The music notation developed by ars nova theorists in the first half of the fourteenth century ushered in possibly the most significant change in the way in which musicians and composers in western Europe thought of musical composition, that is, as an act facilitated by the visual forms of music notation. Brand new possibilities were afforded by the expanded variety of note shapes, rests, dots, and metrical combinations. Jean des Murs (fl. 1312-47), a leading French astronomer and mathematician who first appears on the Parisian scene in the 1310s, was one of the main developers of the new music notation. Of it, he claimed : “Whatever can be sung, can now be written down.” Ars nova notation precipitated a series of new compositional techniques that characterised fourteenth-century music, such as syncopation, counterpoint, and complex repetitive formal structures. It also generated heated debate, especially among music theorists based in Paris. The attitudes of its proponents – the so-called “moderni” – prompted laments like this from Jacobus, a vociferous critic of the ars nova:
Oh, so much abuse, so much illegality, so much vanity, so much insolence, so much uselessness, so much rudeness! Oh, so much presumption in the figuring of the notes, so much confusion! (Speculum musicae, book VII, chapter 27)
My new book Music and the moderni, 1300-1350: The ars nova in theory and practice, which was published by Cambridge University Press last month, explores the writings, music, and interactions of the main ars nova protagonists, and identifies and explains what was at stake for them (and the rest of us). Through contemporaneous accounts, I situate the musical developments of this crucial half century within the cultural and intellectual context of the time.
The discoveries I present in this book challenge prevailing narratives of the ars nova, which had previously coupled the emergence of the ars nova too specifically to the c. 1320 copying date of the famous Roman de Fauvel manuscript. I propose instead that the musical aesthetics of the fourteenth-century moderni, derided by older theorists like Jacobus, is bound up with ideas of subtilitas and mathematical precision in notation and style only exemplified in writings and music compositions from the 1330s and 1340s. Music and the moderni, 1300-1350 revisits broader questions about both the chronology and character of the ars nova, and how changes in theory and notation can engender changes in style and practice.
In series of blog posts of the next few weeks, I will introduce some of the ideas and music discussed in my book. Next up: the aesthetic of subtilitas (subtlety)–a surprisingly hard-to-pin-down term with both positive and negative connotations in the Middle Ages–and how it was embraced by the fourteenth-century moderni.
A quick post about one of the courses I’ll be teaching at Brandeis this Fall 2018: “Seeing Sound: The History and Practice of Notation to c.1500.” To the left is one of the songs we’ll work on–a two-voice fourteenth-century ballade by Guillaume de Machaut. This is an intense but fun course, and very hands-on. Each week students work on a different medieval notation–from chant through to keyboard tablatures of the fifteenth century–learning how to interpret and transcribe the notation directly from high-quality images of the original manuscript sources, and from the explanations given in contemporaneous theory treatises. Brandeis University is part of a cross-registration consortium within Boston that includes Tufts University, Boston University, and Boston College, so if you are a student in any of these institutions and are interested in taking the course, please get in touch. For a course flyer, click the link below. The course blurb follows.
The late ninth century witnessed a major technological breakthrough in the transmission of music in western Europe that was to have far-reaching consequences. Chants that had been taught orally for several centuries began to be encoded on parchment using signs placed above text syllables that recorded the shape and contour of the chant melodies. Systems of music notation spread rapidly across Europe, and took root as the way to record, archive, share, and (eventually) compose music. This graduate seminar course examines the form and function of music notation, and how the writing down of music transformed music practice. Students will transcribe music from a variety of early notation systems, working with high quality facsimiles and online reproductions of the original manuscripts.
The ‘Measuring Polyphony’ project presents, for the first time, digitisations of polyphonic compositions written during the thirteenth and fourteenth centuries in mensural notation, and linked directly, in most cases, to high-resolution images of the original manuscript sources, and live playback of the mensural scores as MIDI. I began this project at McGill, and I’m now continuing it at Brandeis University, with the support of a Provost’s Innovation Grant. The project leverages the potential of the rich digital image repositories of music manuscripts and the community-based standards for encoding music notation of the Music Encoding Initiative (MEI) and newly tools available libraries and tools for rendering MEI on the web, such as Verovio.
‘Measuring Polyphony’ has three goals:
makes scored-up transcriptions in the original notation and audio (MIDI) freely available online to performers, scholars, and the general public, presented alongside images of the original music manuscripts;
encodes the medieval notation in a standardised machine-readable format so that the music data can potentially be be searched or analysed using current tools, and through this interoperability make the data available to other websites and applications; and
makes the processes and tools for digitally encoding mensural polyphony in mensural notation freely available so that other stakeholders can easily and rapidly enlarge the dataset.
For more information on the encoding process, and to view and listen to the repertoire already encoded, bookmark http://www.measuringpolyphony.org. Be sure to check out the acknowledgments to the entire project team and the project advisory board. Please be in touch if you have a repertoire of mensural compositions that you would like to encode and make available online.
In book 7 of his Speculum musicae, the fourteenth-century music theorist Jacobus structures a defense of music as it had been practiced in the thirteenth century by such eminent musicians and theorists as Lambertus, Franco, and Petrus de Cruce against the practices of certain unnamed moderni active at the time of Jacobus’s writing. While Jacobus’s quotations from various theoretical works by Jehan des Murs have long been recognized, it previously had been supposed that the remaining quotations were jumbled references from many different theorists. With specific reference to Philippe de Vitry only two quotations from the text edited in vol. 8, Corpus scriptorum de musica, had been identified previously. In fact, there is substantial sustained treatment of a single author, whom I have termed the doctor modernus and who is not Jehan des Murs, that occupies at least five contiguous central chapters of book 7. Following Jacobus’s practice in the previous six books of commentary on a handful of specific works, the writing of Book 7 appears to have been structured around the written works of just four theorists: Lambert, Franco, Jehan des Murs, and the doctor modernus. Furthermore, Jacobus’s vehemence toward the doctor modernus was particularly pronounced and may indicate a personal relationship between the two men. His treatise is quoted with reference to some fundamental ars nova theories, such as extension of long notes beyond the duplex long, remote imperfection, the use of imperfect longs, and imperfect measure in general, and his treatise is described as outlining the precepts of both the old and new arts. The similarities between the treatise of the doctor modernus and many ars nova theory texts (some of which were attributed to Vitry) hints at the possibility that the treatise of the doctor modernus may have been the ancestor text that these other texts had in common, and hence also that Philippe de Vitry may have been the author of the text known to Jacobus, whose subject was the Ars vetus et nova .
As a two-year Banting Fellow at McGill University, along with finishing my monograph, I’m also working on a project to digitally encode a core repertory of French motets from c. 1300 to 1340. With members of the SIMSSA team based at the Schulich School of Music, we eventually plan to present these transcriptions in an online web application side-by-side with images from the manuscript sources (or at least those with publicly-available images), and also to provide some web-based analytical tools for this repertory. Because the motets will be encoded in a standardised machine-readable format, and made available for download to the general public, other researchers and programmers will be able to access this data to conduct their own digital analyses of this repertory.
SIMSSA’s web-based tools can display, search, and browse manuscripts containing neumatic notation, and analyse interval successions in countrapuntal music (using a programming framework based on Michael Scott Cuthbert’s music21 toolkit). These projects work with repertories written in neumatic notation and common practice notation: the goal of this project is to encode the details of the mensural notation of fourteenth-century motets. These specific notational details (such as note shape, mensuration, dots, plicas, etc.) and details of layout (such as staff or page ends) could eventually inform any online editions based on these encodings, and the subsequent digital analysis of this mensural repertory.
First things first, however. The initial phase of this project, already underway, aims to create critical encodings of a selected repertory of motets. The transcriptions are taken from a single manuscript source (and thus are diplomatic transcriptions and not editions), and the digital encodings of these transcriptions follow the MEI schema (a core set of rules developed for encoding music notation documents as XML). The MEI schema has a basic mensural notation module: some modifications and additions will probably have to be made to this module in order to capture and describe the features of thirteenth- and fourteenth-century mensural notations. Once the files are encoded in compliant MEI, it should be relatively straightforward to display the encoded transcriptions on the fly using a tool like Verovio, or to write programs to search or analyse the repertory using VIS or music21. At present, the encodings capture the following notational features:
the shape of each note (duplex long, breve, semibreve, minim, etc.);
rests as notated in the manuscript;
the actual duration of each note (whether notes are perfected, imperfected, or altered);
the implied mensuration (in some motets this is clearer than in others);
ascending and descending plicas added to longs or breves;
dots of division and perfection;
downward stems on semibreves;
staff breaks and page breaks;
ligature groupings and ligature type (c.o.p., with or without proprietas, with or without perfectio);
the underlaid text (in diplomatic transcription, expanding abbreviations, but without editing or standardising the spellings).
For the first phase of data entry, I’ve chosen to work with a representative sample of the French motet repertory dating from c. 1300 to c. 1340 (i.e., bridging the ars antiqua and ars nova). From the motets of the eighth fascicle of Mo, Fauv, and Br, I have chosen to encode those motets that have 3 or more syllabic semibreves in one or more voice parts. There are 23 motets in Mo 8 that fit this criteria, and 21 more that are in Fauv and/or Br. Added to this are six more motets specifically cited in the early ars nova theory sources (all six are also found in Iv and Trem). A final category comprises motets that are found in both Iv and Trem, and in one other of the following ars nova sources: either cited in a fourteenth-century theory treatise (other than the early ars nova treatises previously mentioned), or in one of the following music sources Cambrai, Durham, Pic, Strasbourg or Lbl 41667. This category adds a further 11 motets, bringing the total to 61 motets. These are listed at the end of this blog post.
These encodings, at present, capture the diplomatic transcription of each motet from one manuscript source: however, it would be possible at a later stage to overlay readings from concordant sources (through the Critical Apparatus module of the MEI schema) and thus produce online critical editions of these compositions. For example, a motet such as Tribum/Quoniam in this phase of the project is transcribed from a single source (in this case Fauv), and with all semibreves (whether in groups of two, three, or four) encoded simply as semibreves separated by dots of division (although those semibreves notated with a downward stem are specified in the encoding), if the concordant reading in Br were also to encoded, the semibreves minimae notated in this source (with upward stems) would be distinguished as such. Encoding the motets in this way allows for flexibility in imposing a mensural interpretation, for example, in those Fauv motets where it is unclear whether they are in tempus perfectum or imperfectum. One could envision an online edition where a performer could easily switch back and forth between different editorial interpretations of mensuration or rhythmic duration.
I would welcome any feedback regarding additional suggestions regarding the specific notational details to be encoded, or regarding the repertory chosen for this first phase of the project. We have already transcribed marked-up versions of 43 of these 61 motets into Sibelius, and are now working to convert them into MEI files, and to transcribe the remaining motets. Expanding the repertory in the future will simply be a matter of streamlining the process of transcription and conversion (for example, if we or others wanted to encode the remaining additional fascicles of Mo). Please feel free to leave your comments here, or email me directly.
Mo (fascicle 8)
Mout ai/Li dous/PORTARE (Mo 305)
O presul/ O virtutis/SACERDOTUM (Mo 306)
Dieus comment/ O regina/NOBIS CONCEDAS (Mo 307)
Par une matinee/O clemencie/D’un joli dart (Mo 309)
In sompnis/Amours/IN SOMPNIS (Mo 310)
Se je chante/Bien doi amer/ET SPERA(BIT) (Mo 311)
Au tans nouvel/Chele m’a tolu/J’ai fait tout nouveletement (Mo 312)
Dieus, comment puet/Vo vair/(TENOR) (Mo 314)
Se je sui/Jolietement/OMNES (Mo 316)
Aucun qui/Iure/[VIRGO] MARIA (Mo 317)
On parole/A Paris/ Frese nouvele (Mo 319)
De mes amours/ L’autrier/Defors Compiegne (Mo 321)
Marie assumptio/Huius chori /(TENOR) (Mo 322)
Li savours/ Li grant/Non veul mari (Mo 323)
Amor potest/Ad amorem/(TENOR) (Mo 328)
Virginale/ Descendi/ALMA (Mo 330)
Je cuidoie/ Se j’ai folement/SOLEM (Mo 332)
A maistre Jehan/Pour la plus/ALLELUYA (Mo 334)
Cis a petit/ Pluseur dient/PORTARE (Mo 335)
Amours/Solem iusticie /SOLEM (Mo 338)
Balam inquit/Balam inquit/BAL(L)AAM (Mo 340)
Huic ut placuit/Huic ut placuit/[HUIC MAGI] (Mo 341)
Qui d’amours/Tant me/VIRGA YESSE (Mo 342)
Fauv and/or Br
Trahunt/Ve qui gregi
Orbis orbatus/Vos pastores
La mesnie/J’ai fait nouveletement
Inflammatus/Sicut de ligno
Early ars nova theory (also in Iv and Trem)
Mon chant/Qui doloreus
Iv and Trem and one of the following (Cambrai, Durham, Pic, Strasbourg or Lbl 41667 or a later ars nova theory treatise)
Se grasse/Cum venerint
Fortune mere/Ma dolour
L’amoureuse flour/ En l’estat
Je commence/Et je seray
Mo Montpellier, Bibliothèque Interuniversitaire, Section Médecine, H196
Fauv Paris, Bibliothèque nationale, f. fr. 146
Br Brussels, Bibliothèque royale, Ms. 19606
Iv Ivrea, Biblioteca Capitolare, Ms. 115
Trem Paris, Bibliothèque nationale, nouvelles acquisitions françaises 23190 [olim: ms of the duchess de la Trémoïlle]
Cambrai Cambrai, Mediathèque municipale, Ms B 1328
Durham Durham, Chapter Library, MS C.I.20
Pic Paris, Bibliothèque nationale, Collection de Picardie 67
At the recent 2015 International Medieval Congress in Kalamazoo, Michigan, I had a chance to present some of my new work on Jehan des Murs, astronomer and music theorist, that I’m working on as a chapter of my monograph Greedy for New Things: Novelty in Music c. 1300-1340. Serendipitously, I presented my paper on May 14. On the afternoon of that exact date 632 years ago, Jehan observed a solar eclipse in the presence of Jeanne de France, the twenty-one-year-old Queen of Navarre – I opened my paper with a description of this event. They observed the eclipse in a castle that Jeanne had commissioned to be built in Saint-Germain, Évreux, but reportedly had named ‘Navarre’ (this name might give musicologists pause, given that minims were supposedly ‘invented’ in Navarre!).
In my paper I addressed the chronology of Jehan’s writings on music theory in light of revised chronologies of his astronomical and astrological works, and I also considered two aspects of Jehan’s scientific orientation that contextualise his work in music theory-namely, how he negotiates the novelty of his work with respect to that of prior authorities (his auctoritates), and his propensity for continuously revising and recasting his writings in order to cater for particular audiences. An abstract of the paper follows, and stay tuned for more in my forthcoming book!
Although he wrote at least two and possibly four substantial music treatises in the 1320s, Jehan des Murs’s primary preoccupation at the beginning of his career appears to have been with figuring out and disseminating within Paris a new system of astronomy devised in the court of Alfonso X in Toledo. Recent scholarship on Jehan’s writings on astronomy offers a more accurate chronology of his output and activity than was the case in 1970 when Ulrich Michels published on the music treatises. We now have a greater understanding of Jehan’s central role in the introduction and development of Alfonsine astronomony in Western Europe—a system that was to hold say until the revolution of the Copernican heliocentric model. This paper reconciles the facts we know of Jehan’s biography with these new assessments of Jehan’s activity as an astronomer in the 1320s. Jehan’s tables of the stars’ and planets’ positions, based upon the experiential data gathered from his own observations made using the latest technologies, along with the new methods he devised for allowing others to easily calculate planetary longitudes, finally offered the possibility of a more precise and accurate calendar. And although Jehan nominally acknowledges his debt to older scholars (in much the same way as he does in Notitia artis musicae), he is more concerned with recording the true positions of the celestial bodies and precisely measuring reality than he is in ‘standing on the shoulders’ of those who have gone before. His writing betrays a confidence (and some might say arrogance) about the veracity and importance of his work and his conclusions. This paper contends that a similar attitude pervaded Jehan’s work as a music theorist where the aim is to more precisely measure, and indeed reform the entire system of musical time with one more clearly based upon the physical reality of sound and time as a sensibly perceived, and a lack of concern with appeasing those invested in older, less accurate, systems.
My review of Yolanda Plumley’s book The Art of Grafted Song: Citation and Allusion in the Age of Machaut (OUP, 2013) has recently been published in Plainsong and Medieval Music, who have graciously provided a link to download a free PDF, available here:
The citation and first paragraph of my review follows:
Karen Desmond. Book review of The Art of Grafted Song: Citation and Allusion in the Age of Machaut by Yolanda Plumley (Oxford University Press, 2013) in Plainsong and Medieval Music 24 (2015), pp. 99-103.
Anyone who listened to BBC radio broadcasts from the 1950s to the 1980s should remember the popular game show My Word. The last segment of the show featured the two team captains, Denis Norden and Frank Muir, competing to tell the best story that would explain, and end with, a famous phrase or quotation supplied by the game show host. Listeners delighted in the anticipation of the catchphrase, for as the clock ran down, it seemed that the phrase became more and more incompatible with the story unfolding. Norden and Muir endeavoured to outdo each other’s displays of erudition, constrained by the generic requirements of the show’s format, the imposed time limit on their story’s length, and the need for the story to end with the supplied quotation. Such competitive composition, or ‘poetic jousting’ as Yolanda Plumley terms it, between medieval poets and composers, is at the centre of her new book on fourteenth-century song, The Art of Grafted Song: Citation and Allusion in the Age of Machaut. In it, through a virtuosic display that traverses almost a century of song, and considers almost 350 works (as listed in the ‘Index of Cited Compositions’ at the end of the book), Plumley examines the circumstances in which, and the processes by which, fourteenth-century faiseurs plucked material from other contexts and ‘grafted’ it—to use Plumley’s horticultural metaphor, derived from the medieval verb enter (p. 10)—into new works.
I’m happy to provide a link to a wonderful book by Catherine Saucier that I reviewed recently for Early Music, and which is now available on the website Oxford University Press through advance access. I’d highly recommend reading this book not only to those interested in medieval liturgy or the city of Liège, but to anyone interested in the involvement of individual actors in the structuring the narrative of their past, and in particular their civic past, and in how we as cultural historians might recover that narrative through careful analysis and consideration of the the partial remnants that survive in the archives, liturgical books, literature, art, and architecture of a medieval city. The link to the PDF by kind courtesy of Oxford University Press follows, as does the citation and first paragraph of my review.
Karen Desmond, ‘Clergy and City’. Book Review of A Paradise of Priests: Singing the Civic and Episcopal Hagiography of Medieval Liège by Catherine Saucier (University of Rochester Press, 2014) in Early Music (April 2015).
In A Paradise of Priests, Catherine Saucier weaves a compelling narrative centered on the lives of Liège’s founder-bishops as celebrated in the hagiography, art, rituals, and music made, enacted, and reenacted by the medieval clerical population of the Liège. Through an expert examination of an impressively vast array of sources—including archival, liturgical, artistic, and hagiographic—Saucier analyses the changing image of the city and its founder-bishops through nine centuries of documentary record. The story centers on the celebration of the lives and deaths of three bishops—Theodard (d. ca. 668), Lambert (d. ca. 700), and Hubert (d. 727)—who were credited with the foundation, promotion, and protection of Liège. Liège, a large and wealthy city, and capital of the prince-bishopric from 985, had large numbers of secular clergy, encompassing a cathedral and seven collegiate church chapters totaling (in the fourteenth century) between 700 to 800 canons (p. 32). It was, as Petrarch observed, a ‘place noted for its clergy’ (p. 4).
Jennifer Saltzstein’s revisionist study of refrains in the music and poetry of medieval France examines why specific refrains were invoked in specific compositional contexts and within specific communities. The results of this study bring into question the long-held assumption that surviving refrains are the remnants of a lost repertoire of orally transmitted popular song. Through her examination of refrain transmission and distribution, Saltzstein instead relocates refrain quotation within the literate traditions of their clerical poet-composers in order to reveal “an intellectual framework for the practice of refrain usage that has largely been overlooked, namely, the conception of auctoritas in medieval writing” (29).
I delivered a paper on a newly-transcribed fourteenth-century treatise on mensurable music (possibly written by Jehan des Murs) at the 42nd Medieval and Renaissance Music Conference held at the University of Birmingham, 3-6 July 2014. I hope to put the text of the treatise online when I have it fully transcribed: so far I have about 85% of its text deciphered. What follows below is the paper abstract, a works cited list, and a video of my presentation slides with an audio recording of my paper.
Within the mid-fourteenth century Parisian manuscript F-Pn lat. 7378A, three as yet unedited music treatises are found, copied in a tiny, highly abbreviated script in a section of the manuscript devoted mostly to the music treatises of Jehan des Murs. The incipits of the three treatises are as follows: ‘Omnes homines scire desiderant’; ‘Partes prolationis quot sunt’ and ‘Celebranda divina sunt officia in ecclesia’. Lawrence Gushee suggested that Jehan des Murs may be their author, since des Murs listed a book loan of a work authored by him with incipit ‘Omnes homines’ in the Escorial manuscript (O.II.10) that contains his autograph annotations. This paper considers the content of the second treatise, which appears to be closely related to Jehan des Murs’s own Compendium artis musicae. The Compendium begins: ‘Partes prolationis quot sunt? Quinque’ whereas the answer to the same opening question posed in the F-Pn lat. 7378A treatise is ‘Quattuor’. The text of this treatise is considered as a witness to early Ars nova theory as it relates to des Murs’s early works and to the transmission of these texts within the layer of F-Pn lat. 7378A that is devoted to works by des Murs (on both music and astronomy) and his contemporaries in these fields.
Chabás, José, and Bernard Goldstein. “John of Murs Revisited: The Kalendarium solis et lune for 1321.” Journal for the History of Astronomy 43 (2012): 412-37.
Chabás, José, and Bernard R. Goldstein. The Alfonsine Tables of Toledo. Vol. 8, Archimedes: New Studies in the History and Philosophy of Science and Technology. Dordrecht: Kluwer, 2003.
———. “Early Alfonsine Astronomy in Paris: The Tables of John Vimond (1320).” (2009): 207-94.
———. “John of Murs’s Tables of 1321.” Journal of Astronomy 40, no. 3 (2009): 297.
Desmond, Karen. “New Light on Jacobus, Author of Speculum musicae.” Plainsong and Medieval Music 9, no. 1 (2000): 19-40.
Goldstein, Bernard R., and David Pingree. Levi ben Gerson’s Prognostication for the Conjunction of 1345. Vol. 80/6, Transactions of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge. Philadelphia: The American Philosophical Society, 1990.
Gushee, Lawrence. “Jehan des Murs and his Milieu.” In Musik – und die Geschichte der Philosophie und Naturwissenschaften im Mittelalter, edited by Frank Hentschel, 339-72. Leiden: Brill, 1998.
———. “New Sources for the Biography of Johannes de Muris.” Journal of the American Musicological Society 22 (1969): 3-26.
Hugonnard-Roche, H. L’oeuvre astronomique de Thémon Juif, maître parisien du XIVe siècle. Genève/Paris1973.
Jacquart, Danielle. “Rapport de la Table ronde Les disciplines du quadrivium (Paris et Oxford, XIIIe-XVe siècles).” In L’Enseignement des disciplines à la Faculté des arts, edited by Olga Weijers et al., 239-47. Leuven: Brepols, 1997.
Michels, Ulrich. Die Musiktraktate des Johannes de Muris. Vol. 8, Beihefte zum Archiv für Musikwissenschaft. Wiesbaden: F. Steiner, 1970.
Poulle, Emmanuel. “The Alfonsine Tables and Alfonso X of Castille.” Journal for the History of Astronomy 19 (1988): 97-113.
———. “Jean de Murs et les tables alphonsines.” Archives d’histoire doctrinale et littéraire du moyen âge 47 (1980): 241-71.
Rico, Gilles. “Music in the Arts Faculty of Paris in the Thirteenth and Early Fourteenth Centuries.” D. Phil. diss., Oxford University, 2005.
Ristory, Heinz. Denkmodelle zur französischen Mensuraltheorie des 14. Jahrhunderts. Vol. 81, Musicological Studies. Ottowa: The Institute of Mediaeval Music, 2004.
Thomson, Ron B. Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera. Toronto: Pontifical Institute of Mediaeval Studies, 1978.
Tomasello, Andrew. Music and Ritual at Papal Avignon, 1309-1403. Ann Arbor, Michigan: UMI Research Press, 1983.
Wathey, Andrew, and Margaret Bent. “Vitry, Philippe de.” Grove Music Online. Oxford Music Online, http://www.oxfordmusiconline.com/subscriber/article/grove/music/29535.
Werner, Eric. “The Mathematical Foundation of Philippe de Vitri’s “Ars nova”.” Journal of the American Musicological Society 9 (1956): 128-32.