aliqui \textit{γ}}} leges appellantur.} Nosce quod \edtext{per}{\Cfootnote{\textit{om}. \textit{π}}} leges hic audio cantus in quibus \edtext{mox aliqui}{\Cfootnote{aliqui mox \textit{V\textsuperscript{o}} | statim aliqui \textit{γ}}} edocentur cum incipiunt cognitionem habere, sicut illi qui ordinati sunt a lege vt cantentur legaliter. \edtext{Tamen}{\Cfootnote{Cum \textit{G\textsuperscript{u}P\textsuperscript{p}}}}, \edtext{vt visum est .15.}{\Cfootnote{vt visum .15. \textit{βE\textsuperscript{a}} | \textit{om}. \textit{π}}}, in laudem dei \edtext{ceu in lege nostra}{\Cfootnote{seu in lege nostra \textit{E\textsuperscript{a}V\textsuperscript{m}} | \textit{om}. \textit{π}}} docentur pueruli salutationem virginis \edtext{vel}{\Cfootnote{aut \textit{V\textsuperscript{o}}}} \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}V\textsuperscript{z}}}} cantare.
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}}  Soluit dicens causam esse \textit{quia cantantes leges \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}V\textsuperscript{a}}}} puerili antequam aliquid sciuissent de litteris et cantibus choricis, iam viderant leges seu audiuerant \edtext{scientes etiam}{\Cfootnote{scientes et \textit{E\textsuperscript{a}V\textsuperscript{a}}}} ipsas cantare ad hoc vt non lateant}. Sed reddantur manifeste \edtext{note}{\Cfootnote{voce \textit{κ}}}, velut ``principia in scientia et locus ianue in domo''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Met.} II.1, 993b.4–5}} \edtext{vt ostensum est .18\textsuperscript{a}}{\Cfootnote{vt ostensum .18\textsuperscript{a} \textit{β} | \textit{om}. \textit{π}}}. Nam quod primitus discitur optime imprimitur cum anima sit tunc \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{P\textsuperscript{p}V\textsuperscript{b}} | tamquam \textit{G\textsuperscript{u}P\textsuperscript{y}}}} tabula rasa, et \edtext{perfecte}{\Cfootnote{bene \textit{π}}} retinetur, et \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}}}} quidem consuetudo potissime fuit \textit{apud quosdam nominatos agathyrsos.} Quod autem \edtext{primum quis}{\Cfootnote{primum \textit{κ}}} adipiscitur prius nomen suscipit, tanquam \edtext{notum}{\Cfootnote{motum \textit{P\textsuperscript{y}V\textsuperscript{z}}}} cum anima nomen \edtext{imponatur}{\Cfootnote{imponat vt \textit{απ}}} cognoscit. \textit{Itaque cum \edtext{alii}{\Cfootnote{aliis \textit{V\textsuperscript{b}mv}}}, cantus \edtext{vt}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{chorici}{\Cfootnote{theorici \textit{V\textsuperscript{o}}}} siue conuersiui et vniuersaliter tragici, \edtext{sint}{\Cfootnote{sicut \textit{γP\textsuperscript{y}V\textsuperscript{m}}}} posterius cantibus legalibus cantati, puta cum \edtext{sciuerint}{\Cfootnote{sciuerunt \textit{E\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{o}}}} litteras et \edtext{fuerint}{\Cfootnote{fuerit \textit{E\textsuperscript{a}V\textsuperscript{a}}}} adulti leges, primos cantus seu primas melodias quod idem est \edtext{appellarunt}{\Cfootnote{appellauerunt \textit{M\textsuperscript{a}V\textsuperscript{o}V\textsuperscript{z}}}},} sicut scientie de sermone cum prius \edtext{instruantur in hiis}{\Cfootnote{instruantur in eis \textit{βE\textsuperscript{a}} | in eis instruantur \textit{π}}} pueri dicuntur primitiue.
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\subsection[Vigesimumnonum problema … ]{Vigesimumnonum problema}\label{part29}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID RITHIMI ET CANTUS, VOX EXISTENS, CONSUETUDINIBUS ASSIMILANTUR, SAPORES AUTEM NON, SED NEQUE COLORES ET ODORES? \textsuperscript{2}AUT QUIA MOTUS SUNT QUEMADMODUM ET OPERATIONES? IAM VTIQUE ACTIO QUIDEM MORALE ET FACIT CONSUETUDINEM, SAPORES AUTEM ET COLORES NON FACIUNT SIMILITER}{\lemma{PROPTER \dots\ SIMILITER}\Cfootnote{PROPTER QUID RITHIMI \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} Repetit \edtext{.27. problema dicens}{\Cfootnote{.27\textsuperscript{m}. dicens \textit{β} | .27\textsuperscript{m}. particulariter dicens \textit{γE\textsuperscript{a}} | problema \textit{π}}} \textit{quare rithimi, quantum ad speciem musice rithimicam, et cantus et vox \edtext{pronunciatiua}{\Cfootnote{pronunciatam \textit{P\textsuperscript{x}V\textsuperscript{z}}}} quantum ad armonicam \edtext{similitudinem}{\Cfootnote{similem \textit{β} | \textit{om}. \textit{π}}} acquirunt \edtext{consuetudinis}{\Cfootnote{consuetudinem \textit{π}}}? Redduntur enim consueti, sed sapores aut \edtext{color vel odores}{\Cfootnote{colores vel odores \textit{βE\textsuperscript{a}V\textsuperscript{b}} | odores vel colores \textit{πV\textsuperscript{a}}}} non proportionantur consuetudini.}
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quia motus qui sunt circa sensibilia \edtext{auditus}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{m}}}} tanquam operationes existunt}. Quod declarat quia sicut \edtext{videmus}{\Cfootnote{nos videmus \textit{π}}} \textit{in actione morali quod actus eius inducit consuetudinem et inclinationem ad istud vel \edtext{ad}{\Cfootnote{\textit{om}. \textit{π}}} illud agendum,} ita similiter est in \edtext{sensibus}{\Cfootnote{sensibilibus \textit{γE\textsuperscript{a}M\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{m}}}} auditus. Ex multotiens \edtext{enim audire}{\Cfootnote{quidem audire \textit{V\textsuperscript{o}}}} \edtext{vnum}{\Cfootnote{in vnum \textit{P\textsuperscript{x}V\textsuperscript{m}} | nimium \textit{M\textsuperscript{a}V\textsuperscript{z}}}} cantum inducitur consuetudo, \edtext{qua}{\Cfootnote{que \textit{γλN\textsuperscript{u}}}} habituata cantus ille \edtext{reputatur}{\Cfootnote{redditur \textit{π}}} delectabilis, \edtext{etiam}{\Cfootnote{et \textit{M\textsuperscript{a}P\textsuperscript{z}V\textsuperscript{a}}}} si non sit secundum veritatem, non autem circa sensibilia aliorum sensuum, vt \edtext{declaratum}{\Cfootnote{declaratum est \textit{γπ}}}, accidit. Et hoc est quod subditur sapores qui sunt sensibilia; \textit{gustus et odores qui olfatus non id operantur,} quod ex ipsis consuetudo \edtext{consimilibus}{\Cfootnote{similiter \textit{β} | consimiliter \textit{γE\textsuperscript{a}V\textsuperscript{o}}}} audibilibus acquiratur.
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\subsection{Trigesimum problema}\label{part30}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID NEQUE YPODORISTIUM NEQUE YPOFRIGISTIUM NON EST IN TRAGEDIA ROTATIUUM? \textsuperscript{2}AUT NON HABET CONUERSIUUM? SED A SCENA, IMMUTATIUA ENIM}{\lemma{PROPTER \dots\ ENIM}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} ypodoristium et ypophirgium cantare non \edtext{est}{\Cfootnote{\textit{om}. \textit{γ}}} rotatiuum et conuersiuum in tragedia}, ubi \edtext{obseruentur}{\Cfootnote{obseruantur \textit{π}}} diuersitates sonorum secundum graue et acutum ascensione \edtext{et}{\Cfootnote{ac \textit{γN\textsuperscript{u}}}} descensione per circulos casuum.
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quia \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}}}} cantare non \edtext{habent}{\Cfootnote{habet \textit{πV\textsuperscript{m}}}}, vt expositum \edtext{est}{\Cfootnote{\textit{om}. \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}}, conuersiuum} in quo magis attenditur ad cantus quam ad verba. \textit{Sed potius \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} cantus est \edtext{immutatiuus}{\Cfootnote{imitatiuus \textit{M\textsuperscript{a}V\textsuperscript{o}}}}, \edtext{qui}{\Cfootnote{quod \textit{π} | quam \textit{γP\textsuperscript{x}}}} dicitur ascena} contentus sub legibus et comedia, vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{βV\textsuperscript{b}V\textsuperscript{o}}}} .15\textsuperscript{to}., in quo amplius attenditur ad verba quam ad \edtext{cantum}{\Cfootnote{cantus \textit{M\textsuperscript{a}V\textsuperscript{z}}}}, et id amplius apparebit 48°.
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\textbf{4} Nosce vt apparet 8°. \textit{politicorum} aristotelem tangere tres \edtext{cantus modos}{\Cfootnote{modos cantus \textit{λπ} | cantus mentis \textit{γ}}}: precipue phrigistium, \edtext{doricum}{\Cfootnote{doricum vel doristicum \textit{π}}}, et lidium. \edtext{Phrigistius}{\Cfootnote{Frigisticus vel phigistius \textit{π}}} est troianus, \edtext{qui}{\Cfootnote{que \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} est cantus \edtext{agrestis}{\Cfootnote{agrestis et \textit{γ}}} incitatiuus in iram et passionem, \edtext{ceu}{\Cfootnote{seu \textit{πE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}}}} qui fistule \edtext{in quam}{\Cfootnote{in quem \textit{γE\textsuperscript{a}M\textsuperscript{a}}}} concidit decenter \edtext{ditirantibus}{\Cfootnote{ex tyranbus \textit{πE\textsuperscript{a}}}}. \edtext{Dorista}{\Cfootnote{Doristas \textit{γ}}} siquidem maluaficiacus cantus laudabilis ``tanquam \edtext{stabilis}{\Cfootnote{stabilis et \textit{πE\textsuperscript{a}}}} \edtext{virilis}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{b}}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII, 1324b12}} \edtext{vnde .48.}{\Cfootnote[nosep]{\textit{emend.} | vnde .40. \textit{βV\textsuperscript{a}} | vnde .41. \textit{π} | \textit{om}. \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} ``\edtext{magnificus stabilis}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{b}}}} cytharicissimus omnium''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.-Arist. \textit{Prob.} XIX.48, 922b15}} ``\edtext{et}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{ceu}{\Cfootnote{seu \textit{V\textsuperscript{o}}}} omnium superhabundantiarum \edtext{mediis}{\Cfootnote{medius \textit{βM\textsuperscript{a}}}} laudatus in quo pueruli sunt edocendi.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII, 1324b12–18}} Lidius quidem dicitur a \hyperref[Tor]{chorebo} lidorum rege. Qui omnibus laudabilior \edtext{est}{\Cfootnote{\textit{om}. \textit{απ}}}. ``\edtext{Cum}{\Cfootnote{Cum enim \textit{απ}}} sit aliorum \edtext{medius}{\Cfootnote{mediis \textit{V\textsuperscript{b}mv}}}, tria in se continet quod medium \edtext{rationem habens}{\Cfootnote{commune habens \textit{πE\textsuperscript{a}}}} virtutis quod decens \edtext{et}{\Cfootnote{et quod \textit{απ}}} possibile,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Pol.} VIII, 1324b32–34}} que in cantu et maxime pverorum sunt obseruanda. Vnde \edtext{boecius \textit{musice} .4.}{\Cfootnote{boecius quarto \textit{musice} \textit{π}}} ``si quis totas constitutiones faciat acutiores vel \edtext{graues}{\Cfootnote{grauiores \textit{π}}} totas remittat dyapason consonantie species, efficiet modos .7. \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} sunt ypodorius ypofrigius ypolidius dorius frigius lidius mixolidius''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} IV.15 ((Friedlein p. 342, ll. 10–14))}} \edtext{ypermixolidius}{\Cfootnote{\textit{om}. \textit{π}}}. Duplat \edtext{igitur}{\Cfootnote{enim \textit{E\textsuperscript{a}V\textsuperscript{o}}}} vel triplat quod \hyperref[Ar]{aristoteles} \hyperref[Bo]{boecius}.
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\subsection[Trigesimumprimum problema … ]{Trigesimumprimum problema}\label{part31}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID CIRCA PHRINICUM FUERUNT MAGIS DULCE MELODIA? \textsuperscript{2}AUT PROPTER ID QUOD MULTIPLICES SUNT TUNC CANTUS IN TRAGEDIIS METRORUM}{\lemma{PROPTER \dots\ METRORUM}\Cfootnote{PROPTER QUID CIRCA FRINICHUM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}?}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} est quod \edtext{apud}{\Cfootnote{circa \textit{G\textsuperscript{u}P\textsuperscript{p}}}} phrinicum, \edtext{qui}{\Cfootnote{quod \textit{πP\textsuperscript{x}}}} est modus \edtext{cantantium}{\Cfootnote{cantantium vel cantuum \textit{V\textsuperscript{o}}}} quorundam, melodie fuerunt magis suaues quam apud \edtext{alios}{\Cfootnote{alias \textit{V\textsuperscript{o}}}}?}
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quoniam cantus apud eum fuerunt multiplices et differenter compositi propter quod secundum plurimum fuerunt in tragediis metrorum.} Tales autem sunt suaues maxime, cum eorum sit principaliter \edtext{finis}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} iocunditatem inducere.
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\subsection[Trigesimumsecundum problema … ]{Trigesimumsecundum problema}\label{part32}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DYAPASON APPELLATUR, SED NON SECUNDUM NUMERUM DYAOCTO, QUEMADMODUM ET DYATESSERON ET DYAPENTE? \textsuperscript{2}AUT QUIA .VII. FUERUNT CORDE ANTIQUITUS? POSTEA EXTRAHENS TERTIAM, \hyperref[Ter]{TERPANDERUS} NITEM APPOSUIT. ET IN HOC APPELLATA EST. DYAPASON, SED NON DYAOCTO, DYASEPTEM ENIM ERAT}{\lemma{PROPTER \dots\ ERAT}\Cfootnote{PROPTER QUID DYAPASON \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare consonantia}{\Cfootnote{Querit quare consonantia \textit{λ} | \textit{om}. \textit{V\textsuperscript{o}}}} dupla dyapason siue per totum appellatur, et non dicitur dyocto propter numerum \edtext{octonarium}{\Cfootnote{octogenarium \textit{πE\textsuperscript{a}}}},} in quem videtur terminari \edtext{in}{\Cfootnote{\textit{om}. \textit{π}}} parigraphia proportionum? Terminatur enim super .8. Et subdit \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} motiuum dicens \textit{\edtext{quemadmodum}{\Cfootnote{quod deinde \textit{V\textsuperscript{o}}}} dicitur dyatesseron,} quia finitur in quarto loco in predicta descriptione, aut quia fit in quatuor cordis musice. Quinto \textit{et similiter dyapente} \edtext{siue per 5.}{\Cfootnote{\textit{om}. \textit{π}}} terminatur in quinto loco vel in 5. constat cordis.
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse cur dyapason dicitur et non dyaocto \textit{quia corde \edtext{instrumentorum antiquitus 7. tantum fuere}{\Cfootnote{instrumentorum antiquitus 7 fuere \textit{π}}}. Postmodum superuenit \hyperref[Ter]{tripandius}, qui tertiam subtraxit cordam vt paramese cuius quidem loco nitem \edtext{apponit}{\Cfootnote{apposuit \textit{απM\textsuperscript{a}}}}}, ita vt non videatur mutasse numerum, sed proportionem potius, \textit{et propter hoc appellata est dyapason,} \edtext{quoniam}{\Cfootnote{quia \textit{λ}}} ex tali mutatione \edtext{optima}{\Cfootnote{opposita \textit{β}}} \edtext{adinuenta est}{\Cfootnote{est adinuenta \textit{λ} | inuenta est \textit{π} | adiuncta est \textit{V\textsuperscript{b}}}} consonantia. \textit{Non itaque dicitur dyaocto, cum corde non \edtext{sunt}{\Cfootnote{sint \textit{λE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}} nisi 7.} Potius enim \edtext{dicendum}{\Cfootnote{deberet \textit{γ} | debent \textit{V\textsuperscript{m}}}} \edtext{esset}{\Cfootnote{esse \textit{V\textsuperscript{b}V\textsuperscript{m}} | dici \textit{V\textsuperscript{a}mv}}} dyaseptem, cum tantum isto numero \edtext{imitando qui planetarum numerum vt visum est contineantur}{\Cfootnote{imitando qui planetarum vt visum est numero contineantur \textit{λ} | immutando qui planetarum numerum vt visum est contineantur \textit{γE\textsuperscript{a}} | contineantur imitando numerum planetarum vt visum est \textit{G\textsuperscript{u}P\textsuperscript{z}} | contineantur imitando naturam planetarum vt visum est \textit{P\textsuperscript{p}V\textsuperscript{o}}}}. Quia \edtext{tamen}{\Cfootnote{cum \textit{γM\textsuperscript{a}V\textsuperscript{z}}}} \edtext{corde prius .7.}{\Cfootnote{prius corde .7. \textit{πE\textsuperscript{a}}}} fuerunt vt \edtext{apparuit}{\Cfootnote{apparebit \textit{πE\textsuperscript{a}}}}, pauciores pluresque \edtext{demum}{\Cfootnote{deinde \textit{V\textsuperscript{o}}}} adinuente, tamquam nomen stabilius, elegerunt potius dyapason dicere seu per totum et perfectum quam dyaseptem, tanquam nomen magis \edtext{denotatiuum}{\Cfootnote{denominatiuum \textit{V\textsuperscript{o}} | determinatum \textit{G\textsuperscript{u}P\textsuperscript{p}}}} essentie ipsius. Et tactum quidem \edtext{.25. problema}{\Cfootnote{25° problemate \textit{P\textsuperscript{y}V\textsuperscript{b}} | est hoc prius \textit{π}}}.
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\subsection[Trigesimumtertium problema … ]{Trigesimumtertium problema}\label{part33}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID CONSONANTIUS EST IN GRAUE QUAM A GRAUI IN ACUTUM? \textsuperscript{2}UTRUM QUIA A PRINCIPIO FIT INCIPERE? \textsuperscript{3}MEDIA ENIM ET PRINCIPALIS ACUTISSIMA PARACORDE. HOC AUTEM NON A PRINCIPIO, SED A FINE. \textsuperscript{4}AUT QUIA GRAUE AB ACUTO FORTIUS ET MAGIS VOCALE}{\lemma{PROPTER \dots\ VOCALE}\Cfootnote{PROPTER QUID CONSONANTIUS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}?}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} in cantu maior extat consonantia si incipiatur ab acuto et \edtext{terminetur}{\Cfootnote{terminatur \textit{P\textsuperscript{x}V\textsuperscript{o}}}} in \edtext{grauem}{\Cfootnote{graue \textit{α}}}, quam si a graui \edtext{preincipiens}{\Cfootnote{incipiens \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}} terminetur in acutum?}
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\textbf{3} \edtext{\textit{UTRUM.}}{\Cfootnote{Deinde. VTRUM. \textit{V\textsuperscript{a}mv}}} Soluit et assignat duplicem causam dicens primo id esse \textit{quia contingit incipere \edtext{a}{\Cfootnote{in \textit{πE\textsuperscript{a}}}} principio rei in qualibet re}, modo ipsum acutum \edtext{tenet}{\Cfootnote{retinet \textit{γ}}} locum principii in instrumento. Nam prime corde minores sunt ille que \edtext{sonum}{\Cfootnote{\textit{om}. \textit{λ}}} reddunt acutum, a quibus oportet incipere, vt ordo seruetur naturalis in arte.
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\pstart
\textbf{4} \edtext{\textit{MEDIA.}}{\Cfootnote{Deinde. MEDIA. \textit{V\textsuperscript{a}mv}}} Remouet dubitationem, quia crederet aliquis \edtext{quod}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} ab omni corda acuta \edtext{esset}{\Cfootnote{esse \textit{V\textsuperscript{o}}}} incipiendum. \edtext{Vnde}{\Cfootnote{\textit{om}. \textit{π}}} id \edtext{remouens}{\Cfootnote{remouet \textit{γπP\textsuperscript{y}}}} dicens quod \textit{corda media \edtext{et}{\Cfootnote{\textit{om}. \textit{λ}}} principalis est acutissima seu fortissima \edtext{peracorde}{\Cfootnote{peracordis \textit{πE\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{z}} | per .7. cordas \textit{γ}}}, id est \edtext{cordis}{\Cfootnote{corde \textit{α}}}, iuxta ipsam \edtext{existentis}{\Cfootnote{existentibus \textit{πM\textsuperscript{a}}}}. Vnde \edtext{hec}{\Cfootnote{hoc \textit{E\textsuperscript{a}P\textsuperscript{z}}}} non est \edtext{tangenda in principio}{\Cfootnote{in principio tangenda \textit{π}}}, sed potius in fine,} propter eius excessum, \edtext{ista}{\Cfootnote{ita \textit{E\textsuperscript{a}P\textsuperscript{x}} | ita quod \textit{π}}} siquidem aut est media positione recte \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{z}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} mese, aut virtute puta ypate, cum in mediam \edtext{condescendit}{\Cfootnote{condescendat \textit{π}}} cum nete, vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{β}}} 20.
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\pstart
\textbf{5} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Assignat aliam causam dicens id \edtext{etiam}{\Cfootnote{\textit{om}. \textit{γV\textsuperscript{o}V\textsuperscript{z}}}} esse \textit{quoniam graue \edtext{incipiens}{\Cfootnote{descendens \textit{π}}} ab acuto fortius est. Diutius enim permanet et magis vocale existit quam econuerso.} Hoc autem facere magis est \edtext{connaturale}{\Cfootnote{contra naturale \textit{βV\textsuperscript{z}}}} \edtext{ac}{\Cfootnote{hac \textit{V\textsuperscript{o}}}} etiam morale \edtext{quam}{\Cfootnote{quidem \textit{V\textsuperscript{o}}}} suum contrarium. ``Videmus enim in operibus nature quod ipsa in principio a leuibus et debilibus incipit, non enim tunc est, operosa sed in fine,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} X.45, 895b27-896a4}} \edtext{vt dictum est 10\textsuperscript{a}.}{\Cfootnote{\textit{om}. \textit{π}}} Et ideo est videre optimos cantatores naturam \edtext{imitantes}{\Cfootnote{mutantes \textit{P\textsuperscript{x}V\textsuperscript{m}}}} hoc \edtext{seruare}{\Cfootnote{obseruare \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}}, quod in acuto leuiter incipientes in \edtext{grauem}{\Cfootnote{graue \textit{βπV\textsuperscript{a}}}} postea conterminant, et non econtra concordant.
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\subsection[Trigesimumquartum problema … ]{Trigesimumquartum problema}\label{part34}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID BIS QUIDEM PER ACUTOS. ET BIS DYATESSERON NON CONSONAT, BIS DYAPASON AUTEM? \textsuperscript{2}AUT QUIA BIS PER ACUTOS NEQUE BIS DYATESSERON EST, QUOD AUTEM PER QUATUOR AUT PER QUINQUE}{\lemma{PROPTER \dots\ QUINQUE}\Cfootnote{PROPTER QUID BIS QUIDEM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}?}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} \edtext{bis paracutos seu bis paracuta}{\Cfootnote{bis paracuta seu bis paracutos \textit{V\textsuperscript{o}}}} id est bis dyapente et bis dyatesseron siue bis per quatuor id est dyatesseron non consonant et simphoniam constituunt, dyapason vero concordat?} Dicitur autem bis paracutos dyapente, quia constat ex duobus acutis et vno graui seu breuibus et longo, more dactili.
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse \textit{quoniam bis paracutos vel bis dyapente \edtext{vel}{\Cfootnote{et \textit{γλE\textsuperscript{a}N\textsuperscript{u}}}} \edtext{bis dyatesseron}{\Cfootnote{dyatesseron \textit{πP\textsuperscript{x}}}} non est dare, quod constituant aliquam consonantiam, quoniam illud quod per quatuor se habet aut per .5. licet proportionem constituant et per consequens consonantiam, vt per quatuor dyatesseron \edtext{vel}{\Cfootnote{vel per \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{x}}}} 5. dyapente.} Quando \edtext{enim}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{minus extremum prime, et}{\Cfootnote{extremum prime minus \textit{πE\textsuperscript{a}}}} extremum \edtext{magis}{\Cfootnote{maius \textit{απM\textsuperscript{a}}}} secunde comparantur ad inuicem nulla consonantiarum resultat inde, vt videbitur .41., ubi repetitur idem amplioris expositionis gratia.
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\subsection[Trigesimumquintum problema … ]{Trigesimumquintum problema}\label{part35}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DYAPASON OPTIMA CONSONANTIA EST? \textsuperscript{2}AUT QUIA IN OMNIBUS TERMINIS RATIONES SUNT, ALIORUM AUTEM NON IN OMNIBUS? QUONIAM AUTEM DUPLEX NITE YPATES. QUOT NITE DUE YPATEM VNUM, ET QUALIS YPATE DUO NITE QUATUOR, ET SEMPER ITA. \textsuperscript{3}MEDIE AUTEM EMIOLIA. QUOD ENIM PER QUINQUE EMIOLIUM NON IN OMNIBUS NUMERIS EST. NON ENIM VNUM MINUS, MAIUS TANTUM AUTEM ET ADHUC MEDIUM. QUARE NON TOTA AD TOTA COMPARANTUR, SED INSUNT PARTES. \textsuperscript{4}SIMILITER ET IN EO QUOD EST PER IIII. SE HABET, EPITRICON ENIM EST QUANTUM INCIDERE QUOD ET ADHUC IN .IIII. EPITRICON EST.  \textsuperscript{5}AUT PERFECTISSIMA QUE EST EX VTRISQUE, ET QUIA MENSURA MELODIE? PER TOTUM ID QUOD FERTUR QUE EST SECUNDUM MEDIUM MOTUS FORTISSIMUS, INCIPIENTE AUTEM ET FINIENTE MOLLIOR. QUANDO AUTEM FORTISSIMUS MOTUS, ET VOX ACUTIOR EIUS QUΟD FERTUR, PROPTER QUOD ET CORDE TENSE ACUTIUS SONANT, VELOCIOR ENIM MOTUS FIT. \textsuperscript{6}SONUS AUTEM AUT AERIS AUT ALIUS CUIUSDAM MOTUS, CAUSA QUE EST PER MEDIUM PORUM ACUTISSIMUM OPORTET FIERI. QUARE SI HOC NON ACCIDIT, NON ERIT MOTUS ALICUIUS}{\lemma{PROPTER \dots\ ALICUIUS}\Cfootnote{PROPTER QUID DYAPASON \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} dyapason est optima et dulcissima consonantiarum?}
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\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et assignat duplicem causam, \edtext{\edtext{secunda}{\Cfootnote{secunda ibi \textit{γV\textsuperscript{z}}}} \textit{AUT.} \edtext{Adhuc}{\Cfootnote{Ad hoc \textit{N\textsuperscript{u}P\textsuperscript{x}}}} primo dat causam}{\lemma{secunda \dots\ causam}\Cfootnote{\textit{om}. \textit{π}}} \edtext{quare}{\Cfootnote{quare est \textit{γ}}} optima consonantiarum, dicens \edtext{illud}{\Cfootnote{id \textit{βλE\textsuperscript{a}V\textsuperscript{o}}}} esse \textit{quoniam proportiones secundum quas constituitur \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}}}} consonantia \edtext{sumuntur}{\Cfootnote{sumitur \textit{V\textsuperscript{m}V\textsuperscript{z}}}} ex omnibus terminis summis numerorum.} Multi enim sunt modi quibus corde \edtext{ad inuicem comparate}{\Cfootnote{comparate ad inuicem \textit{π}}} hanc efficiunt consonantiam. Quod declarat tam ratione substantie quam \edtext{ex parte}{\Cfootnote{\textit{om}. \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{x}}}} \edtext{quantitatis et qualitatis}{\Cfootnote{qualitatis et quantitatis \textit{V\textsuperscript{a}V\textsuperscript{z}}}}, dicens primo quod \textit{duplex nite, cum accepta fuerit, assumit rationem et substantiam ypates. Et similiter \edtext{ratione}{\Cfootnote{rationem \textit{β}}}} \edtext{quantitatis}{\Cfootnote{quantitatum \textit{β}}}, vt \edtext{querenti}{\Cfootnote{querente \textit{β} | conuenienti \textit{π}}} \edtext{quot}{\Cfootnote{quod \textit{P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{b}}}} nite constituunt ypatem vnam dicendum, \edtext{quoniam}{\Cfootnote{quod \textit{πM\textsuperscript{a}V\textsuperscript{a}}}} due, et secundum \textit{quale etiam due \edtext{nite}{\Cfootnote{vite nete \textit{V\textsuperscript{o}}}} sunt, \edtext{qualis}{\Cfootnote{quales \textit{P\textsuperscript{x}V\textsuperscript{a}}}} est ypate vna, et quatuor nete eodem modo sunt \edtext{due ypate}{\Cfootnote{due ypates \textit{γM\textsuperscript{a}V\textsuperscript{z}}}},} et sex .3. Et ratione substantie \edtext{ac}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}V\textsuperscript{o}V\textsuperscript{z}}}} accidentis \edtext{considerando}{\Cfootnote{consideratio \textit{π}}} \textit{et ita semper} procedendo due nite vnam constituunt ypatem.
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\pstart
\textbf{4} \edtext{\textit{MEDIE.}}{\Cfootnote{Deinde. MEDIE. \textit{V\textsuperscript{a}mv}}} Ostendit quod dyapente et dyatesseron non est optima \edtext{consonantia}{\Cfootnote{consonantiarum \textit{βγπ}}}, primo dicens quod \textit{proportio medietatis est emiola. Et exponit, dicens quod \edtext{illud quod}{\Cfootnote{id quod \textit{λ} | illud \textit{πE\textsuperscript{a}P\textsuperscript{x}}}} est per quinque est emiolum, et tale non reperitur in omnibus numeris;} ymmo in nullis ita vt dicatur omnis et perfecta, \edtext{vt dyapason dicitur perfecta}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}, consonantia. \textit{Non enim perfecte \edtext{totum}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{m}}}} comprehendit numerus maior minorem;} ymmo duorum \edtext{numerorum}{\Cfootnote{minorum \textit{P\textsuperscript{y}V\textsuperscript{a}}}} vnus est sicut \edtext{quid}{\Cfootnote{qui \textit{λV\textsuperscript{b}}}} maius et \edtext{aliud}{\Cfootnote{alius \textit{πV\textsuperscript{z}}}} minus, eo quod talis consonantia consistit in proportione sexquialtera, in qua vnum continet \edtext{alium}{\Cfootnote{aliud \textit{P\textsuperscript{y}V\textsuperscript{z}}}} totum semel \textit{tantum et eius medietatem. Propter quod in hac consonantia non \edtext{fit}{\Cfootnote{sit \textit{E\textsuperscript{a}V\textsuperscript{o}}}} comparatio totius ad totum,} vt quod \edtext{vnus numerus}{\Cfootnote{vnius numeri \textit{β}}} \edtext{bis quod}{\Cfootnote{bisque \textit{V\textsuperscript{o}}}} \edtext{totum}{\Cfootnote{totus \textit{V\textsuperscript{o}}}} mensuret \edtext{alterius}{\Cfootnote{alterum \textit{G\textsuperscript{u}P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{o}}}}, \textit{sed fit comparatio totius ad partes,} quare non erit optima consonantiarum dyapente.
\pend

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\textbf{5} \edtext{\textit{SIMILITER.}}{\Cfootnote{Deinde. SIMILITER. \textit{V\textsuperscript{a}mv}}} Ostendit consequenter quod dyatesseron non est optima consonantiarum, dicens \textit{quod in illa consonantia que est per quatuor vt dyatesseron eodem modo se habet.} Non enim \edtext{est}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{a}V\textsuperscript{m}}}} optima consonantiarum, cum non fiat comparatio totius ad totum, quoniam \textit{epitricam \edtext{proportionem}{\Cfootnote{proportio \textit{λ}}} esse \edtext{quoddam}{\Cfootnote{quod \textit{π}}} est, \edtext{quantum interuenire}{\Cfootnote{cantum inuenire \textit{π}}}, vt cum maior numerus continet minorem totum et amplius cum hoc \edtext{epitritio}{\Cfootnote{epitricon \textit{α} | epitricus \textit{π}}} est in quatuor,} id est continet super hoc eius tertiam partem, ex quo consonantia resultat quaterna. Et sic non fit comparatio totius ad totum, sed totius ad partem, vt dictum \edtext{est}{\Cfootnote{est in \textit{γE\textsuperscript{a}V\textsuperscript{o}} | in \textit{β}}} dyapente.
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\textbf{6} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} \edtext{Assignat}{\Cfootnote{Soluit et assignat \textit{λ}}} aliam causam, vel \edtext{quod}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}P\textsuperscript{y}}}} \edtext{dictum}{\Cfootnote{dictum est \textit{γM\textsuperscript{a}}}} confirmat per amplius, dicens primo \edtext{quod}{\Cfootnote{\textit{om}. \textit{γ}}} \textit{consonantia ex \edtext{hiis}{\Cfootnote{\textit{om}. \textit{π}}} vtrisque composita, \edtext{puta}{\Cfootnote{\textit{om}. \textit{κ}}} ex dyatesseron et \edtext{dyapente}{\Cfootnote{dyapasson \textit{β}}}, est perfectissima.} Ipsa enim vt dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}}}} 13. ex hiis duabus constat proportionibus, cum dyapason sit fere \edtext{sex}{\Cfootnote{sexta \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} tonorum simphonia. Et dyatesseron ex duobus tonis et semitonio \edtext{constet}{\Cfootnote{constat \textit{πV\textsuperscript{a}}}}, ac dyapente ex tribus et semitonio, que \edtext{coniuncta}{\Cfootnote{coniecta \textit{P\textsuperscript{x}V\textsuperscript{z}}}} sex tonos fere constituunt. Dico autem \textit{fere} eo quod semitonium vel limes aut dyesis minor est tono.
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\textbf{7} Quod autem sit perfectissima ostenditur, quoniam \textit{mensura extat et ratio omnium melodiarum est.} ``Namque simplicissima, vt ratio mensure \edtext{perquirit}{\Cfootnote{requirit \textit{π}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Met.} X, 1052b34–37}} \edtext{decimo apparens}{\Cfootnote{vt apparet in decimo \textit{π} | quarto apparens \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{z}}}} \textit{methaphysice}. Vnde secundo \textit{musice} boecius: ``\edtext{dyapason}{\Cfootnote{dyapente \textit{βγ}}} est maxima consonantiarum et simplicitate notissima.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} II.18 ((Freidlein 250, ll. 6–8))}} Quod autem sit mensura apparet quia \textit{motus illius consonantie que secundum medium \edtext{existit}{\Cfootnote{consistit \textit{λ}}} fortissimus est per totum \edtext{id per}{\Cfootnote{id \textit{β} | per \textit{π}}} quod fertur siue sit in principio siue in fine.} In medio namque \edtext{constat virtus}{\Cfootnote{virtus constat \textit{γλπE\textsuperscript{a}}}} totius, dyapason enim \edtext{ex}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{cordis}{\Cfootnote{corde \textit{V\textsuperscript{a}V\textsuperscript{m}}}} conficitur extremis et ad inuicem confractis et mediis, non autem alie consonantie. Et quamquam vt \edtext{dictum est}{\Cfootnote{predictum est \textit{π} | dictum \textit{E\textsuperscript{a}N\textsuperscript{u}}}} predicta consonantia sit perfectissima. \textit{Tamen quando motus \edtext{quo causatur}{\Cfootnote{cantus \textit{π}}} fuerit fortissimus et velocissimus, efficietur hec \edtext{dulcior}{\Cfootnote{delectatior \textit{γ}}} consonantia quam quando fuerit minus fortis et velox, eo quod tunc vox eius \edtext{quod}{\Cfootnote{que \textit{κ}}} mouetur \edtext{erit}{\Cfootnote{est \textit{κ}}} \edtext{acutior}{\Cfootnote{acutior et \textit{π}}} sonus, autem \edtext{is}{\Cfootnote{hiis \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{c}V\textsuperscript{z}}}} dulcior obtuso, propter quod corde tense velociter sonant, eo quod motus earum est velox cito pertransiens.} Cum enim corda fuerit tensa, \edtext{non}{\Cfootnote{\textit{om}. \textit{γ}}} potest \edtext{in tantum}{\Cfootnote{iterum \textit{λ}}} moueri et vagari a medio sicut si fuerit \edtext{relaxata}{\Cfootnote{laxata \textit{π}}}.
\pend

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\textbf{8} \edtext{\textit{SONUS.}}{\Cfootnote{Deinde. SONUS. \textit{V\textsuperscript{a}mv}}} \edtext{Et}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}} quia fecerat mentionem de sono ultimo, vnde causatur ostendit, dicens \textit{sonum esse aut motum aeris \edtext{in se}{\Cfootnote{\textit{om}. \textit{π}}} aut causa \edtext{cuiusdam}{\Cfootnote{eiusdem \textit{π}}} alterius quod in aere mouetur. \edtext{Cuius}{\Cfootnote{Cum \textit{V\textsuperscript{o}}}} autem fuerit causa motus qui per medietatem pori vel meatus existit tunc et acutissimum oportet \edtext{sonum fieri}{\Cfootnote{fieri sonum \textit{β}}} in poro.} ``\edtext{Sic}{\Cfootnote{Sicut \textit{κE\textsuperscript{a}P\textsuperscript{y}}}} enim aer constringitur et fortissimo motu \edtext{velocissime}{\Cfootnote{\textit{om}. \textit{κV\textsuperscript{a}}}} cogitur moueri,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Prob.} XI.15, 900b10–12}} \edtext{vt declaratum est}{\Cfootnote{vt declaratum \textit{βV\textsuperscript{a}} | \textit{om}. \textit{π}}} \edtext{xi\textsuperscript{a},}{\Cfootnote{\textit{om}. \textit{π}}} et sonum tunc reddit acutum. Et ideo instrumenta perforantur, que foramina si claudantur minime sonant. \textit{Quare \edtext{contingit}{\Cfootnote{concludit \textit{βγ}}} quod quando non \edtext{est}{\Cfootnote{erit \textit{βγ}}} porus per medium, vt \edtext{dictum}{\Cfootnote{dictum est \textit{γE\textsuperscript{a}}}}, existens, non causabitur maneries \edtext{alicuius soni}{\Cfootnote{aliqua soni vel alicuius \textit{π}}}.}
\pend

\pstart
\textbf{9} Sciendum ptolomeum in \textit{armonica} taliter declarare ``dyapason esse optimam et perfectissimam \edtext{consonantiarum}{\Cfootnote{consonantiam \textit{πP\textsuperscript{y}}}}, fere siquidem \edtext{vna}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{y}}}} existit \edtext{vocula}{\Cfootnote{notula \textit{πP\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{z}}}}, vnum quodammodo sonum componens, puta \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} si \edtext{denario}{\Cfootnote{denarius \textit{M\textsuperscript{a}V\textsuperscript{m}}}} aliquis numerorum addatur, vt duo, semper remanet \edtext{denarius}{\Cfootnote{binarius \textit{πE\textsuperscript{a}}}} integer illesus,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} V.10 ((Friedlein 360, ll. 6–10. Cf. Ptol. Harm. I.7, where Ptolemy actually says that the octave is a homophone, not a concord.))}} ``non enim ultra ipsum numerus quis eleuatur,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Prob.} XV.3, 910b23–30}} vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{βV\textsuperscript{a}}}} .15. Quod tamen non contingit in ceteris proportionibus, nam si tribus duo coniungantur statim fiunt .5. et mutatur species numeri. Sed \edtext{si}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}V\textsuperscript{o}}}} duo \edtext{addas}{\Cfootnote{addit \textit{P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{o}}}} \edtext{denario}{\Cfootnote{denarium \textit{V\textsuperscript{o}}}} duodecim \edtext{faciens}{\Cfootnote{facies \textit{M\textsuperscript{a}P\textsuperscript{y}}}}, \edtext{et}{\Cfootnote{\textit{om}. \textit{γ}}} \edtext{vtrique}{\Cfootnote{vterque \textit{M\textsuperscript{a}V\textsuperscript{z}}}} \edtext{numerus}{\Cfootnote{videtur \textit{πE\textsuperscript{a}}}} integer \edtext{conseruabitur}{\Cfootnote{conseruari \textit{πE\textsuperscript{a}}}}, et similiter si addas ternarium et ceteros eodem modo. \edtext{Similiterque}{\Cfootnote{Similiter quia \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{x}}}} dyapason simphonia \edtext{quancunque}{\Cfootnote{quantumcunque \textit{γ}}} consonantiam aliam sumat, seruat suam consonantiam neque \edtext{immutat}{\Cfootnote{imitatur \textit{V\textsuperscript{o}}}}, vt ex \edtext{consona}{\Cfootnote{consonantia \textit{γ}}} reddat se dissonam, quod in aliis non accidit simphoniis.
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\subsection[Trigesimumsextum problema … ]{Trigesimumsextum problema}\label{part36}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID SI QUIDEM MEDIA MOUEBITUR ET ALIE CORDE SONANT, SI AUTEM ITERUM HEC QUIDEM MANET ALIARUM AUTEM QUEDAM MOUEATUR MOTA SOLA SONAT? \textsuperscript{2}AUT QUIA CONCORDARE EST OMNIBUS HABERE SE AUTEM QUODAMMODO AD MEDIAM OMNIBUS, ET ORDINATIO QUE EST VNIUSCUIUSQUE VTIQUE PER ILLAM EST? ELEUATA IGITUR CAUSA CONCORDANDI ET CONTINENTIS, NON ADHUC SIMILITER APPARERE EXISTERE. VNA AUTEM DISCORDANTE EXISTENTE MEDIA AUTEM MANENTE, RATIONABILITER QUOD EST SECUNDUM IPSAM DEFICIENS, ALIIS ENIM EXISTIT CONCORDARE}{\lemma{PROPTER \dots\ CONCORDARE}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} si media \edtext{cordarum}{\Cfootnote{corda \textit{β}}} moueatur alie corde consonant illi, sed si postea fiat quod media immobilis permaneat et quedam de numero \edtext{aliarum a media}{\Cfootnote{aliarum \textit{V\textsuperscript{a}V\textsuperscript{o}} | aliarum armoniarum \textit{G\textsuperscript{u}P\textsuperscript{p}}}} \edtext{moueatur}{\Cfootnote{moueantur \textit{V\textsuperscript{o}}}} solum illa que fuerit mota sonat,} \edtext{ita quod}{\Cfootnote{itaque \textit{V\textsuperscript{o}}}} neque alia sonat propter motum ipsius, neque etiam media.
\pend

\pstart
\textbf{3} \edtext{Nosce \edtext{quod}{\Cfootnote{quod de \textit{γ}}} \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} \edtext{problemate}{\Cfootnote{problematis \textit{M\textsuperscript{a}N\textsuperscript{u}V\textsuperscript{z}}}} iam sermo aliqualiter precessit in 20\textsuperscript{a}. problemate}{\lemma{Nosce \dots\ problemate}\Cfootnote{\textit{om}. \textit{π}}}.
\pend

\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse quoniam media cordarum concordat et consonat cum omnibus cordis. Quod ostenditur \textit{quoniam concordare cum omnibus nichil aliud est quam se habere aliqualiter ad mediam \edtext{omnis}{\Cfootnote{omnes \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{b}}}}, ordinatio enim vniuscuiusque aliarum cordarum per illam existit. Quod probatur quoniam \edtext{ipsa}{\Cfootnote{illa \textit{V\textsuperscript{o}}}} media remota ex instrumento, non amplius erit sentire cordam aliquam que existat ratio concordandi alias concorditerque retinendi. Sed vna aliarum \edtext{remota}{\Cfootnote{remoueatur \textit{π}}} ita quod causa existat discordandi dummodo media cordarum \edtext{recte maneat}{\Cfootnote{remaneat \textit{G\textsuperscript{u}P\textsuperscript{p}} | remoueat \textit{V\textsuperscript{o}}}}, sicut debet, illud quod erat deficiens, vt corda ablata, suppletur per mediam, ita quod illa remota adhuc aliis cordis inerit concordare et consonantiam reddere per virtutem medie.} Media enim est vt magistra et regula \edtext{aliarum omnium}{\Cfootnote{omnium aliarum \textit{λ} | omnium ceterarum \textit{P\textsuperscript{p}P\textsuperscript{u}} | omnium ceterorum \textit{V\textsuperscript{o}}}}.
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\subsection[Trigesimumseptimum problema … ]{Trigesimumseptimum problema}\label{part37}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID QUOD EST IN VOCE ACUTO EXISTENTE MODICUM, GRAUI AUTEM SECUNDUM MULTUM GRAUE? QUIDEM ENIM PROPTER MULTITUDINEM GRAUE, ACUTUM AUTEM PROPTER PAUCITATEM VELOX EST. \textsuperscript{2}MOLESTUM MAGIS CANTARE ACUTA QUAM GRAUIA, ET PAUCI ASCENDENTIA POSSUNT CANTARE, ET LEGES DIRIGUNT, ET ACUTI DIFFICILES DIMITTERE PROPTER ID QUOD TENSI SUNT. \textsuperscript{3}MINUS OPUS MODICUM MOUERE QUAM MULTUM, QUARE ET AEREM. \textsuperscript{4}AUT IDEM ACUTE SONANS ESSE NATURA ET ACUTE CANERE, SED NATURA QUIDEM ACUTE VOCIS OMNIA PROPTER IMBECILLITATEM PRO EO QUOD NON POTEST MOUERE MULTUM AEREM, SED MODICUM. MODICUS AUTEM VELOCITER FERTUR. IN CANTANDO AUTEM ACUTUM POTENTIE SIGNUM, QUOD QUIDEM ENIM VEHEMENTER FERTUR VELOCITER, PROPTER QUOD ET ACUTUM POTENTIE SIGNUM. PROPTER QUOD ET QUI SUNT BONI HABITUS ACUTE VOCIS, ET MOLESTUM QUE ELEUANTUR CANTARE, GRAUIA AUTEM DEORSUM}{\lemma{PROPTER \dots\ DEORSUM}\Cfootnote{PROPTER QUID QUOD EST IN VOCE \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \edtext{Querit}{\Cfootnote{Quare \textit{G\textsuperscript{u}P\textsuperscript{z}}}} quesitum cuius terminorum primitus tangit expositionem, dicens quod \textit{cum in voce acutum existat secundum modicum temporis, graue autem secundum multum. Nam graue causatur propter multitudinem aeris moti qui tarde mouetur, acutum vero propter paucitatem ipsius, ex \edtext{qua}{\Cfootnote{quo \textit{βπE\textsuperscript{a}}}} velocitas causatur,} ``sicut ex multitudine tarditas,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XI.16, 900b24–28}} vt visum \edtext{est .xi\textsuperscript{a}}{\Cfootnote{.xi\textsuperscript{a} \textit{β} | est 11\textsuperscript{a} particula. \textit{γ} | \textit{om}. \textit{π}}} 
\pend

\pstart
\textbf{3} \edtext{\textit{MOLESTUM.}}{\Cfootnote{Deinde. MOLESTUM. \textit{V\textsuperscript{a}mv}}} \edtext{Secundo}{\Cfootnote{\textit{om}. \textit{π}}} format quesitum, dicens \textit{quare magis \edtext{difficile}{\Cfootnote{difficile est \textit{λπ}}} cantare \edtext{illa}{\Cfootnote{ea \textit{πP\textsuperscript{x}}}} que sunt acuta quam \edtext{ea}{\Cfootnote{illa \textit{P\textsuperscript{y}V\textsuperscript{a}}}} que sunt grauia. Quod ostenditur \edtext{quoniam}{\Cfootnote{quia \textit{λV\textsuperscript{a}}}} pauci sunt qui \edtext{possunt }{\Cfootnote{possint \textit{N\textsuperscript{u}V\textsuperscript{b}V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{z}}}} cantus ascendentes cantare maxime, cum sint acutissimi. Et ideo \edtext{huius}{\Cfootnote{huiusmodi \textit{βV\textsuperscript{b}V\textsuperscript{z}}}} non cantant, sed potius dirigunt leges}, legales enim cantus constituunt. \textit{Sunt enim tanquam graues remissi faciles ad cantandum,} vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{β}}} \edtext{.15}{\Cfootnote{in 15 \textit{V\textsuperscript{o}}}}. Et non solum difficile est in acutum tendere et seruare cantus, sed etiam difficile mox cessare a cantibus acutis, eo quod \edtext{tensi}{\Cfootnote{remissi \textit{γ}}} sunt valde \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} equum extensum et velocissime currentem molestum \edtext{est}{\Cfootnote{\textit{om}. \textit{αV\textsuperscript{o}}}} confestim sistere, atque similiter nauem velocissime inpulsam.
\pend

\pstart
\textbf{4} \edtext{\textit{MINUS.}}{\Cfootnote{Deinde. MINUS. \textit{V\textsuperscript{a}mv}}} Tertio arguit in contrarium, quasi dicat \edtext{quod}{\Cfootnote{quia \textit{βγM\textsuperscript{a}V\textsuperscript{z}}}} \textit{minus est opus et sic \edtext{maior}{\Cfootnote{minor \textit{E\textsuperscript{a}V\textsuperscript{o}}}} virtus requiritur ad mouendum illud quod est modicum quam \edtext{quod}{\Cfootnote{quod est \textit{β}}} multum}, non videtur quod acutum mouere \edtext{cantando}{\Cfootnote{euitando \textit{E\textsuperscript{a}V\textsuperscript{o}}}} \edtext{sit}{\Cfootnote{fit \textit{E\textsuperscript{a}P\textsuperscript{x}}}} difficile, cum sit modicum, et graue facile, quia multum ymmo econuerso, quare sicut est. Vniversaliter in aliis ita \edtext{etiam}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{x}}}} videtur esse in aere qui est subiectum sonorum, eius enim medium proprium est ipse in quo causatur et recipitur primitus.
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\textbf{5} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit dicens causam esse sumendam ex hoc, \textit{quia idem videtur esse naturaliter acute sonare et \edtext{acute cantare}{\Cfootnote{sonare \textit{π}}}, \edtext{et}{\Cfootnote{\textit{om}. \textit{κV\textsuperscript{o}}}} non differt nisi \edtext{secundum quod}{\Cfootnote{sicut \textit{π}}} \edtext{superius differt}{\Cfootnote{differt superius \textit{λ} | superius \textit{π}}} ab inferiori.} Hoc autem assumit vt \edtext{declaret}{\Cfootnote{declararet \textit{P\textsuperscript{x}V\textsuperscript{b}}}} per \edtext{id}{\Cfootnote{illud \textit{E\textsuperscript{a}P\textsuperscript{x}}}} quod est in voce \edtext{vniversaliter}{\Cfootnote{videtur \textit{π}}} \edtext{sic esse}{\Cfootnote{est \textit{κ}}} in sono. Vnde subdens ait ``omnia naturaliter acute \edtext{vocis}{\Cfootnote{vocem \textit{γM\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} fieri propter debilitatem suam, eo quod non \edtext{possunt}{\Cfootnote{possint \textit{λE\textsuperscript{a}V\textsuperscript{b}}}} multum aerem mouere, qui \edtext{grauis vocis est causa}{\Cfootnote{est causa grauis vocis \textit{γ} | grauis est causa \textit{πE\textsuperscript{a}}}}. Sed modicum possunt, modicus autem aer motus velociter fertur,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XI.13, 900a21–27, 901a.31–32}} \edtext{vt visum est .xi.}{\Cfootnote{vt visum .xi. \textit{β} | \textit{om}. \textit{π}}} \edtext{Sepe atque}{\Cfootnote{Sepe at quia \textit{E\textsuperscript{a}N\textsuperscript{u}} | Aut quia \textit{π}}} acutum \edtext{sic velociter mouetur}{\Cfootnote{velociter mouetur sic \textit{π}}} si quis cantauerit secundum acutum. Signum quidem ex hoc suscipitur \edtext{cuiusdam}{\Cfootnote{eiusdem \textit{π}}} vigoris \edtext{in}{\Cfootnote{\textit{om}. \textit{γ}}} eo quod mouet aliquid velociter, quod non fieret nisi vigor adesset. Et ideo ex hoc \edtext{confirmatione}{\Cfootnote{confirmatiue \textit{P\textsuperscript{x}V\textsuperscript{o}}}} concluditur \edtext{quod}{\Cfootnote{quod si \textit{κ}}} illi qui sunt bone complexionis quorum pulmones non redundant humiditatibus superfluis existere vocis acute eo quod organum est \edtext{exsiccatum}{\Cfootnote{excitatum \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}V\textsuperscript{o}}}} solidum. Et quia difficile est \edtext{cantare}{\Cfootnote{cantare ea \textit{π}}} que eleuantur, cum acuta extollantur, ipsa \edtext{fiet}{\Cfootnote{fiat \textit{P\textsuperscript{y}V\textsuperscript{a}}}} cantare difficile, nam ``\edtext{eleuare}{\Cfootnote{eleuantur \textit{π}}} contra naturam,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} II.38, 870a34. }} \edtext{vt visum est particula secunda.}{\Cfootnote{vt visum particula secunda \textit{βE\textsuperscript{a}} | \textit{om}. \textit{π}}} Et quia grauia naturaliter tendunt deorsum, quod nobis magis \edtext{naturale}{\Cfootnote{naturale est \textit{π}}}, \edtext{ideo facile fit}{\Cfootnote{facile fit ideo \textit{β} | facile igitur sit \textit{πE\textsuperscript{a}}}} ipsa cantare. Sic igitur soluitur adquesitum, neque ambiguitas relinquitur aliqua que in xi\textsuperscript{a}. particula intelligenti dicta, quoniam in acute cantare siue sonare \edtext{duo fit}{\Cfootnote{est duo \textit{π}}} considerare: puta ``modicum aerem motum et quantum ad hoc videtur quod acutum mouere sit facile,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XI.14, 900a34–36}} cum id debiles agant et impotentes. \edtext{Est et}{\Cfootnote{Et est \textit{π} | Est \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}} considerare quod illud modicum aeris velociter mouetur, et quia quod \edtext{huius}{\Cfootnote{huiusmodi \textit{βγ}}} non \edtext{fit}{\Cfootnote{sit \textit{γE\textsuperscript{a}}}} nisi a constanti virtute. \edtext{Ideo}{\Cfootnote{Item \textit{V\textsuperscript{o}} | Ergo \textit{G\textsuperscript{u}P\textsuperscript{p}}}} acutum mouere denotat constantiam virtutis, et difficultatem per consequens, quare apparuit .14°. \edtext{xi\textsuperscript{e}.}{\Cfootnote{\textit{om}. \textit{π}}} ``pueros \edtext{voces acutiores}{\Cfootnote{acutiores voces \textit{πP\textsuperscript{y}}}} habere \edtext{senioribus}{\Cfootnote{senibus \textit{P\textsuperscript{x}P\textsuperscript{y}}}}, propter consistentiam calidi \edtext{virtuosi}{\Cfootnote{unctuosi \textit{γ}}}.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XI.14, 900a33-900b8}}
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\subsection[Trigesimumoctauum problema … ]{Trigesimumoctauum problema}\label{part38}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID ET RITHIMO ET CANTU ET OMNINO CONSONANTIIS GAUDENT OMNES? \textsuperscript{2}AUT QUIA MOTIBUS QUI SUNT SECUNDUM NATURAM GAUDEMUS SECUNDUM NATURAM? \textsuperscript{3}SIGNUM AUTEM PUEROS CONFESTIM NATOS GAUDERE IPSIS. PROPTER CONSUETUDINEM AUTEM MODIS CANTUUM GAUDEMUS. \textsuperscript{4}RITHIMO AUTEM GAUDEMUS PROPTER MOTUM ET ORDINATUM NUMERUM HABERE ET MOUERE NOS ORDINATE. MAGIS PRIUS ENIM EST ORDINATUS MOTUS NATURA INORDINATO, QUARE ET SECUNDUM NATURAM MAGIS. \textsuperscript{5}SIGNUM AUTEM LABORANTES ENIM ET BIBENTES ET COMEDENTES ORDINATE SALUAMUS ET AUGEMUS NATURAM ET VIRTUTEM, INORDINATE AUTEM CORRUMPIMUS ET STUPEFACIMUS IPSAM. EGRITUDINES ENIM ORDINATIONIS CORPORIS NON SECUNDUM NATURAM MOTUS SUNT. \textsuperscript{6}CONSONANTIA GAUDEMUS QUONIAM COMPLEXIO EST CONTRARIORUM HABENTIUM PROPORTIONEM AD INUICEM. PROPORTIO AUTEM ORDINATIO, QUOD ERAT NATURA DELECTABILE. COMPLEXIONATUM AUTEM INCOMPLEXIONATO OMNE DELECTABILIUS ALITER. \textsuperscript{7}QUAMUIS SENSIBILE SIT VTRISQUE EXTREMIS EX EQUALI POTENTIAM HABET IN CONSONANTIA PROPORTIO}{\lemma{PROPTER \dots\ PROPORTIO}\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} omnia,} ``ita quod etiam bruta vtcunque''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XXVIII.7, 949b37-950a16}} \edtext{sicut videbitur .7. 29\textsuperscript{e}.}{\Cfootnote{sicut videbitur .vii°. 29\textsuperscript{e}. particule \textit{γ} | \textit{om}. \textit{π}}}, \textit{\edtext{exultantur}{\Cfootnote{exultant \textit{πE\textsuperscript{a}P\textsuperscript{y}}}} \edtext{rithimo}{\Cfootnote{rithimi \textit{V\textsuperscript{o}}}} et cantu \edtext{ac}{\Cfootnote{ac etiam \textit{λ} | et \textit{π}}} vniuersaliter consonantiarum auditu in hiis}, siquidem \edtext{verbis}{\Cfootnote{\textit{om}. \textit{γ}}} triplex \edtext{pretacta indicatur}{\Cfootnote{iudicatur pretacta \textit{π}}} musica?
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit \edtext{primo dans causam vniversaliter dicens}{\Cfootnote{dicens primo \textit{V\textsuperscript{o}}}} causam \edtext{fore}{\Cfootnote{esse \textit{π}}} communem, \textit{quoniam \edtext{gaudemus}{\Cfootnote{gaudere \textit{P\textsuperscript{x}P\textsuperscript{y}}}} naturaliter motibus naturalibus, cum in temperamento quodam consistant.} Sicut enim ``excellentie sensibilium corrumpunt sensus,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{De an.} III.13, 435b15 ((Cf. 426a30))}} \edtext{vt apparet \textit{de anima} tertio,}{\Cfootnote{vt apparet tertio \textit{de anima}, \textit{V\textsuperscript{a}V\textsuperscript{z}} | vt patet tertio \textit{de anima}, \textit{π}}} \edtext{ita}{\Cfootnote{\textit{om}. \textit{π}}} et ``\edtext{ipsorum}{\Cfootnote{ipsarum \textit{π}}} conmensurationes eos custodiunt,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XΙΧ.2, 917b22–29}} \edtext{vt}{\Cfootnote{sicut \textit{απ}}} etiam tactum \edtext{est}{\Cfootnote{\textit{om}. \textit{βV\textsuperscript{a}}}} in principio huius particule.
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\textbf{4} \edtext{\textit{SIGNUM.}}{\Cfootnote{Deinde. SIGNUM. \textit{V\textsuperscript{a}mv}}} Declarat secundo quomodo in vnoquoque trium \edtext{predictorum}{\Cfootnote{predictorum sicut \textit{κ}}} motus secundum naturam \edtext{existat}{\Cfootnote{existit \textit{P\textsuperscript{x}mv}}} et per consequens gaudere \edtext{fiat}{\Cfootnote{faciat \textit{π}}} \edtext{ex eo}{\Cfootnote{\textit{om}. \textit{π}}}. \edtext{Primoque}{\Cfootnote{Primo \textit{πV\textsuperscript{z}}}} manifestans in \edtext{tactu}{\Cfootnote{cantu \textit{α} | sensu \textit{π}}} \edtext{dicens}{\Cfootnote{dicit \textit{P\textsuperscript{x}mv}}} \textit{huius signum esse quoniam pueri \edtext{mox cum}{\Cfootnote{cum mox \textit{π}}} fuerint nati accidit ipsis gaudere cantu \edtext{tum}{\Cfootnote{cum \textit{V\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}} | tamen \textit{P\textsuperscript{x}mv}}} propter musicam naturaliter \edtext{ipsis}{\Cfootnote{sibi \textit{π}}} \edtext{inditam}{\Cfootnote{indicatam \textit{V\textsuperscript{a}} | indicatiua \textit{V\textsuperscript{b}mv}}}.} Anima enim ``\edtext{maxime nostra}{\Cfootnote{magis vel maxime \textit{π}}} ad similitudinem numeri consoni \edtext{constricta}{\Cfootnote{constructa \textit{αM\textsuperscript{a}} | constituta \textit{V\textsuperscript{o}}}} \edtext{existit}{\Cfootnote{est \textit{π}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Yuḥannā ibn Māsawaiyh \textit{Aph.} f. 39v, ll. 10–11)}} vt \edtext{apparet per}{\Cfootnote{\textit{om}. \textit{β}}} \edtext{damascenum}{\Cfootnote{damascenus \textit{β} | dictum \textit{M\textsuperscript{a}V\textsuperscript{m}}}} in \textit{afforismis}. Quare primo \textit{de regimine sanitatis} ``pueri familiariter dispositi sunt ad musicam.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Gal. \textit{De san. tu.} I.7}} \textit{Tum etiam propter consuetudinem quam adipiscimur deinceps ex maneriebus cantuum gaudere accidit,} quidam enim gaudent vno cantu, alii \edtext{vero}{\Cfootnote{autem \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}} alio secundum quod consueti \edtext{fuerunt}{\Cfootnote{fuerint \textit{πE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{b}V\textsuperscript{z}}}} audire a \edtext{natiuitate}{\Cfootnote{natiuitate audiant \textit{λ}}}. Quod autem infantes \edtext{huius}{\Cfootnote{huiusmodi \textit{γN\textsuperscript{u}}}} cantibus \edtext{delectentur}{\Cfootnote{delectantur \textit{V\textsuperscript{o}}}}, id sciunt nutrices industriose que modis cantuum et motuum \edtext{quibusdam}{\Cfootnote{quidam \textit{V\textsuperscript{o}} | communium \textit{G\textsuperscript{u}P\textsuperscript{p}}}}, cum hii \edtext{tristentur}{\Cfootnote{tristantur \textit{πV\textsuperscript{z}}}} et plorant, ipsorum animos \edtext{demulciunt}{\Cfootnote{demulcent \textit{π}}}, quiescere facientes et gaudere, cum ostensum sit hanc esse potentiam musice. Vnde \edtext{eodem}{\Cfootnote{eodem modo \textit{E\textsuperscript{a}V\textsuperscript{o}}}} primo: ``nutrices tribus modis consolantur et ad sompnum prouocant pueros: vt appositione mamillarum ad os, melodia vocis, et motu.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Gal. \textit{De san. tu.} I.7}}
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\textbf{5} \edtext{\textit{RITHIMO.}}{\Cfootnote{AC HUIC. \textit{π} | Deinde. RITHIMO. \textit{V\textsuperscript{a}mv}}} \edtext{Secundo dat causam}{\Cfootnote{Declarat secundo \textit{V\textsuperscript{a}} | Secundo declarat causam \textit{V\textsuperscript{b}mv} | Dat causam secundo \textit{π}}} \edtext{rithimo quare}{\Cfootnote{quare rithimo \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{z}} | rithimo \textit{P\textsuperscript{x}V\textsuperscript{b}V\textsuperscript{m}V\textsuperscript{o}}}} \edtext{fit}{\Cfootnote{sit \textit{πN\textsuperscript{u}}}} gaudere quod secundo \edtext{signo manifestat}{\Cfootnote{manifestat signo ibi \textit{G\textsuperscript{u}P\textsuperscript{p}} | manifestat signum ibi \textit{V\textsuperscript{o}} | signo inanimat \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} \textit{\edtext{SIGNUM}{\Cfootnote{DIGNUM \textit{γ}}} AUTEM,} \edtext{dicens}{\Cfootnote{dicit \textit{γM\textsuperscript{a}V\textsuperscript{z}}}} causam esse quare \textit{gaudemus rithimo. \edtext{Est}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}} propter motum suauem et numerum ordinatum eius et proportionatum clausularum \edtext{vnius}{\Cfootnote{vniuscuiusque \textit{π}}} \edtext{cum}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} altera concordantium \edtext{quos}{\Cfootnote{quas \textit{πP\textsuperscript{x}}}} habet. Que \edtext{autem huius}{\Cfootnote{autem huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}} | aut huius \textit{V\textsuperscript{o}}}} nos habent mouere amplius paulatiue et ordinate, nam motus ordinatus et vniformis prior est natura opposito,} \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} habitus priuatione. Ordinatum enim habet rationem equalitatis, inordinatum vero \edtext{inequalitatis}{\Cfootnote{rationem inequalitatis \textit{π}}}. Omne \edtext{autem}{\Cfootnote{enim \textit{G\textsuperscript{u}P\textsuperscript{q}V\textsuperscript{z}}}} inequale reducitur ad equale, \edtext{vt}{\Cfootnote{sicut \textit{απ}}} ad primum et perfectum, vt \edtext{probatum est}{\Cfootnote{probant \textit{απM\textsuperscript{a}}}} arismetrici et musici. Et ideo concluditur quod \textit{naturaliter magis \edtext{debeamus gaudere}{\Cfootnote{debemus gaudere \textit{π}}} rithimo et \edtext{iocundari}{\Cfootnote{iocunditati \textit{P\textsuperscript{x}V\textsuperscript{z}}}}, tanquam ordinato.}
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\textbf{6} \edtext{\textit{SIGNUM}}{\Cfootnote{Deinde. SIGNUM \textit{V\textsuperscript{a}mv}}} \textit{AUTEM.} Declarat \edtext{id}{\Cfootnote{illud \textit{V\textsuperscript{m}V\textsuperscript{o}}}} signo dicens quod \textit{\edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} signum in regimine \edtext{vite}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} sumitur tale, quia nos exercitantes et laborantes bibentesque \edtext{et}{\Cfootnote{ac \textit{βE\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{o}}}} comedentes \edtext{conmensurate}{\Cfootnote{mensurate \textit{V\textsuperscript{o}}}} \edtext{et}{\Cfootnote{ac \textit{E\textsuperscript{a}V\textsuperscript{o}}}} ordinate,} \edtext{vt}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} precipit \textit{regimen} \edtext{galeni}{\Cfootnote{caloris \textit{κ}}} \textit{\edtext{sanatium}{\Cfootnote{sanitatem \textit{π}}}}, \textit{saluamus et augmentamus \edtext{nostram naturam}{\Cfootnote{naturam nostram \textit{λπ}}} siue complexionem, quantum ad corpus magis et virtutem quo ad animam plus. Per oppositum vero cum indebite ac inordinate vtimur regimine \edtext{sanitatis}{\Cfootnote{\textit{om}. \textit{π}}} corrumpimus \edtext{ac}{\Cfootnote{et \textit{π}}} stupefacimus \edtext{naturam nostram}{\Cfootnote{nostram naturam \textit{γ} | naturam \textit{πE\textsuperscript{a}}}} \edtext{et}{\Cfootnote{ac \textit{γ}}} virtutem.} Quod ostenditur quia \textit{egritudines sunt motus cuiusdam \edtext{ordinationis}{\Cfootnote{inordinationis \textit{π}}} et concursus corporis nostri non secundum naturam} \edtext{verum}{\Cfootnote{sed \textit{G\textsuperscript{u}P\textsuperscript{p}mv}}} preter naturam existentis. \edtext{Quomodo}{\Cfootnote{Quomodo autem \textit{βπE\textsuperscript{a}V\textsuperscript{b}}}} ``egritudo sit motus''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} VII.4, 886b5–6}} \edtext{ostensum differentia septima}{\Cfootnote{ostensum est differentia septima \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{a}} | ostensum differentia septima alia \textit{βV\textsuperscript{b}} | visum est prima \textit{π}}} problemate quarto.
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\textbf{7} \edtext{\textit{CONSONANTIA.}}{\Cfootnote{CONSONANTIAM. \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}} | Deinde. CONSONANTIA. \textit{V\textsuperscript{a}mv}}} Assignat tertio causam delectationis consonantie, dicens causam \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} esse, quare \textit{gaudeamus consonantia, quia ipsa causatur ex coniunctione contrariorum habentium proportionem ad inuicem secundum acuitatem et grauitatem}. \edtext{Confranguntur}{\Cfootnote{Confrangantur \textit{P\textsuperscript{x}V\textsuperscript{a}}}} enim oppositorum excellentie ita vt ex eis resultet vnum medium contemperatum ac proportionatum extremis, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} opposita ostendunt contrariorum excellentium in copulatione ad inuicem qualitatum. \textit{In \edtext{huius}{\Cfootnote{huiusmodi \textit{N\textsuperscript{u}V\textsuperscript{b}}}} autem proportione \edtext{copulationis}{\Cfootnote{complexionis \textit{αM\textsuperscript{a}V\textsuperscript{o}V\textsuperscript{z}}}} saluatur ordinatio, que naturaliter \edtext{delectabilis existit}{\Cfootnote{est delectabilis \textit{π}}}.} Quod autem hoc sit delectabile ostenditur quoniam habens consonantiam est complexionatum seu mixtum, \textit{omne autem \edtext{complexionatum}{\Cfootnote{complexionatum et mixtum \textit{G\textsuperscript{u}P\textsuperscript{p}V\textsuperscript{o}}}} est delectabilius quam incomplexionatum \edtext{vel}{\Cfootnote{seu \textit{λE\textsuperscript{a}} | et \textit{π}}} simplex, aliter tamen et aliter,} quoniam multo magis est delectabile cum in simplicibus vt in elementis primis remaneant excellentie, in complexionatis autem \edtext{consurgunt}{\Cfootnote{confringantur \textit{βE\textsuperscript{a}} | refranguntur \textit{π}}} et ad medium \edtext{reducte sunt}{\Cfootnote{sunt reducte \textit{β}}}.
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\pstart
\textbf{8} \edtext{\textit{QUAMUIS.}}{\Cfootnote{Deinde. QUAMUIS. \textit{V\textsuperscript{a}mv}}} Remouet \edtext{secundo dubitationem}{\Cfootnote{dubium \textit{G\textsuperscript{u}P\textsuperscript{p}}}}, quia crederet aliquis eo quod complexionatum compositum est ex vtrisque \edtext{extremorum non posse delectabile fore. Quod remouet, dicens}{\Cfootnote{\textit{om}. \textit{κ}}} \textit{\edtext{quodlibet sit complexionatum ex vtrisque}{\Cfootnote{\textit{om}. \textit{κ}}} extremorum excellentiis constitutum.} Nichilominus tamen est delectabile, quoniam ipsum constituitur ex contrariis et inequalibus, et non ex \edtext{omnino}{\Cfootnote{non \textit{V\textsuperscript{b}mv}}} similibus et equalibus, sic enim non \edtext{posset}{\Cfootnote{potest \textit{π}}} adeo mouere. Et ideo thimoteus consonantiam fieri ait ex dissimilibus in vnam et eandem vocem venientibus, nam in vocibus que nulla equalitate discordant nulla omnino extat consonantia, cum ea sit vt tactum \edtext{est}{\Cfootnote{\textit{om}. \textit{βE\textsuperscript{a}}}} dissimilium inter se vocum \edtext{in vnum}{\Cfootnote{\textit{om}. \textit{π}}} redacta concordia. Si enim grauis graui misceatur, nulla \edtext{fit}{\Cfootnote{sit \textit{E\textsuperscript{a}G\textsuperscript{u}P\textsuperscript{p}}}} consonantia sed dissimilitudo, \edtext{neque}{\Cfootnote{ne \textit{V\textsuperscript{o}}}} similiter si acutum iungatur acuto. Plato siquidem voluit \edtext{taliter}{\Cfootnote{talem \textit{λ} | tradere taliter \textit{π}}} \edtext{consonantiam}{\Cfootnote{consonantia \textit{E\textsuperscript{a}V\textsuperscript{a}}}} in aure causari, necesse quidem ait sonum velociorem \edtext{esse}{\Cfootnote{esse et \textit{λ}}} acutiorem, qui cum precesserit grauem ingreditur celeriter in aurem ipsius ledens extremitatem intrinsecam. Et ideo quasi pulsus cum \edtext{tanto}{\Cfootnote{canto \textit{P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{o}}}} motu \edtext{retro reuertitur}{\Cfootnote{retrouertitur \textit{βγM\textsuperscript{a}V\textsuperscript{z}}}}, et quia iam pigrescit exire, obuiat graui subintranti et sibi commiscetur consonantiam constituens musicalem. Id tamen vt visum est dictis non conuenit thimotei.
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\subsection[Trigesimumnonum problema … ]{Trigesimumnonum problema}\label{part39}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DELECTABILIUS EST SECUNDUM CONSONANTIAM EO QUOD EST VNIUS VOCIS? \textsuperscript{2}AUT ET ANTIFONUM QUIDEM CONSONUM EST DYAPASON? EX PUERIS ENIM IUUENIBUS, ET NUMERUS FIT ANTIFONUM, QUI DISTANT COMMUNIS SICUT NITE AD YPATEM. CONSONANTIA AUTEM OMNIS DELECTABILIUS SIMPLICI SONO, PER QUE AUTEM DICTUM EST, ET HORUM DYAPASON DELECTABILISSIMA QUIDEM QUAM VNIUS SONI EST SIMPLICEM HABET SONUM. ORGANIZANT AUTEM IN DYAPASON CONSONANTIA. \textsuperscript{3}QUIA QUEMADMODUM IN METRIS PEDES HABENT AD SE IPSOS RATIONEM EQUALEM AD EQUALEM, AUT DUO AD VNUM, AUT ALIQUEM ALIUM, \textsuperscript{4}ET ITA IN CONSONANTIA SONI PROPORTIONEM HABENT MOTUS AD SE IPSOS. \textsuperscript{5}ALIARUM AUTEM CONSONANTIARUM IMPERFECTE, QUE ALTERIUS REUERSIUE SUNT IN MEDIUM FINIUNT, PROPTER QUOD POTENTIA NON EQUALES SUNT. EXISTENTES AUTEM INEQUALES PER CORRUPTIONEM SENSUI. QUEMADMODUM IN ROTIS IN DISSOLUENDO MAIUS SALTANS SONANTIBUS EST.  \textsuperscript{6}AMPLIUS YPATE ACCIDIT EUNDEM FINEM PERIODORUM, QUI SUNT IN SONIS, HABERE; SECUNDA AUTEM NEATIS ICTUS AERIS YPATE EST. FINIENS AUTEM IN IDEM, NON IDEM FACIENTIBUS VNUM ET COMMUNE OPUS ACCIDIT FIERI, QUEMADMODUM SONANTIBUS SUB CANTU. ETENIM HII ALIA NON PARITER FISTULANTUR, SI IN IDEM CONUERTANT DELECTANT MAGIS FINE QUAM CONTRISTANT DIFFERENTIIS QUE SUNT AD FINEM, IN EO QUOD FIT EX DIFFERENTIBUS COMMUNE DELECTABILISSIMUM EX DYAPASON. ORGANIZANT AUTEM EX CONTRARIIS VOCIBUS; PROPTER HOC IN DYAPASON ORGANIZANT}{\lemma{PROPTER \dots\ ORGANIZANT}\Cfootnote{PROPTER QUID DELECTABILIUS \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} cantus et vniuersaliter sonus, existens secundum consonantiam, que est copulatio quedam suauem et mixtum reddens sonum, delectabilior est quam cantus et omnino sonus qui fuerit vnius vocis et percussionis}, et uniuersaliter dyapason ad alias consonantias?
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et primo assignat \edtext{causam vnam}{\Cfootnote{vnam causam \textit{π}}}, secundo \edtext{alteram}{\Cfootnote{aliam \textit{π}}} vel \edtext{priorem}{\Cfootnote{primam \textit{π}}} confirmat \edtext{secunda}{\Cfootnote{ibi \textit{πV\textsuperscript{a}} | ibi secunda \textit{V\textsuperscript{b}mv}}} \textit{AMPLIUS}. Adhuc \edtext{prius}{\Cfootnote{primo \textit{πV\textsuperscript{z}}}} dat causam prime partis, \edtext{dicens}{\Cfootnote{dicens primo \textit{αV\textsuperscript{o}}}} \edtext{id}{\Cfootnote{ideo \textit{P\textsuperscript{z}V\textsuperscript{m}} | hoc \textit{γ} | id ideo \textit{V\textsuperscript{o}}}} esse quoniam \textit{dyapason est \edtext{antiphonum}{\Cfootnote{archisonum \textit{V\textsuperscript{a}mv}}} quoddam consonum.} Secundum enim ipsam solam fit antiphonizare, vt \edtext{apparuit}{\Cfootnote{apparebit \textit{πE\textsuperscript{a}}}}, et organizare cum sit ex pluribus sonis vel vocibus consonantibus \edtext{coaptata}{\Cfootnote{adaptata \textit{λ}}}. Ipsa \edtext{enim}{\Cfootnote{enim est \textit{V\textsuperscript{o}}}} vt dictum \edtext{est}{\Cfootnote{fuit \textit{π}}} \edtext{optima}{\Cfootnote{optima est \textit{λ}}} consonantiarum, \edtext{ac}{\Cfootnote{et \textit{V\textsuperscript{o}}}} \textit{suauissima ex vocibus \edtext{acutis pverorum}{\Cfootnote{pverorum acutis \textit{π}}} et grauibus iuuenum consurgit. Fit enim numerus \edtext{quidam}{\Cfootnote{quidem \textit{π}}} et proportio consonum reddens antiphonum. \edtext{Alias}{\Cfootnote{Aliquando \textit{V\textsuperscript{o}}}} \edtext{et viris fit}{\Cfootnote{etiam fit viris \textit{π}}} et tunc iuuenibus veniat \edtext{indeterminatio}{\Cfootnote{in determinationem \textit{απM\textsuperscript{a}V\textsuperscript{z}}}} \edtext{de li}{\Cfootnote{de lii \textit{G\textsuperscript{u}P\textsuperscript{p}V\textsuperscript{b}}}} pueris, prior tamen scriptura melior. Qui \edtext{siquidem}{\Cfootnote{sic quidem sit \textit{π}}} differunt \edtext{ab}{\Cfootnote{ad \textit{P\textsuperscript{y}V\textsuperscript{o}}}} inuicem ratione \edtext{acuitatis et grauitatis}{\Cfootnote{grauitatis et acuitatis \textit{P\textsuperscript{x}P\textsuperscript{y}}}} vocis acquisite \edtext{secundum}{\Cfootnote{sed \textit{πE\textsuperscript{a}}}} tempus, quanta est differentia communis \edtext{uel}{\Cfootnote{et \textit{π}}} vniversalis et ultime \edtext{ad ypatem}{\Cfootnote{dyapente \textit{λ}}} communem supremam.} Que quidem \edtext{due}{\Cfootnote{secunde \textit{V\textsuperscript{b}mv}}} corde tacte reddunt consonantiam delectabilem, cum sonus earum sit compositus. \textit{Omnis autem consonantia \edtext{delectabilior}{\Cfootnote{delectabilis \textit{π}}} \edtext{est}{\Cfootnote{est magis \textit{π}}} quam sonus simplex. Et ideo dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}} prius quod dyapason de numero \edtext{sonorum}{\Cfootnote{consonantiarum \textit{π}}} est delectabilissima per respectum ad sonos simplices et vnum tantum. Vnde secundum eam solum \edtext{vt}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}P\textsuperscript{y}}}} \edtext{prescriptum}{\Cfootnote{scriptum \textit{γ}}} est organizare accidit} 18, quod in alia consonantiarum non contingit. Dicit autem \textit{\edtext{communis}{\Cfootnote{\textit{om}. \textit{π}}}} quia ypate et nete sicut et \edtext{aliarum}{\Cfootnote{aliorum \textit{π}}} quedam preponuntur et alie supponuntur, repetuntur enim cum determinatione cum plures septem ponuntur corde vt policorda edocent \edtext{\hyperref[Bo]{boecii}}{\Cfootnote{boni \textit{M\textsuperscript{a}V\textsuperscript{m}} | \textit{om}. \textit{πE\textsuperscript{a}}}}.
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\textbf{4} \edtext{\textit{QUIA QUEMADMODUM.}}{\Cfootnote{QUARE QUODDAM. \textit{V\textsuperscript{o}} | Deinde. QUIA QUEMADMODUM. \textit{V\textsuperscript{a}mv}}} Declarat \edtext{deinde id}{\Cfootnote{id demum \textit{V\textsuperscript{o}V\textsuperscript{z}}}}, dicens hoc esse \textit{quia sicut in carminibus pedes obtinent debitam proportionem ad inuicem, vt sit proportio aut equalis ad \edtext{equale}{\Cfootnote{equalem \textit{P\textsuperscript{p}P\textsuperscript{y}}}}}, vt \edtext{dicatur}{\Cfootnote{dicitur \textit{G\textsuperscript{u}P\textsuperscript{p}}}} in dactilo \edtext{constante}{\Cfootnote{constare \textit{γ}}} ex tribus sillabis \edtext{equalium}{\Cfootnote{equalium et \textit{E\textsuperscript{a}V\textsuperscript{o}}}} tandem temporum, quia ex prima longa, que est duorum temporum, et duabus breuibus, quarum alter \edtext{vterque}{\Cfootnote{vtram \textit{λ}}} vnius, \edtext{ubi}{\Cfootnote{vt \textit{V\textsuperscript{o}}}} duo ad vnum \edtext{comparantur}{\Cfootnote{comparatur \textit{V\textsuperscript{o}}}}, vt due breues ad vnam longam, vel econtra, \edtext{\edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}} | sicut \textit{P\textsuperscript{x}}}} in anapesto huic opposito, vel secundum alium quendam modum sicut vna sillaba duorum temporum ad aliam consimilem, ceu in spondeo. \textit{Aut vnam longam \edtext{ad}{\Cfootnote{et \textit{M\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}}}} alteram breuem vel econtra}}{\lemma{ceu \dots\ econtra}\Cfootnote{\textit{om}. \textit{πV\textsuperscript{z}}}}, \edtext{velut}{\Cfootnote{vt \textit{λP\textsuperscript{y}}}} in trocheo et iambo, spondeus siquidem sono proportionatur graui et iuuenum voci. Tribrachus \edtext{vero}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} \edtext{constans ex tribus breuibus}{\Cfootnote{ex tribus breuibus constans \textit{π}}} acute, sed dactilus anapestus trocheus et iambus similantur \edtext{vtrique}{\Cfootnote{vtraque \textit{λ} | vterque \textit{V\textsuperscript{o}}}}. Similiter etiam pedes quibus vtimur diuersa optinent loca in metro vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} .20°.
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\textbf{5} \edtext{\textit{ET}}{\Cfootnote{Deinde. ET \textit{V\textsuperscript{a}mv}}} \textit{ITA.} \edtext{Tunc}{\Cfootnote{\textit{om}. \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{y}}}} applicat ad propositum, dicens \textit{eodem modo in \edtext{consonantia}{\Cfootnote{consonantiam \textit{γE\textsuperscript{a}}}} \edtext{euenire soni}{\Cfootnote{euenit soni \textit{κ} | euenire \textit{E\textsuperscript{a}V\textsuperscript{o}}}} et proprie dyapason, nam motus sonorum \edtext{causatiui}{\Cfootnote{cantantiui \textit{M\textsuperscript{a}V\textsuperscript{m}}}} proportionem habent \edtext{ad}{\Cfootnote{ad se \textit{απ}}} inuicem.}
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\textbf{6} \edtext{\textit{ALIARUM.}}{\Cfootnote{ALIORUM. \textit{πE\textsuperscript{a}} | Deinde. ALIARUM. \textit{V\textsuperscript{a}mv}}} Αssignat \edtext{secundo}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{z}}}} \edtext{causam partis alterius}{\Cfootnote{causam alterius partis \textit{V\textsuperscript{m}V\textsuperscript{z}}}}, dicens quod, \textit{de numero aliarum consonantiarum \edtext{ad}{\Cfootnote{a \textit{P\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{o}}}} dyapason quia sunt alterius proportionis ab ea, sunt imperfecte}, non enim sunt per totum ambientes omnia signa .7. vel .8. monocordi prima sicut dyapason. \textit{Sed potius reuersiue facte circa medium terminantur, } vt dyatesseron \edtext{in quarto}{\Cfootnote{medio \textit{M\textsuperscript{a}V\textsuperscript{z}}}} \edtext{signo}{\Cfootnote{signo et \textit{αV\textsuperscript{o}}}} dyapente in quinto, vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{m}V\textsuperscript{z}}}} .13°. \edtext{et 32°}{\Cfootnote{\textit{om}. \textit{π}}}. \textit{Et ideo non habent virtutem equaliter \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} dyapason, \edtext{verum}{\Cfootnote{sed \textit{π}}} potius \edtext{inequaliter}{\Cfootnote{inequalitantur \textit{V\textsuperscript{m}V\textsuperscript{z}}}} propter ipsas, namque sensus horret auditus et corrumpitur.} Sunt enim respectu dyapason aspere ac horride, quod declarat, quia sicut accidit in rotis cantantibus, \edtext{id est}{\Cfootnote{id est in \textit{πP\textsuperscript{y}}}} cantibus rotatiuis seu \edtext{conuersis}{\Cfootnote{conuersiuis \textit{P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}}, \edtext{quod si aliquis}{\Cfootnote{\textit{om}. \textit{π}}} eleuando sonum ultra id quod debet, saltem in maius sursum ad aliquas notas in quas \edtext{non}{\Cfootnote{nunc \textit{π}}} debet peruenire, accidit sonum totum dissolui, vt apparet in organis ita et in aliis consonantiis \edtext{ad}{\Cfootnote{a \textit{πE\textsuperscript{a}M\textsuperscript{a}} | \textit{om}. \textit{γP\textsuperscript{y}}}} dyapason contingit suo modo et maxime in dyatesseron, vel rotis, id est instrumentis \edtext{illis}{\Cfootnote{\textit{om}. \textit{π}}} antiquioribus habentibus duas acies cordarum, \edtext{vnam}{\Cfootnote{vna \textit{E\textsuperscript{a}M\textsuperscript{a}}}} quidem in vno \edtext{laterum}{\Cfootnote{latere \textit{γP\textsuperscript{x}}}} et reliquam in altero, que simul manibus tanguntur ambabus. \edtext{Nunc}{\Cfootnote{Non \textit{π}}} autem si inequaliter tetigerit eas, magis \edtext{pulsans}{\Cfootnote{pulsans vnam quam aliam vel \textit{π}}} cordas vnius partis quam alterius, adulteratur simphonia, primum tamen \edtext{melius et verius}{\Cfootnote{melius et uniuersalius \textit{α} | vniuersalius et melius \textit{π}}}. Dicit autem \textit{corruptio sensui} quia \edtext{dictum est}{\Cfootnote{est dictum \textit{λ}}} quod musica non solum \edtext{rationi}{\Cfootnote{rationem \textit{E\textsuperscript{a}V\textsuperscript{b}}}} innititur, vt \edtext{sensit}{\Cfootnote{sensui \textit{V\textsuperscript{o}}}} pictagoras, neque tantummodo sensui, \edtext{sicut}{\Cfootnote{sic \textit{P\textsuperscript{y}V\textsuperscript{m}}}} aristoxenus, sed vtrumque contractat.
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\textbf{7} \edtext{\textit{AMPLIUS.}}{\Cfootnote{Deinde. AMPLIUS. \textit{V\textsuperscript{a}mv}}} Ponit aliam solutionem vel magis pretactam confirmat et declarat dicens \textit{\edtext{rursus}{\Cfootnote{\textit{om}. \textit{π}}} in ypate accidere quod habeat \edtext{vnum}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{p}P\textsuperscript{u}}}} et eundem finem omnium periodorum siue circuitionum existentium in sonis.} In \edtext{ipsa}{\Cfootnote{ipsam \textit{E\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}V\textsuperscript{z}}}} enim, tanquam in principaliorem, \edtext{terminantur}{\Cfootnote{terminatur \textit{λP\textsuperscript{y}V\textsuperscript{a}}}} vel melius finem periodorum id est \edtext{resonationum}{\Cfootnote{resonantium \textit{γ}}} post ictum \edtext{remanentium}{\Cfootnote{resonantium \textit{πE\textsuperscript{a}}}} diu in aere, quod tinnitus \edtext{potest dici}{\Cfootnote{dici potest \textit{π}}}. Et ideo vocatur magistra, ipsa enim suscipit omnes et obtundit cum eius sit \edtext{proprius}{\Cfootnote{proprium \textit{πV\textsuperscript{a}V\textsuperscript{z}}}} antiphonizare \edtext{et}{\Cfootnote{atque \textit{αλ} | ac \textit{V\textsuperscript{o}}}} organizare. \textit{Sed secunda corda, correspondens ypati per oppositum omnino, est neatis siue netes vel neatis, \edtext{id est}{\Cfootnote{siue \textit{κ}}} resonantie siue tinitus, pretacti.} Dupliciter enim accipitur neate .42. apparens, \edtext{secundo}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}} cuius quidem ictus et sonus \edtext{causatus}{\Cfootnote{cantus \textit{E\textsuperscript{a}V\textsuperscript{o}}}} in aere. \textit{\edtext{In idem}{\Cfootnote{\textit{om}. \textit{γ}}} finit cum sono qui \edtext{fit}{\Cfootnote{fuit \textit{αλ}}} ab ypate,} ex \edtext{ambabus}{\Cfootnote{ambobus \textit{γ} | amborum \textit{π}}} enim consurgit, vt visum est sepius \edtext{in}{\Cfootnote{\textit{om}. \textit{λE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}} simphonia dyapason. Et merito, \edtext{quia}{\Cfootnote{\textit{om}. \textit{π}}} licet non eundem sonum reddant secundum se. Ymmo vna acutum et altera grauem. \textit{\edtext{Tandem}{\Cfootnote{Et idem \textit{πE\textsuperscript{a}}}} cum ipsarum soni \edtext{coniungantur}{\Cfootnote{coniunguntur \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}} ad inuicem, contingit fieri vnum et eundem sonum omnifariam, quemadmodum accidit illis qui sonant instrumenta sub cantu cantantium. Vel quia aliqui \edtext{precinunt}{\Cfootnote{ponunt et precinunt \textit{G\textsuperscript{u}P\textsuperscript{p}}}} et alii subcinunt licet,} \edtext{et}{\Cfootnote{\textit{om}. \textit{πP\textsuperscript{y}}}} enim videantur discordare in principio, \edtext{concidunt}{\Cfootnote{concidunt tamen \textit{α} | concordant tamen \textit{G\textsuperscript{u}P\textsuperscript{z}} | contra tamen \textit{V\textsuperscript{o}}}} in \edtext{finem}{\Cfootnote{fine \textit{P\textsuperscript{x}V\textsuperscript{a}}}} vnum. Et \edtext{illud est}{\Cfootnote{id est \textit{E\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{m}} | hoc est \textit{π} | idem \textit{V\textsuperscript{a}V\textsuperscript{z}}}} quod subiungitur, \textit{tales \edtext{namque}{\Cfootnote{enim \textit{π}}} \edtext{non}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} pariter et vna \edtext{fistulantur}{\Cfootnote{fistulantur et cantant \textit{E\textsuperscript{a}P\textsuperscript{y}} | fistulant et cantant \textit{π} | similantur et cantant \textit{γ}}}, sed secundum alia et alia. \edtext{Si tamen}{\Cfootnote{Tamen et \textit{π}}} quamuis \edtext{sint}{\Cfootnote{sicut \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}}}} alii \edtext{et alteri}{\Cfootnote{et aliquando \textit{γE\textsuperscript{a}} | tamen \textit{V\textsuperscript{o}}}} \edtext{concurrant}{\Cfootnote{concurrunt \textit{κπE\textsuperscript{a}}}} \edtext{tandem}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{fine in vnum}{\Cfootnote{in finem vnum \textit{πV\textsuperscript{b}} | finem in vnum \textit{E\textsuperscript{a}V\textsuperscript{a}}}} \edtext{quendam}{\Cfootnote{aliquando \textit{π}}} sonum, magis habent inducere delectationem quam tristitiam, propter hoc quod differentie seu diuersitates que sunt in eis secundum principium reperte sopiuntur et concurrunt \edtext{demum}{\Cfootnote{deinde \textit{P\textsuperscript{y}V\textsuperscript{o}}}} ad eundem finem \edtext{siue}{\Cfootnote{seu \textit{P\textsuperscript{x}V\textsuperscript{z}}}} terminationem. In eo enim, vt \edtext{declaratum est}{\Cfootnote{declaratum \textit{E\textsuperscript{a}P\textsuperscript{y}} | declaratum prius> \textit{π}}} \edtext{antea}{\Cfootnote{\textit{om}. \textit{π}}}, quod ex \edtext{differentiis}{\Cfootnote{differentibus \textit{απ}}} diuersisque extremis efficitur \edtext{vnum}{\Cfootnote{vnum et \textit{πE\textsuperscript{a}}}} quoddam commune \edtext{proportionatum}{\Cfootnote{proportionaliter \textit{V\textsuperscript{o}}}}. Cum dyapason \edtext{sit}{\Cfootnote{sic \textit{P\textsuperscript{y}V\textsuperscript{z}}}} \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} quod \edtext{relinquitur}{\Cfootnote{derelinquitur \textit{π}}} ab ea simphonum delectabilissimum \edtext{extat}{\Cfootnote{est \textit{π}}}, quod ostenditur quia secundum eam organizare ex \edtext{oppositis}{\Cfootnote{diuersis \textit{π}}} vocibus causatur et sonis \edtext{tandem in unum}{\Cfootnote{in vnum tandem \textit{πE\textsuperscript{a}}}} redactis. Et quia dyapason \edtext{ex}{\Cfootnote{in \textit{G\textsuperscript{u}P\textsuperscript{p}}}} \edtext{huius}{\Cfootnote{huiusmodi \textit{γE\textsuperscript{a}}}} consistit, vt expositum \edtext{est}{\Cfootnote{\textit{om}. \textit{α}}}, sepius ideo secundum \edtext{ipsam}{\Cfootnote{huius \textit{λ}}} dictum est \edtext{solam}{\Cfootnote{solum \textit{V\textsuperscript{o}}}} nullamque \edtext{aliarum}{\Cfootnote{aliam \textit{P\textsuperscript{p}P\textsuperscript{x}}}} \edtext{organizari}{\Cfootnote{organizare \textit{G\textsuperscript{u}P\textsuperscript{p}mv}}},} .20. problemate.
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\textbf{8} Nosce quod sicut se habet vox composita ad simplicem, ita dyapason ad reliquas consonantias.
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\subsection[Quadragesimum problema … ]{Quadragesimum problema}\label{part40}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DELECTABILITER AUDIUNT CANTANTES QUOSCUNQUE VTIQUE PRESCIENTES ATTINGANT CANTUS QUAM SI NESCIUNT? \textsuperscript{2}VTRUM QUIA MAGIS MANIFESTUS EST ATTINGENS QUEMADMODUM INTENTIONEM QUANDO COGNOSCUNT QUOD CANITUR? COGNOSCENTIBUS AUTEM DELECTABILE CONSIDERARE. \textsuperscript{3}AUT COMPATIENS EST AUDITOR CANTANTI NOTUM? CONSONAT AUTEM IPSO, CANTAT AUTEM OMNIS CONSONANS, QUI NON PROPTER ALIQUAM NECESSITATEM FACIT HOC}{\lemma{PROPTER \dots\ HOC}\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} \edtext{Et}{\Cfootnote{\textit{om}. \textit{λπ}}} \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} \edtext{quidem}{\Cfootnote{autem \textit{G\textsuperscript{u}P\textsuperscript{q}} | \textit{om}. \textit{V\textsuperscript{a}V\textsuperscript{o}}}} expositio sufficiens iam \edtext{quinto}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{apparuit}{\Cfootnote{apparebit \textit{E\textsuperscript{a}P\textsuperscript{x}}}} \edtext{problemate}{\Cfootnote{\textit{om}. \textit{π}}}. Repetitur \edtext{vero}{\Cfootnote{autem \textit{V\textsuperscript{o}}}} vt \edtext{memorie amplius commendetur}{\Cfootnote{amplius memorie comendetur \textit{λ} | memorie commendetur amplius \textit{π}}}.
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\subsection[Quadragesimumprimum problema … ]{Quadragesimumprimum problema}\label{part41}
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\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID BIS PARACUTA AUT BIS QUATUOR NON CONCORDANT, BIS DYAPASON AUTEM ITA? \textsuperscript{2}AUT QUIA QUOD QUIDEM DYAPENTE EST IN EMIOLIA PROPORTIONE, QUOD VERO PER QUATUOR IN EPITRITO? EXISTENTIBUS AUTEM EMIOLIBUS TRIBUS CONSEQUENTER NUMERIS, QUE EST EPITRICON, EXTREMA AD SE INUICEM NULLAM PROPORTIONEM HABEBUNT, NEQUE ENIM PARTICULARIA NEQUE MULTIPLICIA ERUNT. \textsuperscript{3}DYAPASON AUTEM, QUONIAM EST IN DUPLICI PROPORTIONE, BIS HOC FACTO, IN QUADRUPLICI PROPORTIONE SI FUERINT EXTREMA AD INUICEM. \textsuperscript{4}QUARE QUONIAM CONSONANTIA RATIONABILEM HABENTIUM SONUM AD SE INUICEM EST, RATIONEM AUTEM QUI QUIDEM HABENT BIS DYAPASON INTERDECISIONEM AD SE INUICEM SONI HABENT, QUI AUTEM BIS PER QUATUOR AUT BIS PER QUINQUE NON HABENT. QUI QUIDEM BIS DYAPASON CONSONABUNT, VTIQUE ALII VERO NON PER EA QUE DICTA SUNT}{\lemma{PROPTER \dots\ SUNT}\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}} | PROPTER QUID BIS PARACUTA \textit{V\textsuperscript{m}V\textsuperscript{o}}}}.}
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\textbf{2} Repetit .34. vt amplius exponat.
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\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et primo assignat causam prime partis, secundo secunde \edtext{secunda}{\Cfootnote{secunda ibi \textit{γ}}} \textit{DYAPASON}, 3° concludit \edtext{intentum}{\Cfootnote{intentum ibi \textit{γ}}} \textit{QUARE QUONIAM}. \edtext{Dicens causam esse}{\Cfootnote{Dicit causam esse \textit{M\textsuperscript{a}V\textsuperscript{z}} | Primo querit quare \textit{V\textsuperscript{o}}}} \textit{quoniam dyapente est in sexquialtera proportione, \edtext{dyatesseron in}{\Cfootnote{dyatesseron \textit{P\textsuperscript{x}V\textsuperscript{m}}}} epitrica. Hiis \edtext{igitur}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{a}V\textsuperscript{m}}}} ita dispositis \edtext{numeris}{\Cfootnote{numerus \textit{κ}}}, vt \edtext{duobus}{\Cfootnote{tribus \textit{V\textsuperscript{o}}}} sexquialteris et tribus, \edtext{et}{\Cfootnote{etiam \textit{αV\textsuperscript{o}}}} numeris consequenter \edtext{reiteratis}{\Cfootnote{reiterari \textit{V\textsuperscript{o}}}}, vt tria ad quatuor, ex quibus erat epitrica, vt dictum est proportio cum erant \edtext{simplices}{\Cfootnote{simplices vt \textit{γ}}} inrepetiti, tunc si extrema \edtext{huius duarum}{\Cfootnote{huiusmodi duarum \textit{γ}}} consonantiarum duplentur, ad inuicem \edtext{coniuncta}{\Cfootnote{coniecta \textit{P\textsuperscript{x}V\textsuperscript{z}}}} nullam habebunt proportionem consurgentem. Inde quod probatur, quia talia non \edtext{redduntur}{\Cfootnote{reddunt \textit{V\textsuperscript{o}}}} \edtext{in}{\Cfootnote{neque \textit{γλE\textsuperscript{a}V\textsuperscript{o}}}} superparticularia neque multiplicia,} penes que, vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{b}}}} 13°., proportiones \edtext{assumuntur}{\Cfootnote{accipiuntur \textit{V\textsuperscript{o}}}} musice. Vnde apparet quod \hyperref[Ar]{aristoteles} consentit \edtext{pictagoricis}{\Cfootnote{\hyperref[Pyth]{pictagoras} \textit{E\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}}}}, has ponentibus solum, et non concidit cum \hyperref[Pt]{ptholomeo}, ponente \edtext{fore}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} proportionem compositam ex dyatesseron et dyapason consurgere ex multiplici et \edtext{superparticulari}{\Cfootnote{superpartienti \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}} | superuenienti \textit{P\textsuperscript{x}V\textsuperscript{m}}}}. Si namque sumatur bis dyatesseron puta prima dyatesseron \edtext{sit}{\Cfootnote{fit \textit{E\textsuperscript{a}V\textsuperscript{o}}}} proportio \edtext{trium}{\Cfootnote{.3. \textit{P\textsuperscript{x}V\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} ad quatuor, \edtext{et}{\Cfootnote{\textit{om}. \textit{γP\textsuperscript{x}}}} secunda \edtext{sex ad octo}{\Cfootnote{quinque ad quatuor \textit{V\textsuperscript{o}}}}, et coniungantur extrema, vt tria comparando ad .8. nulla consurgit proportio pretactarum multiplicis aut superparticularis. \edtext{Similiter}{\Cfootnote{Similiter et \textit{γλE\textsuperscript{a}}}} \edtext{in}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{a}V\textsuperscript{m}}}} dyapente, vt si sumatur exempli gratia vna dyapente, puta cum duo comparantur ad tria, et secunda. Quando 6 ad 9, ita vt duo \edtext{comparantur}{\Cfootnote{comparentur \textit{E\textsuperscript{a}V\textsuperscript{a}}}} ad .9. Modo eodem nulla consurgit proportio cum omnis vt sic constet in numero epitrico emiola duplari triplari et quadruplo, vt ostensum \edtext{est}{\Cfootnote{est in \textit{P\textsuperscript{y}V\textsuperscript{b}}}} \edtext{anterius}{\Cfootnote{superius \textit{V\textsuperscript{o}V\textsuperscript{z}}}}.
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\textbf{4} \edtext{\textit{DYAPASON.}}{\Cfootnote{Deinde. DYAPASON. \textit{V\textsuperscript{a}mv}}} Assignat causam secunde partis, dicens \textit{quia dyapason se habet in duplici proportione, si bis assumatur ea erit in quadruplici extremis comparatis ad inuicem proportionum \edtext{minoribus et maioribus}{\Cfootnote{minoris et maioris \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}}.} Puta sumatur bis dyapason \edtext{vt in vna quarum}{\Cfootnote{vt in unaquaque \textit{V\textsuperscript{o}} | vt quarum \textit{κ}}} comparetur vnum ad duo et \edtext{in}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} reliqua duo ad quatuor, \edtext{tunc si extrema comparentur ad inuicem vt vnum ad quatuor}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}} resultabit numerus quadruplus iuxta \edtext{quem}{\Cfootnote{quod \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}} dictum est bis dyapason \edtext{siue}{\Cfootnote{fieri \textit{mv} | \textit{om}. \textit{V\textsuperscript{b}}}} 
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\textbf{5} \edtext{\textit{QUARE}}{\Cfootnote{Deinde. QUARE \textit{V\textsuperscript{a}mv}}} \textit{QUONIAM.} Concludit intentum tertio, dicens et ideo \textit{quia consonantia est compositio quedam tonorum habentium \edtext{rationabilem}{\Cfootnote{rationalem \textit{V\textsuperscript{a}V\textsuperscript{o}}}} et \edtext{proportionatum}{\Cfootnote{proportionalem \textit{V\textsuperscript{o}}}} sonum \edtext{ad se inuicem}{\Cfootnote{ad inuicem \textit{E\textsuperscript{a}P\textsuperscript{y}}}}, et hoc habet bis dyapason, non autem bis dyatesseron aut dyapente et reliqua. Quod \edtext{declaratur}{\Cfootnote{declarat \textit{E\textsuperscript{a}P\textsuperscript{x}}}} quoniam proportio \edtext{quam quidem}{\Cfootnote[nosep]{\textit{emend.} | quam \textit{V\textsuperscript{o}} | quam siquidem \textit{ακ}}} habent soni ad inuicem est habere inter se \edtext{decisionem}{\Cfootnote{decisiones \textit{V\textsuperscript{o}}}} et confractionem ad inuicem quandam, per quam \edtext{resultant}{\Cfootnote{resultent \textit{γE\textsuperscript{a}}}} ipsi proportionales armonici, vt bis dyapason, quod non habet \edtext{consonantia}{\Cfootnote{consonantiam \textit{λE\textsuperscript{a}P\textsuperscript{y}}}} per quatuor aut .5. siue dyatesseron \edtext{vel}{\Cfootnote{et \textit{E\textsuperscript{a}V\textsuperscript{o}}}} dyapente. \edtext{Sed}{\Cfootnote{Cumque \textit{E\textsuperscript{a}V\textsuperscript{o}}}} vt dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}}, tantum soni qui fuerint secundum dyapason duplici consonabunt, alii vero minime, sicut ostensum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{z}}}} \edtext{per}{\Cfootnote{propter \textit{P\textsuperscript{y}V\textsuperscript{b}}}} antedictam causam.}
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    {\Large \ding{167}}
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\subsection[Quadragesimumsecundum problema … ]{Quadragesimumsecundum problema}\label{part42}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID SI QUIS ALTERA NITEM ACCIPIAT AUT YPATEM ACCIPIAT, YPATE SOLA VIDETUR SUBSONARE? \textsuperscript{2}AUT QUIA NEATE FINIENS ET DEPERDITA YPATE FIT? \textsuperscript{3}SIGNUM AUTEM POSSE AB YPATE NEATEN POSSE CANERE. SICUT ENIM EXISTENTE IPSA CAUSAM NEATE, SIMILITUDINEM ACCIPIUNT AB IPSA. QUONIAM AUTEM ET ECHO CANTUS QUIDAM EST, TACTUS EST VOCIS NEATIS FINIENTIS SONUS EXISTENS IDEM SONO YPATES MOUET CONUENIENTER, SIMILITUDINI YPATEM NITE VIDETUR MOUERE. NEATEN ENIM SCIMUS VBI MOUETUR ASSUMPTA. YPATEN AUTEM IPSAM VIDENTES INCOMPREHENSIBILEM EXISTENTEM ET SONUM IPSIUS AUDIENTES, HANC PUTAMUS SONARE. QUOD IN MULTIS NOBIS ACCIDIT, IN QUIBUS NEQUE RATIONE. NEQUE SENSU POSSUMUS CERTIFICARE CERTUM. \textsuperscript{4}AMPLIUS SI PERCUSSA NEATE MAXIME TENSA ACCIDIT IUGUM MOUERI NICHIL VTIQUE ERIT MIRABILE. MOTO AUTEM OMNES CORDAS PARITER MOUERI ET SONUM FACERE NON IRRATIONABILE. ALIIS QUIDEM IGITUR NEATIS SONUS ALIENUS EST ET FINIENS ET INCIPIENS, YPATE AUTEM FINIENS IDEM. NON APPOSITO PROPRIO SUO MOTU, ILLIUS VIDEBITUR OMNIA EORUNDEM ESSE NON EST INCONUENIENS. ERIT AUTEM MAIOR QUAM RELIQUARUM CORDARUM SONUS, QUOD HEE QUIDEM A NEATE SICUT IMPULSE MOLLITER SONUERUNT, NEATE AUTEM, TOTA SUI IPSIUS POTENTIA EXISTENS, IPSARUM FORTISSIMA EST. \textsuperscript{5}QUARE MERITO ET SECULARIUM IPSIUS MAIUS ERIT QUAM QUOD EST ALIARUM, QUARE BREUES MOUERI, SICUT IPSIS}{\lemma{PROPTER \dots\ IPSIS}\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} si quis cantu aut \edtext{tactu}{\Cfootnote{tacta \textit{γ}}} \edtext{altera}{\Cfootnote{\textit{om}. \textit{π}}} suscipiat nitem \edtext{siue}{\Cfootnote{aut \textit{π}}} extremam, quia multiplex vt \edtext{visum}{\Cfootnote{dictum \textit{λ}}} est, aut pro et ypatem accipiat, simul coniungens alias quidem \edtext{alteram}{\Cfootnote{altera \textit{M\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{m}} | altam \textit{E\textsuperscript{a}V\textsuperscript{o}}}} vocem, tunc sola ypate sentitur ultimo subsonare,} non quidem quod minorem sonum reddat, \edtext{sed}{\Cfootnote{seu \textit{π} | \textit{om}. \textit{κ}}} \edtext{sonat}{\Cfootnote{ac \textit{π}}} posterius cum sonus eius finitur, ipsa enim est \edtext{illa}{\Cfootnote{ista \textit{P\textsuperscript{y}V\textsuperscript{z}}}} ex qua relinquitur sonus, ille faticius diutius resonatio dicta vel neate seu tinitus.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{AUT QUIA. \textit{γ} | Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit \edtext{primo assignans}{\Cfootnote{et primo assignat \textit{λπE\textsuperscript{a}P\textsuperscript{y}}}} causam, secundo confirmat et declarat ipsam magis. \edtext{Secunda}{\Cfootnote{ibi secunda \textit{γV\textsuperscript{z}}}}: \textit{AMPLIUS}. Primo igitur \edtext{causam dat}{\Cfootnote{dat causam \textit{γπV\textsuperscript{z}}}}, \edtext{dicens hoc esse}{\Cfootnote{dicens ideo esse \textit{π}}} \textit{quoniam cum sonus nites finitur \edtext{deperditur}{\Cfootnote{de proportione \textit{γ}}} et occultatur propter \edtext{resonationem}{\Cfootnote{resonantiam \textit{π}}} ypates, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} maior motus depellit minorem, ea namque sola videtur posterius subsonare}.
\pend

\pstart
\textbf{4} \edtext{\textit{SIGNUM.}}{\Cfootnote{Deinde. SIGNUM. \textit{V\textsuperscript{a}mv}}} Declarat 2° per signum, dicens \textit{signum nobis \edtext{ideo}{\Cfootnote{id \textit{α} | hoc \textit{V\textsuperscript{o}}}} \edtext{demonstrans}{\Cfootnote{demorans \textit{P\textsuperscript{x}V\textsuperscript{m}} | denotans \textit{G\textsuperscript{u}P\textsuperscript{u}}}} est quoniam neate potest amplius subsonare \edtext{tanta}{\Cfootnote{causata \textit{αV\textsuperscript{o}V\textsuperscript{z}}}} ab ypate diutius remanendo post tactum corde quam neate alicuius alterius corde.} Vnde cum ipsa potissime sit causa, vt accidat neate, omnes relique corde \edtext{in terminatione}{\Cfootnote{iteratione \textit{γ}}} suorum sonorum suscipiunt similitudinem ab ypate.
\pend

\pstart
\textbf{5} Nosce quod neate interdum sumitur \edtext{pro hypate}{\Cfootnote[nosep]{\textit{emend}. | pro nete \textit{γκπE\textsuperscript{a}}}}. Vt visum est .13°. \edtext{et 39°.}{\Cfootnote{et 19°. \textit{λ}}}: ``est enim quasi corda iuuenilis et inferior accipitur etiam pro resonatione secunda que \edtext{sentitur}{\Cfootnote{retinetur \textit{γ}}} diu post percussionem, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{P\textsuperscript{p}V\textsuperscript{b}}}} hic.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XIX.13, 918b3–6; XIX.39, 921a7–31}} Sicut enim neate corda est inferior, ita resonatio etiam ypates est debilior sono priori, \edtext{sic vt}{\Cfootnote{sicut \textit{γ}}} concidant.
\pend

\pstart
\textbf{6} Quod autem ita sit ostenditur etiam per echo de quo dictum est \edtext{.xi.}{\Cfootnote{14 \textit{V\textsuperscript{o}}}}, quod est quidam \edtext{cantus}{\Cfootnote{tinitus \textit{P\textsuperscript{p}P\textsuperscript{u}P\textsuperscript{z}V\textsuperscript{o}}}} seu sonus \edtext{reflexus. Est enim hic echo cantus aut sonus}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}P\textsuperscript{y}}}} vocis siue soni \edtext{neatis vel}{\Cfootnote{\textit{om}. \textit{γ}}} resonationis finientis existens idem cum sono ypates principaliori, quando \edtext{fuit}{\Cfootnote{fuerit \textit{π}}} \edtext{tactus eius}{\Cfootnote{eius tactus \textit{απ}}}, differens, \edtext{tamen}{\Cfootnote{cum \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{m}}}} priori et posteriori, fortiori et debiliori. Nam cum aliquid mouetur ita vt decenter \edtext{adaptetur}{\Cfootnote{aptetur \textit{P\textsuperscript{x}V\textsuperscript{a}}}} similitudini soni qui fit secundum ypatem, postremo tunc apparet \edtext{ac si}{\Cfootnote{sicut \textit{π}}} \edtext{nite}{\Cfootnote{vite \textit{V\textsuperscript{o}}}} actu tangeretur resonatio siquidem \edtext{ypates}{\Cfootnote{ypate \textit{πE\textsuperscript{a}}}} videtur sono principaliori netes adaptari. Tam enim immensus est ypates sonus quod \edtext{deficiens}{\Cfootnote{\textit{om}. \textit{π}}} equiparatur sono netes principaliori. Quod ostenditur quia neatem alterius \edtext{corde}{\Cfootnote{corde alterius \textit{V\textsuperscript{o}}}} ab ypate scire dicimur quando tangitur \edtext{presignata}{\Cfootnote{prelibata \textit{γ}}} corda et mouetur ab ea altera. Sed secus accidit circa ypatem quoniam etiam si videmus quod ypates non \edtext{comprehendatur}{\Cfootnote{comprehenditur \textit{π}}} ita vt manifeste tangi eam \edtext{scimus}{\Cfootnote{sentimus \textit{πE\textsuperscript{a}}}}, sonum \edtext{tamen}{\Cfootnote{tantum \textit{M\textsuperscript{a}V\textsuperscript{m}}}} audientes in communi \edtext{extimamus}{\Cfootnote{existimamus \textit{V\textsuperscript{o}}}} propter ipsius diuturnam \edtext{resonationem}{\Cfootnote{resonantiam \textit{π}}} fore ipsius. Quod declarat eo quod id non solum \edtext{hic accidit}{\Cfootnote{accidit hic \textit{π}}}, verum \edtext{et in multis}{\Cfootnote{etiam in multis \textit{λπ}}} aliis rebus de quibus non habemus perfectam rationem neque sensum perspicacem. Ex quibus \edtext{possumus certificari omnino}{\Cfootnote{non possumus omnino certificari \textit{π}}} in talibus tunc accipimus inde aliquam opinionem \edtext{estimatione}{\Cfootnote{existimatione \textit{V\textsuperscript{o}}}} debili \edtext{constrictam}{\Cfootnote{constructam \textit{πE\textsuperscript{a}P\textsuperscript{y}} | coniectiua \textit{γM\textsuperscript{a}}}}. Sic itaque ypate \edtext{resonat}{\Cfootnote{sonat \textit{γ}}} pre aliis. Neate etiam, \edtext{que}{\Cfootnote{quam \textit{κ}}} ultimo relinquitur, ipsius extat que \edtext{aliarum}{\Cfootnote{aliorum \textit{π}}} occultat et deperdit neatem.
\pend

\pstart
\textbf{7} \edtext{\textit{AMPLIUS.}}{\Cfootnote{Deinde. AMPLIUS. \textit{V\textsuperscript{a}mv}}} Secundo confirmat causam predictam vel alteram subiungit, dicens quod \textit{non est mirabile aliquod si \edtext{contingat}{\Cfootnote{contingit \textit{γ}}} \edtext{iugum}{\Cfootnote{istam \textit{π}}} moueri \edtext{subito}{\Cfootnote{cito \textit{π}}} a \edtext{situ suo}{\Cfootnote{suo situ \textit{π}}} cum ypate fuerit maxime tensa. Si accidat quod \edtext{premissum est}{\Cfootnote{premissum \textit{E\textsuperscript{a}P\textsuperscript{y}} | premisit \textit{π}}}, neque etiam est mirum. Si omnes tunc corde alie \edtext{pariter moueantur}{\Cfootnote{moueantur pariter \textit{π}}} \edtext{et}{\Cfootnote{in \textit{P\textsuperscript{x}V\textsuperscript{m}}}} sonum reddant \edtext{cum iugum}{\Cfootnote{coniugum \textit{V\textsuperscript{o}}}} cui innituntur a sua mox \edtext{mouetur}{\Cfootnote{moueantur \textit{π}}} positione, licet tamen sic accidat sonus neates qui fit ab aliis cordis preterquam ab ypate extraneus est et debilis in principio ac etiam in fine, \edtext{cum}{\Cfootnote{cum in \textit{πE\textsuperscript{a}V\textsuperscript{b}}}} ypates concidat sonum. Sed \edtext{huius}{\Cfootnote{huiusmodi \textit{V\textsuperscript{b}mv}}} sonus incipiens et finiens idem permanet, non enim sono apponitur seu adiungitur aliarum, \edtext{seu}{\Cfootnote{ceu \textit{γM\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{z}}}} econtra, ipsa namque proportio concitata motu videbitur quod omnes soni sint earundem cordarum aliarum eius, cum in eam concidant occultati. Quod etiam, cum sit \edtext{soni}{\Cfootnote{soli \textit{γ}}} ceteris fortioris, \edtext{id}{\Cfootnote{\textit{om}. \textit{π}}} non \edtext{reputatur}{\Cfootnote{reparatur \textit{V\textsuperscript{o}}}} inconueniens, quare tunc sonus \edtext{ipsius}{\Cfootnote{eius \textit{π}}} \edtext{est}{\Cfootnote{erit \textit{απ}}} \edtext{maior}{\Cfootnote{manifestior \textit{π}}} quam sonus aliarum cordarum. Nam alie quidem impulse mollem \edtext{reddunt}{\Cfootnote{reddiderunt \textit{γπE\textsuperscript{a}}}} sonum et paruum, ac si esset quedam neate ypates. Cuius \edtext{equidem}{\Cfootnote{eadem \textit{γ}}} est \edtext{neate}{\Cfootnote{neates \textit{πV\textsuperscript{z}}}} tota, cum extet fortissima virtute ceterarum cordarum.} \edtext{Quod}{\Cfootnote{Quod sit \textit{λ}}} principalior ostendit \edtext{eius}{\Cfootnote{ipsius \textit{αλ}}} sonus, per iugum siquidem audio scabellum super quod substentantur corde aut \edtext{illud}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{m}}}} vniversaliter super quod \edtext{innixe}{\Cfootnote{in nixeto \textit{P\textsuperscript{x}V\textsuperscript{m}} | inuise \textit{V\textsuperscript{a}mv} | in mese \textit{V\textsuperscript{b}} | \textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{extenduntur}{\Cfootnote{coextenduntur \textit{P\textsuperscript{y}V\textsuperscript{a}}}}. Vel per iugum \edtext{intelligit}{\Cfootnote{intelligitur \textit{V\textsuperscript{o}}}} \edtext{sinzeusim siue}{\Cfootnote{Initeus in sui \textit{π}}} \edtext{coniunctionem}{\Cfootnote{coniectionem \textit{P\textsuperscript{x}V\textsuperscript{z}}}} \edtext{duorum}{\Cfootnote{duarum \textit{P\textsuperscript{x}V\textsuperscript{z}}}} tetracordarum in mese, vt videbitur \edtext{.44}{\Cfootnote{in .44. innixe coexuntur \textit{πE\textsuperscript{a}}}}.
\pend

\pstart
\textbf{8} \edtext{\textit{MERITO.}}{\Cfootnote{Deinde. MERITO. \textit{V\textsuperscript{a}mv}}} Concludit tertio intentum, dicens \textit{quare non immerito \edtext{secularium}{\Cfootnote{\textit{om}. \textit{κ}}} ipsius ypates vel \edtext{condeterminatio}{\Cfootnote{conterminatio \textit{α} | confirmatio \textit{π}}} \edtext{seu}{\Cfootnote{vel \textit{P\textsuperscript{x}V\textsuperscript{a}}}} finis,} solent enim musici, et maxime psallentes, in diuinis \edtext{in fine}{\Cfootnote{\textit{om}. \textit{π}}} dicere seculorum amen. Est maius et diutius perdurans quam secularium reliquarum cordarum, quapropter \edtext{reliquis sicut}{\Cfootnote{reliquis sic \textit{π}}} \edtext{ipsis}{\Cfootnote{\textit{om}. \textit{π}}} existentibus breuis motus in ypate \edtext{longi}{\Cfootnote{longum \textit{V\textsuperscript{o}}}} \edtext{neatem}{\Cfootnote{neate \textit{πE\textsuperscript{a}P\textsuperscript{y}}}} omnium \edtext{conterminatur}{\Cfootnote{conterminantur \textit{M\textsuperscript{a}V\textsuperscript{z}}}} in \edtext{ipsam}{\Cfootnote{ipsa \textit{P\textsuperscript{y}V\textsuperscript{a}}}}. Et ideo eam solam accidit subsonare, vt componatur \edtext{huius}{\Cfootnote{huiusmodi \textit{V\textsuperscript{b}mv}}} sermo. \edtext{Ipsa}{\Cfootnote{illa \textit{E\textsuperscript{a}G\textsuperscript{u}P\textsuperscript{a}P\textsuperscript{z}} | ita \textit{γ}}} corda videtur sola subsonare \edtext{seu}{\Cfootnote{siue \textit{γπE\textsuperscript{a}}}} post alias resonare, cuius neate diutius permanet post cessationem aliarum neates, \edtext{talis}{\Cfootnote{tales \textit{P\textsuperscript{y}V\textsuperscript{a}}}} quidem, vt declaratum \edtext{est,}{\Cfootnote{est, est \textit{γV\textsuperscript{o}}}} ypate.
\pend

\pstart
\textbf{9} Nosce quod potest causa \edtext{huius}{\Cfootnote{huiusmodi \textit{V\textsuperscript{b}mv}}} \edtext{breuitatis}{\Cfootnote{breuiter \textit{λE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}} \edtext{assignari}{\Cfootnote{talis assignari \textit{E\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}}, \edtext{quia}{\Cfootnote{quod \textit{π}}} ypate sonum reddit grauem, qui extenditur secundum longum latum et profundum, reliqua vero ad ipsam acutum, qui secundum longum, \edtext{vt}{\Cfootnote{ceu \textit{γM\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{z}} | seu \textit{E\textsuperscript{a}V\textsuperscript{m}} | sicut \textit{V\textsuperscript{o}}}} visum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} \edtext{xi\textsuperscript{a}}{\Cfootnote{prius \textit{π}}}. Tardius \edtext{autem}{\Cfootnote{aut \textit{γ}}} quietatur sonus aeris secundum plures moti dimensiones quam secundum \edtext{unam}{\Cfootnote{unum \textit{π}}} cum dictum etiam sit graue tarde et multum moueri acutum econtra, vt \edtext{visum est sepius antea}{\Cfootnote{visum sepius antea \textit{V\textsuperscript{o}}}}.
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\subsection[Quadragesimumtertium problema … ]{Quadragesimumtertium problema}\label{part43}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID DELECTABILE VNIUS CANTUS EST SI AD LYRAM ET FISTULAM CANTANT? \textsuperscript{2}AUT QUIA OMNE DELECTABILE MIXTUM DELECTABILI VNUM EST? \textsuperscript{3}FISTULA ENIM DELECTABILIUS EST LYRA, QUARE ET CANTUS HOC MIXTO LYRA DELECTABILIUS VTIQUE ERIT, QUONIAM MIXTUM IMMIXTO DELECTABILIUS EST, SI VTRORUNQUE SENSUM ALIQUIS SIMUL ACCIPIAT. \textsuperscript{4}UINUM ENIM DELECTABILIUS EST OXIMELLE. PROPTER ID QUOD MIXTA SUNT MAGIS IPSIS QUE A NATURA MIXTA QUAM QUE A NOBIS, EST ENIM ET VINUM QUOD EST MIXTUM EX ACUTO ET DULCI HUMORE. MANIFESTANT AUTEM ET GRANATA VOCATA MUZA. \textsuperscript{5}CANTUS QUIDEM IGITUR ET FISTULA MISCENTUR SIBI IPSIS PROPTER SIMILITUDINEM, SPIRITU ENIM VTRAQUE FIUNT. SONUS AUTEM LYRE, QUONIAM AUTEM NON SPIRITU FIT. \textsuperscript{6}AUT MINUS SENSIBILE QUAM QUI EST FISTULE INCOMMIXTIBILIS EST VOCE? FACIENS AUTEM DIFFERENTIAM SENSUI MINUS DELECTAT, QUEMADMODUM IN SAPORIBUS DICTUM EST. \textsuperscript{7}AMPLIUS FISTULA QUIDEM MULTA SUO SONO ET SIMILITUDINE OCCULTAT PECCATORUM QUE SUNT CANTUS. LYRE AUTEM SONI, EXISTENTES TENUES ET INCOMMIXTIBILES VOCI SECUNDUM SE CONSIDERATI ET EXISTENTES SIBI IPSIS MANIFESTUM FACIUNT CANTUS PECCATUM, QUEMADMODUM REGULE EXISTENTES IPSORUM. INIUSTIS AUTEM IN CANTU CULPATIS, COMMUNE EX VTRISQUE NECESSARIE PEIUS FIERI}{\lemma{PROPTER \dots\ FIERI}\Cfootnote{PROPTER QUID DELECTABILE \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} cantus vnius cum altero mixtus vel cantus \edtext{vnicus}{\Cfootnote{uel \textit{V\textsuperscript{o}}}} delectabilis \edtext{et}{\Cfootnote{vel \textit{π}}} suauis existit, puta, si fuerit aliquis \edtext{cantans}{\Cfootnote{cantus \textit{γ}}} in sono lyre vel fistule?} Et adhuc delectabilior est cantus fistule quam lyre, \edtext{et}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{x}P\textsuperscript{y}}}} per se \edtext{et}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}V\textsuperscript{z}}}} \edtext{coniectus}{\Cfootnote{coniunctus \textit{πE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}}}} cantui, ita vt triplex \edtext{fiat}{\Cfootnote{fiat cantus \textit{π}}} comparatio: puta, cantus simplicis ad \edtext{compositum, et simplicis ad}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} simplicem, vt fistule ad lyram, et compositi ad compositum, sicut fistule et cantus ad lyram et cantum.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et primo assignat causam quare cantus \edtext{vnius}{\Cfootnote{vnus \textit{V\textsuperscript{o}}}} cum sono alterius vel cantus vnicus \edtext{sit}{\Cfootnote{fit \textit{P\textsuperscript{y}V\textsuperscript{b}}}} delectabilis, dicens \edtext{prius}{\Cfootnote{primo \textit{G\textsuperscript{u}V\textsuperscript{a}}}} causam esse \textit{quoniam delectabile \edtext{omne}{\Cfootnote{esse \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}} commixtum est, et \edtext{vnitum}{\Cfootnote{vnicum \textit{P\textsuperscript{x}V\textsuperscript{o}} | mixtum \textit{γ}}} ex diuersis}, vt declaratum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{o}V\textsuperscript{z}}}} \edtext{.38}{\Cfootnote{prius \textit{π}}}. Quod autem \edtext{sit}{\Cfootnote{sic \textit{V\textsuperscript{b}mv} | \textit{om}. \textit{π}}} delectabile vnum existit, cum \edtext{tunc}{\Cfootnote{tamen \textit{V\textsuperscript{o}}}} \edtext{vniformius}{\Cfootnote{uniformis \textit{κ}}} \edtext{sensum moueat}{\Cfootnote{moueat sensum \textit{π}}}.
\pend

\pstart
\textbf{4} \edtext{\textit{FISTULA}}{\Cfootnote{Deinde. FISTULA \textit{V\textsuperscript{a}mv}}} \textit{ENIM.} Comparat secundo fistulam ad lyram penes \edtext{delectationem}{\Cfootnote{delectationem et \textit{V\textsuperscript{o}}}}. Secundo assignat causam quare sonus fistule \edtext{est}{\Cfootnote{sit \textit{V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} delectabilior \edtext{eo qui}{\Cfootnote{quam \textit{π}}} lyre. \edtext{Secunda}{\Cfootnote{Secunda ibi \textit{γV\textsuperscript{z}} | Ibi \textit{π}}}: \textit{AUT.} Dicit \edtext{ergo}{\Cfootnote{igitur \textit{λE\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}}}} primo quod \textit{fistula est delectabilius instrumentum \edtext{auditu}{\Cfootnote{auditui \textit{π}}} quam lyra. Vnde etiam cantus commixtus \edtext{ex}{\Cfootnote{cum \textit{V\textsuperscript{o}}}} voce et fistula \edtext{est}{\Cfootnote{erit \textit{γE\textsuperscript{a}V\textsuperscript{o}}}} iocundior commixto ex voce et lyra. Et hoc quia illud \edtext{quod}{\Cfootnote{\textit{om}. \textit{γ}}} \edtext{est}{\Cfootnote{\textit{om}. \textit{γ}}} \edtext{mixtum}{\Cfootnote{conmixtum \textit{πM\textsuperscript{a}}}} \edtext{delectabilius}{\Cfootnote{delectabilius est \textit{λπE\textsuperscript{a}P\textsuperscript{y}}}} \edtext{eo quod}{\Cfootnote{eo quod est \textit{G\textsuperscript{u}P\textsuperscript{p}V\textsuperscript{z}}}} simplex et immixtum, dummodo mixtura vtrorumque mixtibilium sentiatur vtcunque existens \edtext{tamen}{\Cfootnote{\textit{om}. \textit{π}}} in quadam \edtext{conmixtione}{\Cfootnote{coniunctione \textit{γP\textsuperscript{y}} | soni notione \textit{πE\textsuperscript{a}}}}.}
\pend

\pstart
\textbf{5} \edtext{\textit{UINUM.}}{\Cfootnote{VNDE. \textit{π} | Deinde. VNUM. \textit{V\textsuperscript{a}mv}}} Declarat secundo \edtext{id}{\Cfootnote{\textit{om}. \textit{γπ}}} in mixtione nature \edtext{et artis}{\Cfootnote{naturis \textit{κ}}}, dicens quod \edtext{\textit{vinum} quod}{\Cfootnote{unum quod \textit{P\textsuperscript{p}V\textsuperscript{a}} | vnumquodque \textit{V\textsuperscript{b}mv} | illud quod \textit{G\textsuperscript{u}P\textsuperscript{q}}}} est simplex arte, conmixtum tamen natura, \textit{\edtext{est delectabilius}{\Cfootnote{delectabilius est \textit{P\textsuperscript{y}V\textsuperscript{a}}}} \edtext{oximelle}{\Cfootnote{ex melle \textit{E\textsuperscript{a}V\textsuperscript{o}}}} seu acetimelle}, quod \edtext{est}{\Cfootnote{est in \textit{γ}}} arte constructum ex melle aceto et aqua. \textit{Propter quod \edtext{vino}{\Cfootnote{vinum \textit{M\textsuperscript{a}V\textsuperscript{m}}}} ipsi sunt amplius \edtext{mixta}{\Cfootnote{conmixta \textit{π}}} ea que sunt nature quam in mixtione artificiali per nos \edtext{constructa}{\Cfootnote{considerata \textit{γ}}},} sicut \edtext{illa}{\Cfootnote{ea \textit{π}}} que \edtext{oximellis}{\Cfootnote{oximelle \textit{γ} | ex mellis \textit{πP\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{m}}}}. In priori causa \edtext{vera extat}{\Cfootnote{extat vera \textit{E\textsuperscript{a}V\textsuperscript{o}}}} \edtext{mixtio}{\Cfootnote{commixtio \textit{π}}} in oximelle, \edtext{vero}{\Cfootnote{vero secundum \textit{π}}} \edtext{sensualis}{\Cfootnote{sensum \textit{V\textsuperscript{o}}}} tantum. \textit{Nam \edtext{vinum, quod}{\Cfootnote{vnumquodque \textit{G\textsuperscript{u}P\textsuperscript{q}}}} est \edtext{mixtum quoddam}{\Cfootnote{quoddam mixtum \textit{π}}} \edtext{nature}{\Cfootnote{naturale \textit{π}}}, componitur ex humore acuto et dulci \edtext{mixtione}{\Cfootnote{commixtione \textit{πE\textsuperscript{a}}}} sufficienter. \edtext{Circa}{\Cfootnote{Certa \textit{λ}}} quod manifestant granata nominata musa seu messa}, id est media, sunt enim tria specie: dulcia, \edtext{acida, et}{\Cfootnote{acuta, et \textit{λ}}} media (conmixta ex dulci et acetoso sapore). Et \edtext{ideo}{\Cfootnote{\textit{om}. \textit{π}}} magis delectant \edtext{quando}{\Cfootnote{quam \textit{λ}}} in vno tertio facta sunt natura talia quam si miscerentur arte dulcia cum acidis. Quidam quoque \edtext{habet}{\Cfootnote{habent \textit{πM\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{z}}}} \edtext{ex melle}{\Cfootnote{oximelle \textit{G\textsuperscript{u}P\textsuperscript{p}} | \textit{om}. \textit{V\textsuperscript{o}}}} loco eius quod \edtext{expositum}{\Cfootnote{expositum est \textit{γM\textsuperscript{a}V\textsuperscript{o}}}} \edtext{ex melle}{\Cfootnote{oximelle \textit{αM\textsuperscript{a}V\textsuperscript{z}} | melle \textit{V\textsuperscript{o}}}}, quod non \edtext{sic}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}V\textsuperscript{o}}}} approbo, \edtext{quamuis id}{\Cfootnote{quamuis \textit{V\textsuperscript{o}}}} \edtext{exposuerim}{\Cfootnote{exposuerint \textit{P\textsuperscript{y}mv} | expositum sit \textit{π}}} quandoque.
\pend

\pstart
\textbf{6} \edtext{\textit{CANTUS.}}{\Cfootnote{Deinde. CANTUS. \textit{V\textsuperscript{a}mv}}} Applicat ad propositum \edtext{concludendo}{\Cfootnote{\textit{om}. \textit{π}}}, dicens itaque \textit{cantus \edtext{in}{\Cfootnote{et \textit{γP\textsuperscript{y}}}} fistula \edtext{commiscetur}{\Cfootnote{commiscentur \textit{απ}}} sibi ipsis propter similitudinem quam habent ad inuicem. Quod ostenditur \edtext{quia}{\Cfootnote{quoniam \textit{G\textsuperscript{u}P\textsuperscript{p}} | quando \textit{V\textsuperscript{o}}}} vtrique, vt cantus et fistule, soni perficiuntur flatu ex ore, namque prodeunt. \edtext{Sed}{\Cfootnote{Quod \textit{P\textsuperscript{u}P\textsuperscript{z}}}} sonus lyre non, sed \edtext{potius}{\Cfootnote{\textit{om}. \textit{π}}} tactu \edtext{perficitur}{\Cfootnote{construitur \textit{λE\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}} rotatio.} Que autem similia sunt ad inuicem delectant amplius, cum \edtext{delitia}{\Cfootnote{delitie \textit{P\textsuperscript{x}V\textsuperscript{z}} | delectatio \textit{π}}} \edtext{causentur}{\Cfootnote{causetur \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{o}}}} ex coniunctione conuenientis cum \edtext{conueniente}{\Cfootnote{conuenienti \textit{V\textsuperscript{o}V\textsuperscript{z}}}} quarto \textit{de \edtext{accidenti}{\Cfootnote{accidente \textit{πP\textsuperscript{y}}}}}.
\pend

\pstart
\textbf{7} Nosce quod mixtio vini proportionatur sono composito \edtext{et}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} facto ex fistula et voce, que autem \edtext{ex oximellis}{\Cfootnote{oximellis \textit{M\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{o}} | ex melle \textit{E\textsuperscript{a}P\textsuperscript{u}}}} simplici aut composito ex voce et \edtext{lyra}{\Cfootnote{lyra composito \textit{λ}}}, respectu alterius qui ex fistula.
\pend

\pstart
\textbf{8} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit \edtext{assignans}{\Cfootnote{et assignat \textit{π}}} \edtext{causam}{\Cfootnote{aliam causam \textit{π}}} quare sonus lyre non \edtext{vt qui fistule est}{\Cfootnote{vt qui fistule \textit{E\textsuperscript{a}P\textsuperscript{y}} | est vt qui fistule \textit{π}}} delectabilis. \edtext{Secunda: \textit{AMPLIUS}.}{\Cfootnote{Secunda ibi: \textit{AMPLIUS}. \textit{γ} | \textit{om}. \textit{π}}} \edtext{Dicit}{\Cfootnote{Dicens \textit{πE\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{m}}}} \edtext{primo}{\Cfootnote{prius \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{b}V\textsuperscript{o}V\textsuperscript{z}}}} \edtext{id}{\Cfootnote{ideo \textit{π}}} esse \textit{quoniam sonus lyre minus est sensibilis quam sonus fistule, \edtext{quam}{\Cfootnote{quare \textit{M\textsuperscript{a}V\textsuperscript{a}}}} minus, \edtext{enim}{\Cfootnote{eius \textit{M\textsuperscript{a}V\textsuperscript{m}}}} \edtext{componibilis}{\Cfootnote{ponibilis \textit{V\textsuperscript{o}}}} est cum voce, \edtext{habet}{\Cfootnote{hec \textit{πE\textsuperscript{a}}}} enim \edtext{differentiam}{\Cfootnote{dicta \textit{π}}}, vt \edtext{iam}{\Cfootnote{\textit{om}. \textit{π}}} visum est ad ipsam. Ex illo autem quod differentiam et diuersitatem ad sensum habet minor sentitur delectatio, sicut apparuit in saporibus arte \edtext{et natura mixtis}{\Cfootnote{et natura in mixtis \textit{γE\textsuperscript{a}} | commixtis et natura \textit{π}}}.}
\pend

\pstart
\textbf{9} \edtext{\textit{AMPLIUS.}}{\Cfootnote{Deinde. AMPLIUS. \textit{V\textsuperscript{a}mv}}} Assignat causam \edtext{secundam}{\Cfootnote{duplam \textit{γP\textsuperscript{x}}}}, dicens quod \textit{fistula propter suum sonum et eius similitudinem,} quia sua fortitudine soni \edtext{extollit}{\Cfootnote{excellit \textit{G\textsuperscript{u}P\textsuperscript{p}V\textsuperscript{b}}}} vocem cantantis, \textit{obtegit multa peccata in cantu contingentia. Sed soni lyre, \edtext{quia}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} subtiles sunt et parui et incommiscibiles voci, cum \edtext{secundum}{\Cfootnote{per \textit{απ}}} se \edtext{considerantur}{\Cfootnote{considerentur \textit{π}}}, \edtext{prout}{\Cfootnote{cum \textit{G\textsuperscript{u}P\textsuperscript{p}}}} in seipsis extant et non in aliis, \edtext{quibus}{\Cfootnote{qui hiis \textit{πE\textsuperscript{a}}}} \edtext{occultentur}{\Cfootnote{occultantur \textit{γπM\textsuperscript{a}}}}, \edtext{manifestum est quod}{\Cfootnote{manifestum quod \textit{E\textsuperscript{a}V\textsuperscript{o}} | Minor quia \textit{γ}}} circa ipsos peccatum accidit in cantu, \edtext{nam}{\Cfootnote{non \textit{πE\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} in ipso, sicut in aliis scientiis sunt canones quidam per quos armonice canitur, a quibus cum receditur contingit in cantu peccare. Quare subditur quod \edtext{accidit}{\Cfootnote{acutum \textit{π}}} \edtext{iniustis}{\Cfootnote{in instrumentis \textit{P\textsuperscript{y}} | in tristis \textit{V\textsuperscript{o}} | in istis \textit{P\textsuperscript{z}V\textsuperscript{a}}}} non recte predictos canones \edtext{obseruantes}{\Cfootnote{obseruantibus \textit{απ}}} alia littera multis ac melior, vt peccent et culpentur in cantu. Quod enim commune \edtext{existit}{\Cfootnote{ex istis \textit{πE\textsuperscript{a}}}}, vt ex vtrisque \edtext{confectum}{\Cfootnote{cum factum \textit{M\textsuperscript{a}V\textsuperscript{b}}}}, puta voce et instrumento sicut lyra, \edtext{necessarie}{\Cfootnote{necesse est \textit{M\textsuperscript{a}V\textsuperscript{z}}}} deterius perfici} eorum \edtext{consonantia}{\Cfootnote{dissonantia \textit{λE\textsuperscript{a}} | distantia \textit{γ}}}, \edtext{contra}{\Cfootnote{circa \textit{M\textsuperscript{a}P\textsuperscript{y}}}} regulas armonicas factum.
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\subsection[Quadragesimumquartum problema … ]{Quadragesimumquartum problema}\label{part44}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID .VII. QUIDEM MEDIA VOCATUR, OCTO AUTEM NON EST MEDIUM? \textsuperscript{2}AUT QUIA .VII. CORDARUM FUERUNT ANTIQUITUS CONSONANTIE, .VII. AUTEM HABET MEDIUM? \textsuperscript{3}AMPLIUS EORUM QUE SUNT IN MEDIO EXTREMITATUM, MEDIUM SOLUM PRINCIPIUM QUODDAM EST, EST ENIM ALTERAM EXTREMITATUM DECLINANTIUM IN ALIQUA DISTANTIA PER MEDIUM QUOD PRINCIPIUM EST: HOC EST MEDIUM. QUONIAM AUTEM VLTIMA MEDIUM EST CONSONANTIE NEATE ET YPATE, HORUM AUTEM IN MEDIO RELIQUI SONI, QUORUM MEDIA VOCATA SOLA PRINCIPIUM EST ALTERIUS TETRACORDI, IUSTE MEDIA APPELLATUR. \textsuperscript{4}EORUM AUTEM QUE SUNT IN MEDIO QUARUNDAM EXTREMITATUM MEDIUM ERAT PRINCIPIUM SOLUM}{\lemma{PROPTER \dots\ SOLUM}\Cfootnote{PROPTER QUID SEPTEM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\textbf{2} Repetit .25. \edtext{et}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{vtcunque}{\Cfootnote{vtrunque \textit{M\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}}}} \edtext{.33.}{\Cfootnote{\textit{om}. \textit{π}}}, dicens \textit{quare 7 dicitur habere medium et cetera.}
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} \edtext{Soluit}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}}}} et primo assignat causam, \edtext{et iam}{\Cfootnote{et inde \textit{V\textsuperscript{b}mv} | que iam \textit{π}}} apparuit secundo. Quia mentionem fecerat de medio, ostendit \textit{quomodo \edtext{accipiatur}{\Cfootnote{accipitur \textit{V\textsuperscript{a}V\textsuperscript{z}} | accipiuntur \textit{V\textsuperscript{o}}}} medium cum dicitur .7. habere medium} cuius gratia problema repetiit.
\pend

\pstart
\textbf{4} \edtext{Secunda}{\Cfootnote{Secunda ibi \textit{γV\textsuperscript{z}} | Ibi \textit{G\textsuperscript{u}P\textsuperscript{p}P\textsuperscript{u}}}} \textit{AMPLIUS.} \edtext{Dicit}{\Cfootnote{Dicens \textit{π}}} \edtext{igitur}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} quod \textit{vno modo dicitur medium quoddam principium \edtext{existens medium}{\Cfootnote{\textit{om}. \textit{π}}} inter duo extrema, nam illud principium \edtext{quod}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} per quandam distantiam equaliter declinat ab vtroque extremorum dicitur} \edtext{\textit{medium,}}{\Cfootnote{medium, inter duo extrema \textit{π}}} sicut in \edtext{aliqua}{\Cfootnote{alia \textit{πE\textsuperscript{a}M\textsuperscript{a}}}} \edtext{consonantia}{\Cfootnote{distantiam \textit{π}}} est dare duo \edtext{ultima}{\Cfootnote{extrema \textit{π}}} vt vnus \edtext{sit sonus}{\Cfootnote{sonus sit \textit{π}}} cantus ab ypate fortis existens, \edtext{secundus}{\Cfootnote{alius \textit{π}}} a neate siue nete, inter quos est dare plures sonos medios distantes ab extremis secundum \edtext{maius}{\Cfootnote{magis \textit{απM\textsuperscript{a}}}} et minus, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}}}} inter vnum et 7. que intermedia vt 5. reliqua dicantur media.
\pend

\pstart
\textbf{5} \edtext{Notandum}{\Cfootnote{Nota \textit{G\textsuperscript{u}P\textsuperscript{p}}}} quod \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} medium dicitur principium eo quod est habitus quidam virtutis \edtext{qui}{\Cfootnote{que \textit{π}}} principium extat actuum, cum sit diuisibile habens quantitatem. Est \edtext{et}{\Cfootnote{etiam \textit{π}}} aliud medium \edtext{vere}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{indiuisibile}{\Cfootnote{indiuisibile natura \textit{π}}}; quale medium \edtext{dicitur}{\Cfootnote{dicuntur \textit{απP\textsuperscript{x}}}} \edtext{solum illa habere}{\Cfootnote{solum illi habere \textit{γ} | illa sola habere \textit{π}}} quorum medium est principium alterius tetracordi. Quod non appellatur vere principium, sed medium magis, sicut positis .7. cordis, quarta, \edtext{quidem}{\Cfootnote{quidam \textit{P\textsuperscript{y}mv}}} vt mese, repetita cum tribus superioribus et inferioribus etiam tribus, constituuntur duo tetracorda tunc mese vtrobique \edtext{vere}{\Cfootnote{\textit{om}. \textit{π}}} medium extat. \edtext{verbi gratia}{\Cfootnote{verbi gratia si ordinantur \textit{V\textsuperscript{o}}}}
\pend

\pstart
\ledpb
\begin{edarrayc}
& \textrm{Ypate}. & \\
& \textrm{Parypate}. & \\
& \textrm{Lychanos}. & \\
& \textrm{Mese}. &
\edatright[\textrm{primum tetracordum}.]{\}}{2\baselineskip}
\end{edarrayc}

\begin{edarrayc}
& \textrm{Paramese}. & \\
& \textrm{Paranete}. & \\
& \textrm{Nete}. &
\edatright[\textrm{secundum tetracordum}.]{\}}{2\baselineskip}
\end{edarrayc}
\pend

\pstart
Et \edtext{huius}{\Cfootnote{huiusmodi \textit{γP\textsuperscript{y}}}} duo \edtext{quidem}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} tetracorda secundum sinzeusim \edtext{ordinantur}{\Cfootnote{\textit{om}. \textit{λπ}}}, cum autem \edtext{fuerint}{\Cfootnote{fuerit \textit{E\textsuperscript{a}V\textsuperscript{o}}}} \edtext{.8.}{\Cfootnote{.4. \textit{π}}} corde secundum dyazeusim computantur.
\pend


\pstart
\textbf{6} \edtext{\textit{EORUM.}}{\Cfootnote{Deinde. EORUM. \textit{V\textsuperscript{a}mv}}} Tunc repetit \textit{medium prius horum}, \edtext{et}{\Cfootnote{etiam \textit{λ}}} apparet sic igitur medium est duplex: vnum quidem geometricum siue quo ad nos aut \edtext{per abnegationem}{\Cfootnote{paranegomon \textit{γ}}} extremorum, aliud vero arismetricum seu \edtext{rei}{\Cfootnote{rei vel \textit{γE\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}}}} per indiuisibilem equidistantiam ab extremis.
\pend

\pstart
\textbf{7} Apparet denique quod secundum primum medium potest dici dyaocto. Vnde primo \textit{musice}: ``in octocordo \edtext{due semper}{\Cfootnote{duo semper \textit{πE\textsuperscript{a}}}} \edtext{medie}{\Cfootnote{\textit{om}. \textit{π}}} reperiuntur, et \edtext{non}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} vna media potest inueniri.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Boeth. \textit{De inst. mus.} Ι.20 ((Friedlein p. 208, ll. 8–10))}}
\pend

\pstart
\vspace{1em}
\begin{center}
    {\Large \ding{167}}
\end{center}
\pend
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\newpage

\subsection[Quadragesimumquintum problema … ]{Quadragesimumquintum problema}\label{part45}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID MULTI CANTANTES SALUANT MAGIS RITHIMUM QUAM PAUCI? \textsuperscript{2}AUT QUIA MAGIS AD VNUM, AUT ET RECTOREM, INTENDUNT ET TARDIUS INCIPIUNT, QUARE FACILIUS IPSUM ADIPISCUNTUR? IN VELOCI QUIDEM IGITUR MAGIS FIT PECCATUM. \textsuperscript{3}ACCIDIT AUTEM DUCI INTENDERE MULTOS. APPROPRIATUS AUTEM NULLUS VTIQUE IPSORUM ILLUSTRABITUR, ELEUANS MULTITUDINEM. IN PAUCIS AUTEM MAGIS ILLUSTRANTUR, PROPTER QUΟD SECUNDUM SE IN SE IPSIS MAGIS EXERCITANTUR QUAM APUD DUCEM}{\lemma{PROPTER \dots\ DUCEM}\Cfootnote{PROPTER QUID MULTIPLICANTES \textit{π} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} Repetit \edtext{.22.}{\Cfootnote{.22. problema \textit{γ}}}.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT QUIA. \textit{V\textsuperscript{a}mv}}} Soluit et primo assignat causam \edtext{et iam visum est}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}. Secundo declarat assignando differentiam inter habentes ducem in cantu et non habentes. Cuius causa reiterauit sermonem: \edtext{secunda}{\Cfootnote{secunda ibi \textit{γV\textsuperscript{z}} | ibi \textit{G\textsuperscript{u}P\textsuperscript{p}}}} \textit{ACCIDIT}. \edtext{Dicit}{\Cfootnote{Dicens \textit{P\textsuperscript{y}V\textsuperscript{a}}}} \edtext{secundo contingere quod}{\Cfootnote{hoc contingere quia \textit{π}}} \textit{multi ad ducem \edtext{attendant}{\Cfootnote{attendunt \textit{π}}}} vt eorum cantus \edtext{saluetur}{\Cfootnote{saluentur \textit{V\textsuperscript{o}}}} per amplius, quoniam cum fuerint multi, \textit{nullus ipsorum determinatur, vt eleuando vocem suam propter multitudinem, non paruam \edtext{poterit}{\Cfootnote{\textit{om}. \textit{π}}} illustrari et super aliorum voces \edtext{suam}{\Cfootnote{suas \textit{γ}}} extollere, cum plurimi magis valeant quam vnus appropriatus. Sed in paucis cantantibus id \edtext{poterit}{\Cfootnote{potest \textit{P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}} sufficienter contingere, quare pauci cantantes plus secundum semet ipsos exercitantur et singulariter \edtext{apparent}{\Cfootnote{apparet \textit{G\textsuperscript{u}P\textsuperscript{y}P\textsuperscript{z}V\textsuperscript{a}}}} in cantu suo \edtext{illustrari}{\Cfootnote{illustrati \textit{απ}}} quam \edtext{qui}{\Cfootnote{\textit{om}. \textit{π}}} apud ducem plures cantantes existunt, \edtext{non}{\Cfootnote{\textit{om}. \textit{π}}} enim \edtext{singulatim}{\Cfootnote{sigillatim \textit{απV\textsuperscript{z}}}} sed ad ducem exercitantur.}
\pend

\pstart
\vspace{1em}
\begin{center}
    {\Large \ding{167}}
\end{center}
\pend
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\newpage

\subsection[Quadragesimumsextum problema … ]{Quadragesimumsextum problema}\label{part46}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID IN ACUTUM DISSONANT PLURIMI? \textsuperscript{2}AUT QUIA FACILIUS ACUTUM DIMITTERE QUAM GRAUE? \textsuperscript{3}CANTANT IGITUR ACUTE MAGIS, ET IN QUO CANTANT PECCANT}{\lemma{PROPTER \dots\ PECCANT}\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} \edtext{Huius}{\Cfootnote{Huius problematis \textit{λ}}} precessit expositio .26\textsuperscript{to}. Et veritas \edtext{illic iam}{\Cfootnote{illic \textit{λ} | hic iam \textit{πE\textsuperscript{a}}}} \edtext{apparuit}{\Cfootnote{apparet \textit{π}}} cuiusdam apparentis contrarietatis inter \edtext{.37.}{\Cfootnote[nosep]{\textit{emend}. .27. \textit{ακπ}}}, et istud cum dictum sit \edtext{ibidem}{\Cfootnote{ibi \textit{π}}} difficilius esse \edtext{acute cantare}{\Cfootnote{cantare acute \textit{π}}} quam grauiter. Hic autem videtur contrarium dici.
\pend

\pstart
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    {\Large \ding{167}}
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\newpage

\subsection[Quadragesimumseptimum problema … ]{Quadragesimumseptimum problema}\label{part47}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID ANTIQUI, .VII. CORDARUM FACIENTES CONSONANTIAS, RELIQUERUNT YPATEM, SED NON NITEM RELINQUERUNT? \textsuperscript{2}AUT NON YPATEM, SED NUNC PARAMESEN APPELLATAM, ABSTULERUNT ET TONILEM DISTANTIAM. VTEBANTUR AUTEM VLTIMA MEDIA DENSI IN ACUTUM, PROPTER QUOD ET MEDIAM IPSAM APPELLAUERUNT. \textsuperscript{3}AUT QUIA ERAT SUPERIORIS TETRACORDI FINIS, INFERIORIS AUTEM PRINCIPIUM, ET MEDIAM HABEBAT PROPORTIONEM TONO EXTREMITATUM}{\lemma{PROPTER \dots\ EXTREMITATUM}\Cfootnote{PROPTER QUID ANTIQUI \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}?}
\pend

\pstart
\textbf{2} Repetit .7., et apparuit iam \edtext{eius}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{o}}}} \edtext{expositio}{\Cfootnote{propositio \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}}}}.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} \edtext{Soluit}{\Cfootnote{Soluit assignans \textit{κ}}} \edtext{dupliciter}{\Cfootnote{causam dupliciter \textit{M\textsuperscript{a}V\textsuperscript{z}}}}, aliter \edtext{tamen}{\Cfootnote{causam \textit{V\textsuperscript{o}}}} quam prius, dicens primo quod \textit{non abstulerunt \edtext{de 7 cordo ypatem, sed non pro neque cordam appellatam \edtext{paramese}{\Cfootnote{paramesem \textit{λV\textsuperscript{a}}}} seu tritem. Sed abstulerunt}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}}, vt propositum est, \edtext{nitem}{\Cfootnote{nitem vel netem \textit{V\textsuperscript{o}}}} et per consequens \edtext{toni}{\Cfootnote{communi \textit{M\textsuperscript{a}P\textsuperscript{x}}}} distantiam,} que attenditur ab vna corda in aliam \edtext{siue}{\Cfootnote{seu \textit{π}}} ab vna voce in \edtext{alteram}{\Cfootnote{aliam \textit{λπE\textsuperscript{a}V\textsuperscript{a}}}}, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} vt re. Constituunt enim due voces tonum ita quod vna corda cassata tollitur vox vna, et per consequens tonus, cum ex duabus constet vocibus. Auferentes autem nete, ne armonie destruerentur, cum media corda paramese, \edtext{puta}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} primo modo \edtext{medii}{\Cfootnote{media \textit{πE\textsuperscript{a}}}} \edtext{dicto}{\Cfootnote{dicta \textit{G\textsuperscript{u}P\textsuperscript{q}}}} 44°., supplebant defectum nete sublate. \textit{Vtebantur enim media, scilicet paramese, loco \edtext{ultime}{\Cfootnote{ultimo \textit{P\textsuperscript{y}mv}}}, \edtext{scilicet nete}{\Cfootnote{videlicet nete \textit{αλV\textsuperscript{o}} | paranete scilicet \textit{G\textsuperscript{u}P\textsuperscript{q}}}}, quod erat \edtext{cum}{\Cfootnote{quod \textit{M\textsuperscript{a}V\textsuperscript{a}}}} extendebant \edtext{eam}{\Cfootnote{eum \textit{V\textsuperscript{o}}}} in tantum vt sonum redderet acutum,} ad modum nete, ita quod cum ypate \edtext{constituerent}{\Cfootnote{constituerunt \textit{P\textsuperscript{y}V\textsuperscript{o}}}} dyapason, sicut cum nete preexistente. Et merito quoniam corda \edtext{extensior}{\Cfootnote{exterior \textit{V\textsuperscript{b}mv}}} acutiorem reddit sonum, grauior autem tardiorem, \textit{quare mediam appellauere,} eo quod medium habet virtutem extremi ad ipsum comparatum. Si namque minus extensa plus tenet locum medii, amplius vero extremi, aut \edtext{erat}{\Cfootnote{erit \textit{πV\textsuperscript{b}}}} policordum diegeugeneum, id est disiunctum, ubi cordarum \edtext{reiterantur alique}{\Cfootnote{alique reiterantur \textit{π}}} vel duplantur, vt \edtext{apparuit}{\Cfootnote{apparebit \textit{V\textsuperscript{o}} | apparebat \textit{P\textsuperscript{a}P\textsuperscript{z}}}} \edtext{tertio}{\Cfootnote{30 \textit{V\textsuperscript{o}} | 13 \textit{α}}}.
\pend

\pstart
\textbf{4} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Assignat aliam causam, dicens \textit{quia media \edtext{corda}{\Cfootnote{cordarum \textit{G\textsuperscript{u}P\textsuperscript{q}}}},} \edtext{vt}{\Cfootnote{vel \textit{G\textsuperscript{u}P\textsuperscript{q}}}} paramese virtute \edtext{quidem}{\Cfootnote{quadam \textit{π}}} et non distantia, vt \edtext{tactum}{\Cfootnote{tactum est \textit{π}}}, \textit{erat finis \edtext{tetracordi superioris et principium alterius}{\Cfootnote{\textit{om}. \textit{V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{b}}}} tetracordi inferioris, vt \edtext{visum}{\Cfootnote{dictum est vel visum \textit{G\textsuperscript{u}P\textsuperscript{p}}}} est. Que, secundum quod \edtext{existit media}{\Cfootnote{media existit \textit{V\textsuperscript{o}} | existit mediam \textit{γ}}}, habet proportionem ad extremitates ambas suo tono,} secundum quod fuerit isti extremorum comparata vel alteri. Nam medium comparatum ad \edtext{extrema}{\Cfootnote{extremum \textit{γ} | vnum extremorum \textit{π}}} videtur alterum, quinto phisicorum, aut \edtext{ordinabatur}{\Cfootnote{ordinabam \textit{γ}}} paramese \edtext{ante mese}{\Cfootnote{ante mesem \textit{V\textsuperscript{o}} | aut in mese \textit{γ}}}, vt \edtext{distantia}{\Cfootnote{destructa \textit{πE\textsuperscript{a}P\textsuperscript{x}}}} \edtext{etiam}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} fere sit media. \edtext{Sic}{\Cfootnote{Sicut \textit{V\textsuperscript{m}V\textsuperscript{z}}}} igitur \edtext{quia}{\Cfootnote{\textit{om}. \textit{π}}} paramese habet \edtext{vicem}{\Cfootnote{nitem \textit{P\textsuperscript{x}V\textsuperscript{m}}}} extremorum vt medium, et quia \edtext{per eam}{\Cfootnote{postea \textit{γ}}} ambo constituuntur tetracordi virtute saltem. Siue nete dubitatur, quia dictum est in 7. quod non abstulerunt nitem, \edtext{sed}{\Cfootnote{secundum \textit{λV\textsuperscript{a}}}} tertiam id est \edtext{tritem}{\Cfootnote{netem \textit{κ}}} siue paramese.
\pend

\pstart
\textbf{5} Dicendum quod antiquitus \edtext{fuit primitus}{\Cfootnote{fuit inuentus \textit{γ} | fuit prius \textit{π}}} paramese quam nete inuenta, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} pretactum \edtext{est}{\Cfootnote{est in \textit{V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} 13°. Deinde \edtext{superueniens}{\Cfootnote{perueniens \textit{V\textsuperscript{o}} | veniens \textit{G\textsuperscript{u}P\textsuperscript{p}}}} tripandius, extrahens tertiam cordam vt \edtext{paramese}{\Cfootnote{paramesem \textit{E\textsuperscript{a}V\textsuperscript{o}V\textsuperscript{z}}}} a cordarum ordine, netem loco illius adiunxit ita vt primitus non fuerit nete, postea vero adinuenta extiterit. Et sic \edtext{vtrunque}{\Cfootnote{vtraque \textit{λ}}} verificabitur \edtext{particula}{\Cfootnote{problema \textit{γπP\textsuperscript{y}}}}.
\pend

\pstart
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\newpage

\subsection[Quadragesimumoctauum problema … ]{Quadragesimumoctauum problema}\label{part48}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID IN TRAGEDIA CHORI NON YPODORISTIUM NEQUE YPOFRIDISTIE CANTANT? \textsuperscript{2}AUT QUIA CANTUM MINIME HABENT HEE CONSONANTIE, QUO INDIGENT MAGIS CHORO? \textsuperscript{3}CONSUETUDINEM AUTEM CONSONANTIAM QUE QUIDEM EST YPOFRIDISTIUM OPERATIUUM, PROPTER QUOD ET IN GERIONEM EXITUS ET DEARMATIO IN HAC FACIEBANT. \textsuperscript{4}YPODORISTIUM AUTEM MAGNIFICUM ET STABILE, PROPTER QUOD ET CITHARISTISSIMA EST CONSONANTIARUM. \textsuperscript{5}HEC AUTEM VTRAQUE CHORO QUIDEM IN CONSONANTIA, EIS AUTEM QUE SUNT A SCENA MAGIS PROPRIA. ILLI ENIM HEROUM MUTATIUI, HII AUTEM DUCES ANTIQUORUM SOLI FUERUNT. \textsuperscript{6}PROPRII AUTEM HOMINES QUORUM EST CHORUS. PROPTER QUOD ET CONUENIT IPSI LAMENTABILIS ET QUIETUS MOS ET MELOS, HUMANA ENIM. \textsuperscript{7}HEC AUTEM HABENT ALIE CONSONANTIE, MINUS AUTEM ILLIS QUE EST YPOFRIGISTIUM, DIUINA VEL FURIOSA AUTEM ET BACHICHA. SECUNDUM HANC QUIDEM PATIMUR ALIQUID, PASSIUI AUTEM IMBECILLES MAGIS FORTIBUS SUNT, PROPTER QUOD ET IPSA CONUENIT CHORIS. SECUNDUM AUTEM EAM QUE EST YPODORISTIUM ET YPOFRIDISTIUM OPERAMUR QUOD NON EST PROPRIUM CHORO. \textsuperscript{8}EST AUTEM CHORUS CURATOR INOPERABILIS, BENIUOLENTIAM ENIM SOLUM PREBET QUIBUS ADEST}{\lemma{PROPTER \dots\ ADEST}\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
\pend

\pstart
\textbf{2} Repetit .30. vt amplius \edtext{exponat}{\Cfootnote{apponat \textit{V\textsuperscript{o}}}} \edtext{dicens}{\Cfootnote{\textit{om}. \textit{π}}} \textit{quare chori secundum \edtext{tragediam}{\Cfootnote{tragedia \textit{E\textsuperscript{a}V\textsuperscript{o}}}} non cantant cantum ypodoristium et multo minus ypofrigium?} Que autem ista \edtext{exposita sunt}{\Cfootnote{expositum est \textit{π}}} .30°. \edtext{antea}{\Cfootnote{\textit{om}. \textit{πV\textsuperscript{a}}}}.
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit et primo in communi dicens causam esse \textit{quia \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} consonantie, puta ypodoristie \edtext{ac}{\Cfootnote{et \textit{V\textsuperscript{o}}}} ypofrigie, non habent modum cantandi quo amplius indigent cantantes cantu chorico.} \edtext{Alie}{\Cfootnote{Alia \textit{E\textsuperscript{a}P\textsuperscript{x}P\textsuperscript{y}V\textsuperscript{o}}}} \edtext{namque}{\Cfootnote{enim \textit{π}}} consonantie proportione indigent chori, et \edtext{alie}{\Cfootnote{alia \textit{βλπV\textsuperscript{a}}}} leges comedice, sub quibus ypofrigius \edtext{continetur}{\Cfootnote{continentur \textit{P\textsuperscript{x}P\textsuperscript{y}}}} cantus et ypodoricus, vt apparuit .15°.
\pend

\pstart
\textbf{4} \edtext{\textit{CONSUETUDINEM.}}{\Cfootnote{CONSUETUDINE. \textit{π} | Deinde. CONSUETUDINEM. \textit{V\textsuperscript{a}mv}}} Solutionem declarat secundo in speciali, postea notificat quid sit chorus. \edtext{Secunda}{\Cfootnote{Secunda ibi \textit{γV\textsuperscript{z}}}} \textit{EST AUTEM.} Adhuc \edtext{ostendit}{\Cfootnote{primo ostendit \textit{V\textsuperscript{o}}}} quales \edtext{sint}{\Cfootnote{sunt \textit{π}}} cantus ypodoristii et ypofrigii, et \edtext{quorum}{\Cfootnote{chorum \textit{γ}}} secundo chorici. \edtext{Secunda}{\Cfootnote{Secunda ibi \textit{γV\textsuperscript{z}} | Ibi \textit{G\textsuperscript{u}P\textsuperscript{p}} | \textit{om}. \textit{V\textsuperscript{o}}}} \textit{\edtext{PROPRII}{\Cfootnote{PROPRIA \textit{π}}} AUTEM.} Dicit primo quod ex consuetudine longa est \edtext{consonantiam}{\Cfootnote{consonantia \textit{E\textsuperscript{a}V\textsuperscript{a}V\textsuperscript{m}}}} quandam percipere in cantu ab infantia, \textit{enim mox audiuntur.} \textit{Et ideo leges cantus \edtext{quidam fuerunt}{\Cfootnote{quidem fuerunt \textit{π}}} appellati,} vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} 28°. Que quidem \textit{consonantia consuetudinaria secundum cantum ypodoristium et ypofrigium attenditur maxime, cum operatiuus existat.} \edtext{Ipso}{\Cfootnote{Ipsi \textit{G\textsuperscript{u}P\textsuperscript{p}}}} enim laborantes \edtext{incitantur}{\Cfootnote{excitantur \textit{V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{z}} | exercitantur \textit{G\textsuperscript{u}P\textsuperscript{p}M\textsuperscript{a}}}} ad opus, vt apparet in sono quem emittunt antennas in malos eleuantes. \textit{Et ideo cantus qui fuere antiquitus apud gerionem et gentem darmaticam faciebant exitus tandem et \edtext{terminationes}{\Cfootnote{terminationem \textit{P\textsuperscript{u}P\textsuperscript{y}}}} in \edtext{consonantia}{\Cfootnote{consonantiam \textit{γπE\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{z}}}}} ypofrigiam, sicut et \edtext{ditirambi}{\Cfootnote{dyterambus \textit{M\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{z}}}} \edtext{demum}{\Cfootnote{deinde \textit{α}}} \edtext{concidunt}{\Cfootnote{decidunt vel concidunt \textit{π}}} in ipsam, vt visum est \edtext{.30°.}{\Cfootnote{.3°. \textit{P\textsuperscript{x}V\textsuperscript{o}}}}
\pend

\pstart
\textbf{5} \edtext{\textit{YPODORISTIUM.}}{\Cfootnote{YPODORISTI \textit{πE\textsuperscript{a}P\textsuperscript{y}} | YPODORISTUS. \textit{P\textsuperscript{x}V\textsuperscript{m}} | Deinde. YPODORISTIUM. \textit{V\textsuperscript{a}mv}}} Ostendit secundo qualis sit cantus ypodoristium. \edtext{Dicit}{\Cfootnote{dicens \textit{E\textsuperscript{a}G\textsuperscript{u}M\textsuperscript{a}V\textsuperscript{a}}}} quod \textit{cantus ypodoristius est magnificus laudabilis virilis stabilis. Vnde inter ceteras consonantias est forte citaricissimus.} \edtext{Existit}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{quod propter}{\Cfootnote{propter quod \textit{λ}}} lidium dictum sit vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{o}}}} \edtext{.30°.}{\Cfootnote{.3°. \textit{P\textsuperscript{x}V\textsuperscript{m}}}}, et in hiis indicatur sue rationis maior.
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\textbf{6} \edtext{\textit{HEC}}{\Cfootnote{Deinde. HEC \textit{V\textsuperscript{a}mv}}} \textit{AUTEM.} Ostendit qualis sit cantus choricus subdens minorem, dicens quod \textit{hec vtraque consonantia, puta \edtext{ypofrigium}{\Cfootnote{ypofrigia \textit{M\textsuperscript{a}V\textsuperscript{o}}}} et \edtext{ypodoristium}{\Cfootnote{ypodoristia \textit{πM\textsuperscript{a}P\textsuperscript{y}}}}, relatione cantus chorici est velut \edtext{in}{\Cfootnote{\textit{om}. \textit{γπE\textsuperscript{a}V\textsuperscript{z}}}} \edtext{consonantia}{\Cfootnote{consonantiam \textit{M\textsuperscript{a}V\textsuperscript{m}}}}. Vnde \edtext{est}{\Cfootnote{est in \textit{π}}} \edtext{illis}{\Cfootnote{istis \textit{π} | ipsis \textit{α}}} extranea, sed magis \edtext{propria}{\Cfootnote{pro \textit{γ}}} \edtext{duobus predictis}{\Cfootnote{predictis duobus \textit{α}}} cantandi modis est consonantia ascema dicta, existens vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} prius \edtext{mutatiua}{\Cfootnote{mutata \textit{π} | mutatam \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}}. Et \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} quidem \edtext{consonantia}{\Cfootnote{consonantie \textit{γπP\textsuperscript{y}}}} modi appellati commutatiui \edtext{erant}{\Cfootnote{\textit{om}. \textit{π}}} in \edtext{vsu}{\Cfootnote{visu \textit{V\textsuperscript{o}}}} apud hereos, ipsi enim \edtext{soli}{\Cfootnote{solum \textit{P\textsuperscript{y}V\textsuperscript{a}}}} fuerunt duces et gubernatores antiquorum,} \edtext{incitantes}{\Cfootnote{excitantes \textit{π}}} ipsos in actus \edtext{huius}{\Cfootnote{huiusmodi \textit{P\textsuperscript{y}V\textsuperscript{b}}}} cantibus, vt in bellum vel aliud. Et proprie \edtext{ypofrigium}{\Cfootnote{ypofrigii \textit{M\textsuperscript{a}V\textsuperscript{z}}}}, \edtext{operatiui}{\Cfootnote{operatim \textit{πE\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{a}V\textsuperscript{m}}}} enim magis.
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\textbf{7} \edtext{\textit{PROPRII.}}{\Cfootnote{Deinde. PROPRII. \textit{V\textsuperscript{a}mv}}} Ostendit qualis et quorum sit cantus choricus, dicens primo quod \textit{homines singulares et priuati sunt illi quorum cantus est choricus,} et \edtext{propterea}{\Cfootnote{propter hoc \textit{πE\textsuperscript{a}}}} \edtext{fruuntur}{\Cfootnote{finitur \textit{π}}} hiis que ipsius. \textit{Sibi namque \edtext{conuenit vt sit}{\Cfootnote{competit vt \textit{πE\textsuperscript{a}}}} lamentabilis, inducens \edtext{in pietatem}{\Cfootnote{impietatem \textit{V\textsuperscript{o}}}} homines, etiam agrestes, et quietus more \edtext{et}{\Cfootnote{\textit{om}. \textit{απ}}} honestus et melodiosus maxime, hec enim humana sunt et mansueta.} Et \edtext{ideo}{\Cfootnote{\textit{om}. \textit{γP\textsuperscript{y}}}} \edtext{amatores tali}{\Cfootnote{admonitores tali \textit{π}}} \edtext{genere cantandi}{\Cfootnote{cantandi genere \textit{π}}} vtuntur in demulciendo \edtext{dominarum primitus animos suarum}{\Cfootnote{primitus dominarum animos suos \textit{γ}}} agrestes.
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\textbf{8} \edtext{\textit{HIC}}{\Cfootnote{HOC \textit{πP\textsuperscript{y}V\textsuperscript{b}} | Deinde. HEC \textit{V\textsuperscript{a}mv}}} \textit{AUTEM.} Ostendit secundo quod talia minus insunt ceteris consonantiis \edtext{achorica}{\Cfootnote{a chorico \textit{π}}}, dicens quod \textit{relique consonantie} \edtext{prefate}{\Cfootnote{prefacte \textit{P\textsuperscript{x}V\textsuperscript{a}} | pretacte \textit{λ}}} minus participant \edtext{cum}{\Cfootnote{\textit{om}. \textit{απ}}} predictis, \textit{nam ypofrigia est diuina} seu legalis, vt apparuit .15. et 28., \textit{vel furiosa incitans homines \edtext{in}{\Cfootnote{ad \textit{π}}} tumultum et furiam. \edtext{Est autem}{\Cfootnote{Aut est \textit{γπE\textsuperscript{a}}}} bachica} praua \edtext{solida}{\Cfootnote{stolida \textit{P\textsuperscript{y}V\textsuperscript{a}} | \textit{om}. \textit{πE\textsuperscript{a}M\textsuperscript{a}}}}, iuxta illud noni 30\textsuperscript{e}., \edtext{propter quod}{\Cfootnote{propter quid \textit{πM\textsuperscript{a}P\textsuperscript{y}}}} ``bachici artifices vt frequenter sunt mali.''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Ps.Arist. \textit{Prob.} XXX.10, 956b11–12. }} \textit{Iuxta siquidem \edtext{huius}{\Cfootnote{huiusmodi \textit{V\textsuperscript{b}mv}}} consonantiarum \edtext{imperfectarum}{\Cfootnote{dictarum \textit{κ} | frigiarum \textit{π}}} \edtext{modum}{\Cfootnote{modo \textit{γE\textsuperscript{a}} | modos \textit{π}}} patimur aliquid, cum autem patimur imbecilliores, amplius fortibus \edtext{reddimur}{\Cfootnote{redduntur \textit{M\textsuperscript{a}V\textsuperscript{o}}}}.} Et ideo non possumus cantum tenere choricum, sed leges facimus \edtext{et}{\Cfootnote{vt \textit{M\textsuperscript{a}V\textsuperscript{m}}}} ascema, vt tactum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}} \edtext{.15°}{\Cfootnote{.55° \textit{π}}}. Sed quia hiis opposite sumus dispositi cum cantus est lamentabilis et cetera, tunc \textit{competit dispositio nostra et potentia cantibus choricis.} Potest enim ipsos \edtext{seruare}{\Cfootnote{conseruare \textit{γ}}}, \edtext{quamuis}{\Cfootnote{licet \textit{G\textsuperscript{u}P\textsuperscript{p}} | \textit{om}. \textit{E\textsuperscript{a}V\textsuperscript{o}}}} extent laboriosi, \textit{non autem conuenit vt dictum est \edtext{ypofrigiam}{\Cfootnote{ypofrigia \textit{απ}}} proprie ipsis,} \edtext{operatiua}{\Cfootnote{operatam \textit{P\textsuperscript{x}V\textsuperscript{m}}}} namque, sicut expositum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}M\textsuperscript{a}P\textsuperscript{y}}}}.
\pend

\pstart
\textbf{9} \edtext{\textit{EST}}{\Cfootnote{Deinde. EST \textit{V\textsuperscript{a}mv}}} \textit{AUTEM.} Vltimo \edtext{manifestat}{\Cfootnote{monstrat \textit{G\textsuperscript{u}P\textsuperscript{p}}}} quid \edtext{sit}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}} \edtext{chorus}{\Cfootnote{choris \textit{V\textsuperscript{o}}}}, dicens quod \textit{chorus est velut quidam \edtext{curator}{\Cfootnote{curat \textit{π}}},} studet enim remouere curas et sollicitudines ab audientibus. \textit{Est \edtext{et}{\Cfootnote{etiam \textit{π}}} inoperabilis,} non enim \edtext{incitat}{\Cfootnote{excitat \textit{π}}} ad actus vt \edtext{ypofrigium}{\Cfootnote{ypofrigius \textit{πM\textsuperscript{a}V\textsuperscript{m}} | ypofrigia \textit{E\textsuperscript{a}V\textsuperscript{z}}}}, \textit{\edtext{sed}{\Cfootnote{licet \textit{P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} solum beniuolentiam et complacentiam inducit quibus auditu \edtext{incurrit}{\Cfootnote{occurrit \textit{γE\textsuperscript{a}}}}.} Nihil enim post factum \edtext{relinquit}{\Cfootnote{reliquid \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{o}V\textsuperscript{z}}}} cum solum animos demulcere laboret.
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\subsection[Quadragesimumnonum problema … ]{Quadragesimumnonum problema}\label{part49}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID, SONORUM FACIENTIUM CONSONANTIAM, IN GRAUIORI EST MOLLIOR? \textsuperscript{2}AUT QUIA IN MELODIA SUA VOCE MOLLIS EST ET QUIETA, EA AUTEM QUE EST RITHIMI MISTIONE ASPERA ET MOTIUA? \textsuperscript{3}QUONIAM AUTEM GRAUIS QUIDEM SONUS MOLLIS ET QUIETUS EST, ACUTUS AUTEM MOTIUUS, ET HABENTIBUS IDEM MELOS, ERIT VTIQUE MOLLIOR GRAUIOR IN EODEM MELOS MAGIS. ERAT AUTEM MELOS IPSI MOLLE}{\lemma{PROPTER \dots\ MOLLE}\Cfootnote{PROPTER QUID SONORUM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} consonantia ex grauibus causata sonis est \edtext{mollior}{\Cfootnote{melior \textit{πV\textsuperscript{a}}}} et delectabilior quam ex acutis constructa?}
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\pstart
\textbf{3} \edtext{\textit{AUT}}{\Cfootnote{Deinde. AUT \textit{V\textsuperscript{a}mv}}} \textit{QUIA.} \edtext{Soluit,}{\Cfootnote{Soluit et \textit{πM\textsuperscript{a}V\textsuperscript{z}}}} primo \edtext{ostendens}{\Cfootnote{ostendit \textit{πκ}}} \edtext{quare}{\Cfootnote{quod \textit{V\textsuperscript{o}}}} consonantia \edtext{vocis}{\Cfootnote{vocum \textit{πV\textsuperscript{m}}}} \edtext{grauioris}{\Cfootnote{grauiori \textit{P\textsuperscript{x}V\textsuperscript{m}} | grauior \textit{π}}} est \edtext{mollior}{\Cfootnote{melior \textit{πV\textsuperscript{a}}}}. Secundo adaptat ad eam que sonorum \edtext{illic}{\Cfootnote{ibi \textit{πP\textsuperscript{y}}}}. \textit{QUONIAM AUTEM.} \edtext{Dicit}{\Cfootnote{Dicens \textit{E\textsuperscript{a}P\textsuperscript{y}P\textsuperscript{z}}}} causam \edtext{esse quoniam huius}{\Cfootnote{huius esse quoniam \textit{V\textsuperscript{o}} | esse quoniam huiusmodi \textit{γ}}} \textit{consonantia in melodia propter \edtext{suam vocem}{\Cfootnote{vocem suam \textit{π}}} \edtext{delectabilem}{\Cfootnote{laudabilem \textit{G\textsuperscript{u}P\textsuperscript{q}}}} ex qua construitur mollis est et quieta, iuxta illud quod \edtext{prius dictum est}{\Cfootnote{dictum est prius \textit{γE\textsuperscript{a}V\textsuperscript{o}}}} de cantu chorico. Sed fere per oppositum consonantia rithimica ob suam mixtionem minus \edtext{consonam}{\Cfootnote{consona \textit{πV\textsuperscript{a}}}} aspera existit,} pungens auditum et immutans mox \edtext{subiecta}{\Cfootnote{subita \textit{απ}}}, quasi alteratione nature \edtext{preliatiua}{\Cfootnote{preliata \textit{π}}}.
\pend

\pstart
\textbf{4} \edtext{\textit{QUONIAM.}}{\Cfootnote{Deinde. QUONIAM. \textit{V\textsuperscript{a}mv}}} Applicat ad sonum, dicens similiter \edtext{esse}{\Cfootnote{est \textit{M\textsuperscript{a}V\textsuperscript{a}}}} in sono \textit{quoniam grauis, vt dictum \edtext{est}{\Cfootnote{\textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} de voce mollis, est \edtext{quietus et}{\Cfootnote{et quietus \textit{γ}}} amenus, sed acutus \edtext{asper est}{\Cfootnote{asper \textit{P\textsuperscript{y}V\textsuperscript{a}V\textsuperscript{z}}}} et conmotiuus. Vnde posito quod \edtext{extat}{\Cfootnote{extet \textit{απ}}} vtrobique eadem melodia illa que \edtext{est}{\Cfootnote{erit \textit{γE\textsuperscript{a}V\textsuperscript{o}}}} grauior \edtext{mollior et}{\Cfootnote{et mollior \textit{π}}} \edtext{delectabilior}{\Cfootnote{delectabilior erit \textit{π}}} in \edtext{eadem}{\Cfootnote{eodem \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{m}}}} melodia magis, nam melos ipsius est \edtext{molle}{\Cfootnote{melle \textit{P\textsuperscript{x}P\textsuperscript{y}}}} \edtext{quasi}{\Cfootnote{id est \textit{G\textsuperscript{u}P\textsuperscript{q}}}} dulce.}
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\pstart
\textbf{5} \edtext{Nota}{\Cfootnote{Notandum \textit{γ}}} quod \edtext{cetera}{\Cfootnote{certa \textit{V\textsuperscript{b}V\textsuperscript{o}}}} consonantia debet constare ex acuto et graui. Mollior tamen est et \edtext{suauior}{\Cfootnote{amor \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}} | a motu \textit{γ} | amenior \textit{π}}} cum in graue tendit quam quando in acutum.
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\subsection[Particula XIX: De harmonia]{Quinquagesimum problema}\label{part50}
\pstart
\textbf{1} \textit{\edtext{\textsuperscript{1}PROPTER QUID EQUALIUM DOLIORUM ET SIMILIUM SIQUIDEM ALTERUM VACUUM FUERIT, ALTERUM VERO AD MEDIUM PLENUM, DYAPASON CONSONAT ECHO? \textsuperscript{2}AUT QUIA DUPLEX FIT ET QUE EST EX MEDIO EIUS QUE EST EX VACUO? \textsuperscript{3}QUID ENIM DIFFERT HOC QUAM IN FISTULIS? VIDETUR ENIM VELOCIOR MOTUS ACUTIOR ESSE, IN MAIORIBUS AUTEM TARDIUS AER OBUIAT. ET IN DUPLICIBUS TANTUM, ET IN ALIIS PROPORTIONALE. CONSONAT AUTEM DYAPASON ET DUPLEX ITER AD MEDIUM}{\lemma{PROPTER \dots\ MEDIUM}\Cfootnote{PROPTER QUID EQUALIUM \textit{V\textsuperscript{m}V\textsuperscript{o}} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}V\textsuperscript{b}V\textsuperscript{z}}}}.}
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\pstart
\textbf{2} \textit{\edtext{Quare}{\Cfootnote{Querit quare \textit{λ}}} \edtext{si}{\Cfootnote{cum \textit{G\textsuperscript{u}P\textsuperscript{p}}}} aliquis accipiat duo dolia eiusdem \edtext{mensure}{\Cfootnote{materie \textit{γE\textsuperscript{a}V\textsuperscript{o}}}} quantitatis \edtext{et}{\Cfootnote{\textit{om}. \textit{γE\textsuperscript{a}}}} figure et vniuersaliter similitudinis omnis,} ``in omnibus \edtext{enim}{\Cfootnote{\textit{om}. \textit{γ}}} sit equalitas nisi in hiis in quibus \edtext{fit}{\Cfootnote{sit \textit{E\textsuperscript{a}P\textsuperscript{y}}}} comparatio,''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{(Pietro expresses this idea in his commentary on problem 10 of particula 15. His reference to 5/25 is obscure.)}} vt visum \edtext{est}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{y}V\textsuperscript{o}}}} \edtext{quinto 25°.}{\Cfootnote{.5°. et 5°, \textit{E\textsuperscript{a}M\textsuperscript{a}V\textsuperscript{z}} | \textit{om}. \textit{π}}}, puta quod \textit{vnum sit omnino corpore solido vt liquore \edtext{aliquo}{\Cfootnote{acuto \textit{λ}}} \edtext{vacuum \edtext{plenum, tamen aere}{\Cfootnote{aere tamen plenum \textit{V\textsuperscript{o}}}}}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}P\textsuperscript{z}V\textsuperscript{z}}}},} ``\edtext{idem namque \edtext{apud vulgus est}{\Cfootnote{est apud vulgum \textit{V\textsuperscript{o}}}}}{\Cfootnote{\textit{om}. \textit{M\textsuperscript{a}P\textsuperscript{z}V\textsuperscript{z}}}} vacuum \edtext{et plenum aere}{\Cfootnote{plenum tamen aere \textit{V\textsuperscript{m}V\textsuperscript{z}}}},''\edtext{\textzwnj}{\lemma{\textzwnj}\Afootnote[nosep]{Arist. \textit{Top.} VII.1, 152b20}} 7 \textit{topicorum}, \textit{et alterum sit semiplenum,} et \edtext{percutiantur}{\Cfootnote{percutiatur \textit{E\textsuperscript{a}V\textsuperscript{m}}}} etiam equaliter quocunque modo \edtext{ambo}{\Cfootnote{\textit{om}. \textit{λ}}}, vel clametur in ipsis; \textit{tunc echo resultans \edtext{ex}{\Cfootnote{in \textit{V\textsuperscript{b}V\textsuperscript{z}}}} \edtext{hiis}{\Cfootnote{ipsis \textit{V\textsuperscript{m}V\textsuperscript{z}}}} per \edtext{refractionem}{\Cfootnote{fractionem \textit{π}}} ad parietes reddit simphoniam \edtext{dyapasonicam}{\Cfootnote{dyapason sonicam \textit{V\textsuperscript{b}} | dyapason causa \textit{V\textsuperscript{o}}}}.}
\pend

\pstart
\textbf{3} \edtext{\textit{AUT.}}{\Cfootnote{Deinde. AUT. \textit{V\textsuperscript{a}mv}}} Soluit \edtext{et}{\Cfootnote{\textit{om}. \textit{π}}} \edtext{dicit primo}{\Cfootnote{primo dicens \textit{G\textsuperscript{u}P\textsuperscript{p}}}} causam esse \textit{quia sonus causatus ex vacuo vase se habet ad sonum causatum \edtext{ex}{\Cfootnote{a \textit{V\textsuperscript{o}}}} semipleno in dupla proportione}, continet enim bis eius totum quod vacuum \edtext{omne}{\Cfootnote{esse \textit{κ}}} id quod semiuacuum. Dictum est \edtext{autem}{\Cfootnote{enim \textit{π}}} \edtext{13.}{\Cfootnote{30. \textit{M\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}}}} quod dyapason ex numero duplici nascitur.
\pend

\pstart
\textbf{4} \edtext{\textit{QUID}}{\Cfootnote{Deinde. QUID \textit{V\textsuperscript{a}mv}}} \textit{ENIM.} Secundo remouet dubitationem, quia crederet aliquis quod ex sonis talium non resultaret simphonia sicut ex sonis aliorum. Et ideo subdit quod \textit{non est aliqua differentia \edtext{quin}{\Cfootnote{quare \textit{V\textsuperscript{o}}}} ex sonis istorum constituatur dyapason consonantia, \edtext{ceu}{\Cfootnote{seu \textit{E\textsuperscript{a}V\textsuperscript{m}V\textsuperscript{o}} | sicut \textit{G\textsuperscript{u}P\textsuperscript{p}}}} ex sono fistularum et \edtext{huius}{\Cfootnote{huiusmodi \textit{γ}}} instrumentorum motorum spiritu \edtext{seu}{\Cfootnote{ceu \textit{P\textsuperscript{y}V\textsuperscript{a}}}} tactu.} Quod ostendit inuentio musice malleorum soni proportione, vt dictum \edtext{est}{\Cfootnote{est in \textit{π} | \textit{om}. \textit{E\textsuperscript{a}P\textsuperscript{y}}}} primo, quod siquidem contingere bene potest cum \edtext{ubique}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} in talibus ratio \edtext{semper proportionis duplicis}{\Cfootnote{proportionis duplicis semper \textit{π} | semper proportionis dupliciter \textit{γ}}} obseruetur. Quod ostenditur quia dyapason causatur ex acuto et graui quod hic accidit, \edtext{nam}{\Cfootnote{est iam \textit{V\textsuperscript{o}}}} \textit{motus velocior factus in semipleno angustia loci percipitur esse acutior, in maiori autem tardius mouetur et obuiat \edtext{aer}{\Cfootnote{aeri \textit{κE\textsuperscript{a}}}}} resistenti, propter plurimam \edtext{quantitatem ipsius}{\Cfootnote{ipsius quantitatem \textit{π}}} \edtext{motam}{\Cfootnote{motum \textit{V\textsuperscript{o}}}} et per longam distantiam, vt visum est \edtext{.xi\textsuperscript{a}.}{\Cfootnote{.xli. \textit{π}}}, ita quod \textit{in hiis que sunt dupla \edtext{ad inuicem}{\Cfootnote{in ad inuicem \textit{γE\textsuperscript{a}}}} et \edtext{in}{\Cfootnote{\textit{om}. \textit{πE\textsuperscript{a}}}} \edtext{etiam}{\Cfootnote{aliis \textit{V\textsuperscript{m}V\textsuperscript{z}} | aliis etiam \textit{απ}}} que aliquam proportionem numeralem habent inter se consonat et efficitur penes duplicia dyapason. Cadit enim duplex \edtext{iter}{\Cfootnote{inter \textit{πP\textsuperscript{y}}}} et \edtext{distantia}{\Cfootnote{distantiam \textit{G\textsuperscript{u}P\textsuperscript{p}}}} extremi ad medium,} vt in pleno, vnum autem \edtext{in}{\Cfootnote{\textit{om}. \textit{P\textsuperscript{p}P\textsuperscript{u}}}} semiuacuo, cum alterum contineat pleni medietatem. Secundum vero \edtext{epitrica}{\Cfootnote{epitricam \textit{π}}} dyatesseron et emiola dyapente \edtext{et cetera}{\Cfootnote{et illic \textit{γ}}}, vt \edtext{monstratum}{\Cfootnote{manifestum \textit{V\textsuperscript{o}}}} est .13°.
\pend

\pstart
\textbf{5} \edtext{Ex}{\Cfootnote{Et \textit{V\textsuperscript{o}}}} hiis \edtext{vtique}{\Cfootnote{itaque \textit{π}}} quam breuius \edtext{prodierunt}{\Cfootnote{prodierint \textit{V\textsuperscript{o}V\textsuperscript{z}}}} in lucem \edtext{vtcunque}{\Cfootnote{vtrunque \textit{P\textsuperscript{y}V\textsuperscript{m}V\textsuperscript{o}}}} circa liberales artes, et \edtext{maxime circa}{\Cfootnote{maxime \textit{αV\textsuperscript{m}V\textsuperscript{z}}}} mathematica \edtext{problemata}{\Cfootnote{probabilis \textit{γ}}}, grandi textuali caligine cum eius corruptione inuoluta quandoque. In \edtext{quibus}{\Cfootnote{quibusdam \textit{π}}} etiam quia plurimis extranea non grandis \edtext{vtilitas}{\Cfootnote{subtilitas \textit{V\textsuperscript{a}mv}}} que \edtext{hodie}{\Cfootnote{a plurimis \textit{π}}} appetitur summe reperitur, quare ipsa celerius \edtext{percurri}{\Cfootnote{precucurri \textit{E\textsuperscript{a}P\textsuperscript{y}} | procurari \textit{γ}}}. \edtext{Nunc}{\Cfootnote{Non \textit{E\textsuperscript{a}P\textsuperscript{x}V\textsuperscript{m}V\textsuperscript{z}}}} autem deinceps circa reliquas partes physice, naturalemque maxime, problemata tam circa composita \edtext{quam}{\Cfootnote{quam circa \textit{P\textsuperscript{y}V\textsuperscript{m}}}} simplicia existentia tractabuntur.
\pend

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\section{Back Matter}
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\subsection[{List of persons mentioned by Pietro}]{List of persons mentioned by Pietro}\label{back-historical-figures}
\textbf{Historical figures}
\begin{itemize}

\item \label{Al} Aphrodisiensis, Alexander 
\item \label{Ar} Aristoteles
\item \label{Ax} Tarentinus, Aristoxenus 
\item \label{Av} Avicenna
\item \label{Bo} Boethius, Anicius Manlius Severinus 
\item \label{Co} Colophonius, Hestiaeus 
\item \label{Dam} Damascenus, Johannes 
\item \label{Ga} Galenus, Claudius 
\item \label{GAr} Aretinus, Guido 
\item \label{Hip} Hippocrates
\item \label{Hor} Horatius Flaccus, Quintus 
\item \label{Is} Hispalensis, Isidorus 
\item \label{Mac} Theodosius, Macrobius Ambrosius 
\item \label{Mu} Atheniensis, Musaeus 
\item \label{Ni} Gerasenus, Nicomachus 
\item \label{Ov} Ovidius Naso, Publius 
\item \label{Phil} Cytherius, Philoxenus 
\item \label{Phr} Phrynichus (tragicus)
\item \label{Pl} Plato
\item \label{Pt} Ptolemaeus, Claudius 
\item \label{Pyth} Pythagoras
\item \label{Ter} Antissenis, Terpander 
\item \label{Th} Antiochenus, Theophilus 
\item \label{Ti} Locrus, Timaeus 
\item \label{Ty} Milesius, Timotheus 
\end{itemize}

\textbf{Mythical figures}
\begin{itemize}

\item \label{Cay} Cain
\item \label{Ju} Jupiter
\item \label{Mem} Memoria
\item \label{Mer} Mercurius
\item \label{Mos} Moses
\item \label{Or} Orpheus
\item \label{Tor} Torebus 
\item \label{Tub} Tubal Cain
\end{itemize}

\subsection[{List of titles cited by Pietro}]{List of titles cited by Pietro, with editions supplied}\label{back-list-works-cited}
\begin{bibitemlist}{1}
\bibitem  [d'Abano 1504]{AbCo}  d'Abano, Pietro. \textit{Conciliator differentiarum philosophorum}. Venetiis: n.p., 1504. URL: \url{http://data.onb.ac.at/rep/10B710EE}
\bibitem  [AlPr]{AlPr}  Pseudo-Alexander of Aphrodisias.  \textit{Problemata}. Ed. Robert W. Sharples and Sophia Kapetanaki as \textit{Pseudo-Aristoteles (Pseudo-Alexander), Supplementa Problematorum: A new edition of the Greek text with introduction and annotated translation}. Berlin: De Gruyter, 2006; trans. Pietro d'Abano in Vatican Biblioteca Apostolica reg. lat. 747, ff. 62r–104r, URL: \url{https://digi.vatlib.it/view/MSS\textunderscore Reg.lat.747}
\bibitem  [Arist. De an.]{ArAn}  Aristotle. \textit{De anima}. Ed. J. Decorte and J. Brams as \textit{De Anima}. \textit{Aristoteles Latinus Database}  XII.1 URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MIV11ANIM\textunderscore }
\bibitem  [Arist. Cael.]{ArCa}  Aristotle. \textit{De caelo} Ed. F. Bossier as \textit{De caelo et mundo}. \textit{Aristoteles Latinus Database}  VIII URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MGM11CAMU\textunderscore }
\bibitem  [Arist. Eth. Nic.]{ArEth}  Aristotle. \textit{Liber ethicorum ad Nicomachum}. Ed. R.A. Gauthier as \textit{Analytica posteriora (Iacobi Venetici translationis recensio)}. \textit{Aristoteles Latinus}  XXVI.1–3. Leiden and Bruxelles: Brill and Desclée De Brouwer, 1972–1974, URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MBUPIETN1\textunderscore }
\bibitem  [Arist. Metaph.]{ArMet}  Aristotle. \textit{Metaphysica}. Ed. G. Vuillemin-Diem as \textit{Metaphysica, lib. I-XIV. Recensio et Translatio Guillelmi de Moerbeka}. \textit{Aristoteles Latinus}  XXV.3. Leiden, New York, and Köln: E.J. Brill, 1995, URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MGM55META\textunderscore }
\bibitem  [An. Post.]{ArPA}  Aristotle. \textit{Analytica posteriora}. Ed. L. Minio-Paluello and B. G. Dod as \textit{Analytica posteriora (Iacobi Venetici translationis recensio)}. \textit{Aristoteles Latinus}  IV.1–4. Bruges and Paris: Desclée De Brouwer, 1968, URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MGM55ANPO\textunderscore }
\bibitem  [Arist. Int.]{ArPe}  Aristotle. \textit{Peri hermeneias}. Ed. L. Minio-Paluello and G. Verbeke as \textit{Periermenias (Peri hermeneias) (uel De Interpretatione)}. \textit{Aristoteles Latinus}  II.2. Bruges and Paris: Desclée De Brouwer, 1965, URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MGM11PEHE\textunderscore }
\bibitem  [Arist. Ph.]{ArPh}  Aristotle. \textit{Physica}. Ed. F. Bossier and J. Brams as \textit{Physica (translatio 'uetus')}. \textit{Aristoteles Latinus}  VII. Leiden and New York: E.J. Brill, 1990, URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=MIV11PHYS\textunderscore }
\bibitem  [Arist. Pol.]{ArPo}  Aristotle. \textit{De politicorum}. Ed. Franciscus Susemihl as \textit{Aristotelis Politicorum libri octo cum vetusta translatione Guilelmi de Moerbeka}. Leipzig: Teubner, 1872.
\bibitem  [Arist. Top.]{ArTo} Aristotle. \textit{Topica}. Ed. L. Minio-Paluello as \textit{Topica. Translatio Boethii, Fragmentum Recensionis Alterius et Translatio Anonyma}. \textit{Aristoteles Latinus}  V.1–3. Bruges and Paris: Desclée De Brouwer, 1969, URL: \url{http://clt.brepolis.net/ALD/pages/TextSearch.aspx?key=PARBOTOPI\textunderscore }
\bibitem  [ArPr]{ArPr}  Pseudo-Aristotle. \textit{Problemata physica}. Ed. Carolus Ruelle, Hermannus Knoellinger and Iosephus Klek as \textit{Aristotelis quae feruntur Problemata physica}. Leipzig: Teubner, 1922.
\bibitem  [AvCa]{AvCa}  Avicenna. \textit{Canon medicinae}. Trans. Gerardus Cremonensis as \textit{Avicennae Arabum medicorum principis, Canon medicinae}. Venice: Iuntas, 1608.
\bibitem  [BoAr]{BoAr}  Anicius Manlius Severinus Boethius. \textit{De institutione arithmetica}. Ed. Godofredus Friedlein in \textit{Anicii Manlii Torquati Severini Boetii De institutione arithmetica libri duo, De institutione musica libri quinque}, 1–173. Leipzig: Teubner, 1868.
\bibitem  [BoMu]{BoMu}  Anicius Manlius Severinus Boethius. \textit{De institutione musica}. Ed. Godofredus Friedlein in \textit{Anicii Manlii Torquati Severini Boetii De institutione arithmetica libri duo, De institutione musica libri quinque}, 175–371. Leipzig: Teubner, 1868.
\bibitem  [GaAc]{GaAc}  Galen. \textit{De morbo et accidenti}. Ed. C. G. Kühn as \textit{De symptomatum causis libri iii} in \textit{Claudii Galeni opera omnia}, vol. 7, 83–272. Leipzig: Knobloch, 1824.
\bibitem  [GaPu]{GaPu}  Galen. \textit{De differentia pulsuum libri iv}. Ed. C. G. Kühn in \textit{Claudii Galeni opera omnia}, vol. 8, 493–765. Leipzig: Knobloch, 1824.
\bibitem  [GaSa]{GaSa}  Galen. \textit{De sanitate tuenda libri vi}. Ed. K. Koch in \textit{Corpus medicorum Graecorum}  5.4.2, 3–198. Leipzig: Teubner, 1923.
\bibitem  [HipAph]{HipAph}  Hippocrates. \textit{Aphorismi}. Ed. É. Littré in \textit{Oeuvres complètes d'Hippocrate}, vol. 4, 458–608. Paris: Baillière, 1844.
\bibitem  [IsEt]{IsEt}  Isidore of Seville. \textit{Etymologiae}. Ed. W. M. Lindsay in \textit{Isidori Hispalensis Episcopi Etymologiarum sive originum libri xx}, vol. 1. Oxford: Clarendon Press, 1911.
\bibitem  [MacSS]{MacSS}  Macrobius Ambrosius Theodosius. \textit{Commentarium in somnium Scipionis}. Ed. Franciscus Eyssenhardt in \textit{Macrobius}, 465–570. Leipzig: Teubner, 1868.
\bibitem  [MasAph]{MasAph}  Yuḥannā ibn Māsawaiyh. \textit{Aphorismorum libri}. Ed. Danielle Jacquart and Gérard Troupeau as \textit{Yuhannà ibn Màsawayh (Jean Mésué), Le livre des Axiomes médicaux (Aphorismi), édition du texte arabe et des versions latines avec traduction française et lexique}. Geneva: Droz, 1980. See also Biblioteca Apostolica vat. lat. 2460, f. 39r–42r, URL: \url{https://digi.vatlib.it/view/MSS\textunderscore Vat.lat.2460}
\bibitem  [NiGer]{NiGer}  Nichomachus (Dramatist). \textit{Geryones}.  (Concerning this lost play, see URL: \url{http://www.apgrd.ox.ac.uk/ancient-performance/performance/491})
\bibitem  [Psalm]{Psalm} \textit{Liber psalmorum}. Ed. Robertus Weber in \textit{Biblia sacra iuxta vulgatam versionem}, 767–955. Stuttgart: Deutsche Bibelgesellschaft, 1994.
\bibitem  [PtHar]{PtHar}  Claudius Ptolemy. \textit{Harmonica}. Ed. I. Düring as \textit{Die Harmonielehre des Klaudios Ptolemaios} in \textit{Göteborgs Högskolas Arsskrift}  36, 2–111. Göteborg: Elanders, 1930.
\bibitem  [SePr]{SePr}  Pseudo-Hermes Trismegistus. \textit{De sex rerum principiis}. Ed. Paolo Lucentini and Mark D. Delp in \textit{Corpus Christianorum, Continuatio Mediaeualis}  142. Turnhout: Brepols, 2006.  (Sometimes attributed to Gilbert of Poitiers.)

\end{bibitemlist}

\end{document}