¹ ‘Et dividitur nota secundum musicam planam in sex, videlicet ut, re, mi, fa, sol, la. Nam per istas sex notas tota musica noscitur. Ratio huius est haec, quia secundum philosophos talis scientia inter liberales artes sextum tenet gradum. Et quia sicut dicit apostolus Iacobus, Omne datum optimum et caetera, talis scientia repraesentat sextum donum spiritus sancti, quod est donum pietatis. … Sed istae sex notae possunt reduci ad quattuor notas secundum reductionem artis novae, quae sunt ut, re, mi, fa. Et hoc quare: Quia sicut quattuor sunt elementa de quibus totus mundus et ea quae sunt in mundo composita sunt, sic totus cantus per praedictas quattuor notas componitur et versatur.’ J. Vetulus de Anagnia, Liber de musica (henceforth LDM), ed. F. Hammond, Corpus Scriptorum de Musica, 27 (Neuhausen-Stuttgart, 1977), pp. 26–7. Unless otherwise stated, all translations are my own.

² Near the opening of Liber de musica, Vetulus conflates the theory of the rhythmic modes of the 13th-century theorist Franco of Cologne with his own perfections and imperfections (the triple and the duple divisions of notes, respectively), which would arise only in the 14th century: ‘Modus prout spectat ad musicum est cognitio soni cum suis proprietatibus denotata. Nam ubi incipitur modus, potest inciperi divisio seu mensura temporis. Sed proprietates modorum principalium sunt duae, scilicet perfectam et imperfectam, per quas proprietates modorum omnes divisiones reducuntur. Modi vero plurimi sunt et varias habent opiniones. Inter quos Magister Franco, qui fuit primus inventor mensurabilis musicae, assignat quinque modos, alii sex et alii septem.’ LDM, 24.1–5, pp. 34–5. ‘For a musician a mode is a cognition of sound written down with its proprieties. Where a mode begins a division or a measure of the tempus can begin. There are two proprieties of the principal modes, namely the perfect and imperfect. All the divisions are reduced to these proprieties. There are many modes and [music theorists] have various opinions [about them]. Among these Magister Franco, who was the first inventor of measured music, designated five modes; others [designated] six and others seven.’ Later in the treatise, Vetulus emphasises the validity of the rule similis ante similem, whereby like notes before like must be perfect (triple): ‘Sicut longa ante longam valet tria tempora, et brevis ante brevem valet tres semibreves, ita semibrevis ante semibrevem debet valere tres minimas.’ LDM, 54.36, p. 78. ‘Just as a longa before a longa is worth three tempora and a breve before a breve is worth three semibreves, so too must a semibreve before a semibreve be worth three minims.’ In spite of this, Vetulus frequently introduces examples that break this rule. See, for instance, the passage directly preceding Vetulus’s claim that the similis ante similem rule is valid, where he states that a perfect (dotted semibreve) is followed directly by an imperfect semibreve and a perfect semibreve: LDM, 54.34, p. 78.

³ F. Hammond, Introduction to LDM, ed. Hammond, pp. 10–24, at p. 14. Desmond has noted that only Vetulus’s treatise and the Speculum musice by Jacobus link descriptions of the Trinity appearing in human form with mensural notation: K. Desmond, ‘Did Vitry Write an Ars vetus et nova?’, The Journal of Musicology, 32 (2015), pp. 441–89, at p. 459. The only other author to address Vetulus’s theological views is Romolo J. Fisichelli, who identifies several references to scripture: R. J. Fisichelli, ‘John Verulus of Anagni’s Liber de musica: An Introduction to a Study of a Musicologist of the Ars Nova, with Specimen Translations of His Work’ (MA diss., Fordham University, 1953). For Jacobus see Jacobi Leodiensis Speculum musicae, ed. R. Bragard, 7 vols., Corpus Scriptorum de Musica, 3 (Rome, 1973).

⁴ F. Manzari and J. Stoessel, ‘The Intersection of Anglo-French Culture and Angevin Illumination in a Fourteenth-Century Ars Nova Miscellany: A New Dating of Biblioteca Apostolica Vaticana, Barb. Lat. 307 and Sankt Paul im Lavanttal, Archiv des Benediktinerstiftes, MS. 135/6’, Miscellanea Bibliothecae Apostolicae Vaticanae, 25 (2019), pp. 283–311, at p. 298. A central gathering of eight leaves (fols. 17–24) was removed from Vbar307 and is now housed separately (Sankt Paul im Lavanttal, Archiv des Benediktinerstiftes). F. A. Gallo, La teoria della notazione in Italia dalla fine del XIII all’inizio del XV secolo, Antiquae Musicae Italicae subsidia theorica (Bologna, 1966), pp. 68–9, also dated the treatise to c. 1360.

⁵ Attributed to the French composer Philippe de Vitry (1291–1361) in the Middle Ages, the authorship of the body of treatises known as the Ars nova have been the subject of scholarly debate. Sarah Fuller, ‘A Phantom Treatise of the Fourteenth Century? The Ars nova’, The Journal of Musicology, 4 (1985), pp. 23–50, has argued that Vitry’s Ars nova is a ‘phantom’ treatise and that the nebulous assemblage of treatises that transmit Vitriacan theory were compiled by disciples of Philippe de Vitry, but not the composer himself. Karen Desmond, ‘Did Vitry Write an Ars vetus et nova?’, pp. 441–93, has disputed these claims, presenting new evidence that the extant Ars nova treatises were compiled from a now lost Ars vetus et nova by Philippe de Vitry.

⁶ K. Desmond, Music and the Moderni, 1300–1350: The Ars Nova in Theory and Practice (Cambridge, 2018); A. Hicks, Composing the World: Harmony in the Medieval Platonic Cosmos (Oxford, 2017); E. E. Leach, Sung Birds: Music, Nature, and Poetry in the Later Middle Ages (Ithaca, NY, 2018); C. Panti, Filosofia della musica: Tarda Antichità e Medioevo (Rome, 2008).

⁷ This method is informed by the work of Elizabeth Eva Leach, who advocates for an approach that recognises the potential for music to be a silent discipline in Sung Birds, p. 96. As Hicks has shown, Composing the World, p. 69; A. Hicks, ‘“Musica Speculativa” in the Cambridge Commentary on Martianus Capella’s “De nuptiis”’, The Journal of Medieval Latin, 18 (2008), pp. 292–305, at p. 292, the perceived division between practical and speculative music is largely fictitious.

⁸ LDM, 27–27.5, p. 36.

⁹ Gallo, La teoria della notazione, p. 66. Gallo miswrites 1372 in place of 1374: see G. Caetani, Regesta chartarum: Regesto delle pergamene dell’archivio Caetani, 3 vols. (Perugia, 1928), III, p. 21. This is corrected in Manzari and Stoessel, ‘The Intersection of Anglo-French Culture’, p. 295.

¹⁰ Manzari and Stoessel, ‘The Intersection of Anglo-French Culture’, p. 295.

¹¹ P. Montaubin, ‘Entre gloire curiale et vie commune: Le chapitre Cathédral d’Anagni au Xllle siècle’, Mélanges de l’École Française de Rome: Moyen-Âge, 109 (1997), pp. 303–94, at p. 420, provides an overview of references to Jordanus Vetulus de Anagnia. M. Ross, ‘The Papal Chapel 1288–1304: A Study in Institutional and Cultural Change’ (PhD diss. University College London, 2013), p. 266, notes that Jordanus Vetulus de Anagnia was an employee of the Pope in 1295. Jordanus Vetulus de Anagnia is named as the brother of Andreas in L. Paolini and R. Orioli (eds.), Acta S. Officii Bononie ab anno 1291 usque ad annum 1310, Fonti per la Storia d’Italia, 106 (Rome, 1982), p. 139. He is registered as an employee of Pope Boniface VIII in Les registres de Boniface VIII; Recueil des bulles de ce pape publiées ou analysées d’après les manuscrits originaux des archives du Vatican, 4 vols., ed. G. Digard et al. (Paris, 1904–39), IV, p. 140.

¹² Montaubin, ‘Entre gloire curiale’, p. 420.

¹³ Paolini and Orioli (eds.), Acta S. Officii, no. 108, p. 141.

¹⁴ T. Schmidt, ‘Ein Studentenhaus in Bologna zwischen Bonifaz VIII. und den Colonna’, Quellen und Forschungen aus Italienischen Archiven und Bibliotheken, 67 (1987), pp. 108–41, at p. 120.

¹⁵ Manzari and Stoessel, ‘The Intersection of Anglo-French Culture’, p. 309.

¹⁶ ‘Reverendi Magistri Johannis Vetuli de Anagnia musicae doctoris’: Anonymous, De musica mensurabili [= Omnis ars sive doctrina], ed. C. Sweeney; De semibrevibus caudatis, ed. A. Gilles and C. Sweeney, Corpus Scriptorum de Musica, 13 ([Rome]: American Institute of Musicology, 1971), pp. 29–56, at p. 55. Vbar307, fol. 26^v.

¹⁷ For a model approach to the study of music theorists about whom little (or no) biographical information has survived, see R. C. Wegman, ‘The World According to Anonymous IV’, in Qui musicam in se habet: Studies in Honor of Alejandro Enrique Planchart, ed. A. Zayaruznaya et al., Miscellanea, 9 (Middleton, WI, 2015), pp. 1–38.

¹⁸ C. de Coussemaker, Scriptorum de musica medii aevi novam seriem a Gerbertina alteram, 4 vols. (Paris, 1864–76; repr. Hildesheim, 1963), III, p. xxv, went so far as to suggest that Vetulus was a monk.

¹⁹ Hammond, Introduction to LDM, pp. 13–14, proposed that Vetulus was a cleric.

²⁰ P. M. Lefferts, ‘An Anonymous Treatise of the Theory of Frater Robertus de Brunham’, in Quellen und Studien zur Musiktheorie des Mittelalters, ed. M. Bernhard, Veröffentlichungen der Musikhistorischen Kommission, 8 (Munich, 2001), pp. 217–45, at pp. 238–9.

²¹ Vetulus’s work is characterised as French in M. Gozzi, ‘New Light on Italian Trecento Notation’, Recercare, 13 (2001), p. 19; Desmond, Music and the Moderni, p. 196. Desmond discusses similarities between Vetulus’s work and that of the French composer Philippe de Vitry in ‘Did Vitry Write an Ars vetus et nova?’ pp. 460, 466–7. D. Tanay, Noting Music, Marking Culture: The Intellectual Context of Rhythmic Notation, 1250–1400, Musicological Studies and Documents, 46 (Holzgerlingen, 1999), pp. 123–4, characterises Vetulus’s work as Italian, emphasising similarities with that of both the Italian theorist Marchetto da Padova (fl. c. 1317–19) and the English theorist John of Tewkesbury (author of the Quatuor principalia musicae, 1351).

²² For discussion of the differences between the categorisation of cultural boundaries during the later Middle Ages versus today, see A. Butterfield, The Familiar Enemy: Chaucer, Language, and Nation in the Hundred Years War (Oxford, 2009); J. Stoessel, ‘Revisiting Aÿ, mare, amice mi care: Insights into Late Medieval Music Notation’, Early Music, 40 (2012), pp. 455–68, at p. 466.

²³ ‘Dividitur tamen tempus per annum, menses, hebdomodas, dies, quadrantes, horas, punctos, momenta, uncias et atomos. Atomus vero indivisibilis est. Obmissa divisione omnium temporum, videndum est sicut dividitur dies naturalis, ubi cognoscitur mensura temporis secundum musicum. Dicendum est quod in quattuor principales quadrantes dividitur <dies>. Quadrans habet horas sex. De hora nascuntur puncta quattuor. Punctus habet momenta decem. Momentum habet uncias duodecim. Uncia habet atomos 54.’ LDM, 4.1–5.2, p. 28.

²⁴ The ‘natural day’ (dies naturalis), which refers to the temporal span of a complete revolution of the sun (24 hours), may be compared with the ‘artificial day’ (dies artificialis), which refers to the span of time in which the sun is above the horizon between sunrise and sunset. See, for instance, the definition of the natural and artificial days provided in Lectio XI of Robertus Angelus’s commentary (13th century) on Sacrobosco’s De sphaera mundi: ‘The Commentary of Robertus Angelus: Latin Text’, in The Sphere of Sacrobosco and Its Commentators, ed. L. Thorndike, Corpus of Medieval Scientific Texts, 2 (Chicago, 1949), pp. 143–98, at p. 179.

²⁵ A. M. Busse Berger, ‘Notation mensuraliste et autres systèmes de mesure au XIVe siècle’, Médiévales, 32 (1997), pp. 31–41, at pp. 40–1. For a discussion of Vetulus’s musical atomism, see Tanay, Noting Music, Marking Culture, pp. 114–24.

²⁶ Jacobus, Speculum musice, 7.44, VII, p. 85; Busse Berger, ‘Notation mensuraliste et autres systèmes’, p. 39. On the dating of the Speculum musice, see A. Zayaruznaya, ‘Old, New, and Newer Still in Book 7 of the Speculum musice’, Journal of the American Musicological Society, 73 (2020), pp. 95–148. The name Jacobus de Ispania was discovered by Margaret Bent in an inventory of the sacristy of Vicenza Cathedral (1457); see M. Bent, Magister Jacobus de Ispania, Author of the Speculum musicae, Royal Musical Association Monographs, 28 (Farnham and Burlington, VT, 2015), p. 64. R. C. Wegman, ‘Jacobus de Ispania and Liège’, Journal of the Alamire Foundation, 8 (2016), pp. 253–74, has argued that ‘Ispania’ in this context most probably refers to Hesbaye, an area encompassing parts of modern-day Liège.

²⁷ G. Conti, Das Pomerium von Marchetto da Padova: Ontologische Hintergrunde der mensurierten Musik, Rombach Wissenschaft: Reihe Voces, 18 (Freiburg, 2017), p. 304, makes a similar observation.

²⁸ For a recent overview of computus in the European Middle Ages, see C. P. E. Nothaft, Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe (Oxford, 2018). See also B. J. Blackburn and L. Holford-Strevens, The Oxford Companion to the Year, Oxford Companions (Oxford, 1999), pp. 801–28.

²⁹ Nothaft, Scandalous Error, p. 73.

³⁰ See C. P. E. Nothaft, ‘Science at the Papal Palace: Clement VI and the Calendar Reform Project of 1344/45’, Viator, 46 (2015), pp. 277–302; Nothaft, Scandalous Error, pp. 205–34.

³¹ J. Moreton, ‘Before Grosseteste: Roger of Hereford and Calendar Reform in Eleventh- and Twelfth-Century England’, Isis, 86 (1995), pp. 562–86, at p. 574; I. Warntjes, The Munich Computus, Sudhoffs Archiv, 69 (Stuttgart, 2010), p. 6. More specifically, as C. P. E. Nothaft, ‘Roman vs. Arabic Computistics in Twelfth-Century England: A Newly Discovered Source (“Collatio compoti romani et arabici”)’, Early Science and Medicine, 20 (2015), pp. 187–208, at p. 194, explains, the fraction of time represented by the atom is the theoretical addition to every synodic (natural lunar) month that results from the insertion of the cycle of 30-day months and bissextile days (additional days to allow the calendar to realign, such as the day added in the leap year) during the Metonic cycle after the subtraction of the saltus lunae.

³² Some of the many authors who advocate for this standard division of the day include pseudo-Alcuin of York, De cursu et saltu lunae ac bissexto, Patrologia Latina (henceforth PL, compiled by J.-P. Migne), 101 (Paris, 1851), cols. 979–1002A (online at https://www.proquest.com/patrologialatina/docview/2684146958/Z400180175/2656A144285C4E52PQ/10), at col. 980C; Roger of Hereford (1176–8), in C. P. E. Nothaft, ‘Between Crucifixion and Calendar Reform: Medieval Christian Perceptions of the Jewish Lunisolar Calendar’, in Living the Lunar Calendar, ed. J. Ben-Dov et al. (Oxford, 2012), pp. 259–68, at p. 261; Moreton, ‘Before Grosseteste’, p. 574; Gerland (11th century), in Nothaft, ‘Roman vs. Arabic Computistics’, p. 194; Der Computus Gerlandi, ed. and trans. A. Lohr, Sudhoffs Archiv, 61 (Stuttgart, 2013), pp. 115–22; and Cunestabulus (12th century), in C. P. E. Nothaft, ‘A Reluctant Innovator: Graeco-Arabic Astronomy in the Computus of Magister Cunestabulus (1175)’, Early Science and Medicine, 22 (2017), pp. 24–54, at p. 52.

³³ B. Anglicus, De proprietatibus rerum, ed. B. van den Abeele, 3 vols., De Diversis Artibus, 78–9, 109 (Turnhout, 2007), 9.9.26–28, III, p. 361. Anglicus’s text enjoyed considerable success long after his death and survives in at least 298 manuscripts, of which around two-thirds were copied in the 14th century. See M. Franklin-Brown, Reading the World: Encyclopedic Writing in the Scholastic Age (Chicago, 2012), p. 348; H. Meyer, ‘Bartholomäus Anglicus, “De proprietatibus rerum”: Selbstverständnis und Rezeption’, Zeitschrift für Deutsches Altertum und Deutsche Literatur, 117 (1988), pp. 237–74, at p. 238.

³⁴ Conti, Das Pomerium von Marchetto da Padova, p. 306 (in an appendix that discusses the work of Vetulus). The anonymous author defined the part of time that Vetulus terms the ounce as the ‘twelfth part of the moment’: De divisionibus temporum liber, PL 90 (1850), cols. 653–664D (online at https://www.proquest.com/patrologialatina/docview/2684150371/Z400007906/4C1E8359A78C4117PQ/65), at col. 654C. See J. Bisagni, From Atoms to the Cosmos: The Irish Tradition of the Divisions of Time in the Early Middle Ages, Kathleen Hughes Memorial Lectures, 18 (Cambridge, 2020), pp. 16–34, 66–104.

³⁵ A. Machabey, ‘Notions scientifiques disséminées dans les textes musicologiques du moyen âge’, Musica Disciplina, 17 (1963), pp. 7–20, at pp. 8, 16; M. Popin, ‘Temps naturel et temps musical chez Vetulus de Anagnia’, in La rationalisation du temps au XIII ^e siècle: Musique et mentalités, ed. C. Homo-Lechner (Grâne, 1998), pp. 25–30, at p. 28; R. Maurus, Liber de computo, PL 107 (1851), cols. 669–728B (online at https://www.proquest.com/patrologialatina/docview/2684146631/Z500173689/7E79E88ABB4640CDPQ/71), at cols. 677D–678B.

³⁶ Nothaft, ‘Roman vs. Arabic Computistics’, p. 192; Nothaft, Scandalous Error, p. 126; J. Moreton, ‘John of Sacrobosco and the Calendar’, Viator, 25 (1994), pp. 229–39, at pp. 238–9.

³⁷ See C. P. E. Nothaft, ‘The Chronological Treatise Autores Kalendarii of 1317, Attributed to John of Murs: Text and Introduction’, Cahiers de l’Institut du Moyen-Âge Grec et Latin, 82 (2013), pp. 1–89, at p. 35.

³⁸ L. Thorndike, ‘The De constitutione mundi of John Michael Albert of Carrara’, The Romantic Review, 17 (1926), pp. 193–216, at p. 206.

³⁹ ‘Et imperfectis divisionibus, videlicet in 12^am et 8^am, requiruntur multae figurae variae et diversae et specialiter semibreves caudatae variis et diversis modis’. LDM, 53.5, p. 75. ‘In the imperfect divisions, the duodenaria and the octonaria, many different and varied note shapes are found, semibreves in particular, caudated by various and diverse means.’ He later urges caution over the use of the semiminim, which he refers to as a minim that is changed in shape. LDM, 64.5, p. 96.

⁴⁰ Popin, ‘Temps naturel et temps musical’, p. 28; Conti, Das Pomerium von Marchetto da Padova, p. 306.

⁴¹ Vetulus’s system was previously discussed by Hammond, Introduction to LDM, pp. 20–1, who provided a summary of the divisions but not a complete consideration of the prolationes or ‘extensions’ of the system (to be discussed in further detail below).

⁴² J. Dyer, ‘Speculative “Musica” and the Medieval University of Paris’, Music & Letters, 90 (2009), pp. 177–204, at p. 184.

⁴³ Vetulus replaces the longest semibreves of Marchetto’s divisiones tertie or ‘third divisions’ – the duodenaria (12 minims) and the octonaria (8 minims) – with breves (see Figure 2). Thus, the duodenaria division can be divided into three quaternaria divisions (4 minims) and the octonaria into two quaternaria divisions. K. von Fischer, Studien zur italienischen Musik des Trecento und frühen Quattrocento: Das Repertoire, II: Repertoire-Untersuchungen, Publikationen der Schweizerischen Musikforschenden Gesellschaft, ser. 2, 5 (Bern, 1956), p. 112, coined the term ‘Longanotation’ to describe this practice. Gozzi, ‘New Light’, pp. 19–20, argues that Vetulus’s treatise exhibits French influence on the grounds that it theorises modus (the temporal level above breves). M. P. Long, ‘Musical Tastes in Fourteenth-Century Italy: Notational Styles, Scholarly Traditions, and Historical Circumstances’ (PhD diss., Princeton University, 1980), pp. 32–3, has challenged the idea that Longanotation must always be used as a distinctive marker of French influence, since modus is also present in much Italian repertory. M. Gozzi, ‘La cosiddetta “Longanotation”: Nuove prospettive sulla notazione italiana del trecento’, Musica Disciplina, 49 (1995), pp. 121–49, disputes this position.

⁴⁴ A. Stone, ‘Che cosa c’è di più sottile riguardo l’Ars Subtilior?’ Rivista Italiana di Musicologia, 31 (1996), pp. 3–31, at p. 13.

⁴⁵ ‘Prolatio’, at ΛΟΓEION (https://logeion.uchicago.edu/prolatio), tabs ‘LewisShort’ (A Latin Dictionary, ed. C. T. Lewis and C. Short (1st edn, Oxford, 1879; often repr. to 1989)), ‘DMLBS’ (Dictionary of Medieval Latin from British Sources, ed. R. E. Latham and D. R. Howlett, 17 vols. (Oxford, 1975–2011)). See also A. Zayaruznaya, ‘A Minor History of Tempus and Prolation’ (unpublished workshop paper, ‘Future Histories of Music Theory’, Frankfurt, 19 July 2018; abstract at https://www.aesthetics.mpg.de/en/research/research-group-histories-of-music-mind-and-body/events/future-histories-of-music-theory.html).

⁴⁶ A similar sentiment is expressed in the Barcelona Anonymous treatise, ed. in H. Anglès, ‘De cantu organico: Tratado de un autor catalan del siglo XIV’, Anuario Musical, 13 (1958), pp. 3–24, at p. 22, whose author states that there are ‘duo … modi cantandi, sive prolationis’ (two ways of singing or uttering). The ‘modus prolixior’ (the longer way) is of the perfect tempus, and the ‘modus brevior’ (the shorter way) is of the imperfect tempus. There is a conceptual analogue between this author’s association of the term prolatio with tempo and the system of Vetulus because Vetulus’s prolatio refers to the durations of notes in atoms, and thus may also be said to determine tempo.

⁴⁷ In addition to the divisions discussed in this paper, which I term the ‘proper’ divisions, Vetulus theorises a parallel set of ‘improper’ divisions. Both of these sets of divisions are introduced briefly in P. Ovenden, ‘Atoms and Music in Late Medieval Philosophy’, in Atomism in Philosophy: A History from Antiquity to the Present, ed. U. Zilioli (London, 2020), pp. 231–52, at pp. 242–5. He also devises a method for the division of abstract spans of time that are independent of notes, called the ‘semi-perfect’ and ‘semi-imperfect’ divisions. To avoid excessive complexity, I do not offer commentary on these divisions in this paper.

⁴⁸ Lefferts, ‘An Anonymous Treatise’, pp. 238–9.

⁴⁹ Tanay, Noting Music, Marking Culture, p. 125, has observed that the gradus system represents a musical application of the latitude of forms thesis. Desmond develops this idea in Music and the Moderni, pp. 175–83.

⁵⁰ Absolutes, comparatives and superlatives such as ‘long’, ‘longer’ and ‘longest’ belong to the vocabulary of the latitude of forms thesis: J. E. Murdoch, ‘From Social into Intellectual Factors: An Aspect of the Unitary Character of Late Medieval Learning’, in The Cultural Context of Medieval Learning: Proceedings of the First International Colloquium on Philosophy, Science, and Theology in the Middle Ages – September 1973, ed. J. E. Murdoch and E. D. Sylla, Boston Studies in the Philosophy of Science, 26; Synthese Library, 76 (Dordrecht, 1975), pp. 271–348, at p. 282.

⁵¹ J. Vecchi, ‘Anonimi Rubrice breves’, Quadrivium, 10 (1969), pp. 125–34, at pp. 125–6. Gallo, La teoria della notazione, p. 65, identifies similarities between these two authors’ work. See also Gozzi, ‘New Light’, pp. 15–17.

⁵² Gozzi, ‘New Light’, p. 41, argues that the Rubrice describes practice.

⁵³ The Italian word tempo, pl. tempi, is translatable as ‘time’ into English, and refers to the speed or pace of a musical composition. It is not to be confused with the Latin tempus, pl. tempora, which refers to the span of the breve and its division into parts.

⁵⁴ These threefold hierarchical systems are discussed in D. J. Bonge, ‘The Theory and Practice of Measure in Medieval Polyphony to the Ars Nova’ (PhD diss., University of Michigan, 1975), pp. 89–118; Gozzi, ‘New Light’, pp. 8–10.

⁵⁵ Jacobus, Speculum musicae, 7.17, VII, pp. 35–6.

⁵⁶ R. de Handlo, Regule, ed. in Robertus de Handlo: Regule, The Rules; and Johannes Hanboys: Summa, The Summa, ed. and trans. P. M. Lefferts, Greek and Latin Music Theory (Lincoln, NE, and London, 1991), pp. 80–179, at 4.7.4, pp. 104–5.

⁵⁷ Fol. 20^v.

⁵⁸ Gozzi, ‘New Light’, p. 18, suggests that Vetulus’s tempi are three times as slow as would be expected.

⁵⁹ Even though the 14th century was a time in which mechanical clocks were becoming increasingly established, these timekeeping devices were notoriously inaccurate. They provided a method for the measurement of equal hours, but not minutes and seconds. See G. Dohrn-van Rossum, History of the Hour: Clocks and Modern Temporal Orders, trans. T. Dunlap (Chicago, 1996), p. 161.

⁶⁰ Gozzi, ‘New Light’, pp. 39–50, argues that there were three distinct periods of tempo in the 14th century and assigns metronomic durations to the musical beat in these three periods. The earliest period is associated with theorists such as Marchetto of Padua, the ‘period of transition’ is associated with the Rubrice breves, and the later system is expounded by Vetulus and Prosdocimus de Beldemandis (d. 1428). The present discussion operates under the premise that Vetulus’s tempo designations may be indicative of a relative but not an absolute theorisation of tripartite tempi. See R. I. Deford, Tactus, Mensuration, and Rhythm in Renaissance Music (Cambridge, 2015), pp. 180–1.

⁶¹ This contradicts Vetulus’s statements that the tempus is the breve. As S. Gullo, Das Tempo in der Musik des XIII. und XIV. Jahrhunderts, Publikationen der Schweizerischen Musikforschenden Gesellschaft, ser. 2, 10 (Bern, 1964), pp. 73–4, acknowledged, even if the primary time unit were a longa, Vetulus’s tempi would be twice as slow as expected.

⁶² E. Segerman, ‘A Re-examination of the Evidence on Absolute Tempo before 1700: Part II’, Early Music, 24 (1996), pp. 681–90, at p. 685.

⁶³ The classic study of the metaphor of the ‘Great Chain of Being’ is A. O. Lovejoy, The Great Chain of Being: A Study of the History of an Idea (Cambridge, MA, 1936; repr. New York, 1960).

⁶⁴ ‘Cum igitur quaeque ipsarum possit dividi in tres partes aequales sicut nomina trinitatis, videlicet in patre et filio et spiritu sancto. Non quoad divisionem personarum, quia qualis pater talis filius, talis spiritus sanctus, tam quoad considerationem differunt, quia pater in quantum pater differt a filio eo quod maior sit, filius differt a patre eo quod minor sit, testante Christo in evangelio, Pater maior me est. Spiritus sanctus differt a patre et filio eo quod tenet medium inter patrem et filium. Et id quod tenet medium sapit naturam maioris et minoris extremitatis. Unde spiritus sanctus qui est medius sapit naturam patris et filii quia in perfectione idem sunt. Ad similitudinem cuius spiritus sancti, larga minor tenet medium inter largam maiorem et minimam quoad mensuram temporis et continet in se valorem novem temporum, sicut novem sunt chori angelorum cantantes inter deum et homines unusquisque per se novies Kyrie eleison.’ LDM, 28.5–10, p. 37.

⁶⁵ See Augustine, De Trinitate, ed. W. J. Mountain, Library of Latin Texts (https://www.brepols.net/series/llt-o), 15.17.27–15.18.32.

⁶⁶ Although Augustine does not use the specific term ‘medius’ (mean) in the same way that Vetulus does, authors who later developed Augustine’s position that the Holy Spirit was the mutual love between the Father and the Son would use the term to describe the relationship between the Holy Spirit and the Father and Son. T. Aquinas, Contra errores Graecorum, ed. Fratres Praedicatores and R. A. Verardo, Library of Latin Texts, chap. 9, explains this position: ‘Non dicitur esse medius secundum ordinem enumerationis, qui respondet ordini originis, sic enim filius medius est inter patrem et spiritum sanctum; sed dicitur medius quasi communis nexus amborum: est enim communis amor patris et filii.’ ‘[The Holy Spirit] is not said to be the mean [between the Father and the Son] according to the order of numeration, which corresponds to the order of origin, for [in this respect] the Son is the mean between the Father and the Holy Spirit. Rather, the Holy Spirit is said to be the mean as a common bond between them both [the Father and the Son], for he is the common love of the Father and the Son.’ For further discussion of Aquinas’s views on the role of the Holy Spirit as the love between the Father and the Son and his reliance on Augustine, see G. Emery, The Trinitarian Theology of Saint Thomas Aquinas, trans. F. A. Murphy (Oxford, 2010), 231–45.

⁶⁷ ‘Unusquisque per se habet in se tria, duo et unum, in quibus tribus, duobus et uno deus est in medio. Tria vero sunt corpus, anima et bona voluntas, duo vero corpus et anima, unum vero id quod procedit a corpore et ab anima ad dei laudem personandum.’ LDM, 30.6, p. 38. ‘Everyone has in themself three, two and one; in these three, two and one, God is in the middle. The three are the body, the soul and goodwill; the two are the body and the soul; the one is that which goes forth from the body and soul for the resounding praise of God.’

⁶⁸ Vbar307, fol. 3^r–v.

⁶⁹ Manzari and Stoessel, ‘The Intersection of Anglo-French Culture’, p. 290.

⁷⁰ Pseudo-Dionysius was probably a student of the Platonist philosopher Proclus (412–485): E. R. Dodds, Introduction to Proclus, The Elements of Theology, ed. and trans. idem (Oxford, 1992), p. xxvii.

⁷¹ It had been suspected that pseudo-Dionysius was not the real Dionysius for hundreds of years by authors including Hypatius of Ephesus (6th century), Peter Abelard (1079–1142), Thomas Aquinas (1225–1274) and Nicholas of Cusa (1401–1464), but the attribution was not seriously disputed until the 16th century, when comments of Lorenzo Valla (c. 1406–1457) that called this attribution into question were distributed by Desiderius Erasmus (c. 1466–1536): P. Rorem, Pseudo-Dionysius: A Commentary on the Texts and an Introduction to Their Influence (Oxford, 1993), pp. 10, 14–17.

⁷² Rorem, Pseudo-Dionysius, pp. 76–7. That pseudo-Dionysius was the authority for the angelic hierarchies in the 14th century is attested famously by Dante in his Paradiso, in which Gregory laughs when he realises that he had ordered the hierarchies of angels incorrectly.

⁷³ L. M. Harington, A Thirteenth-Century Textbook of Mystical Theology at the University of Paris: The Mystical Theology of Dionysius the Areopagite in Eriugena’s Latin Translation, with the Scholia Translated by Anastasius the Librarian, and Excerpts from Eriugena’s Periphyseon (Paris and Dudley, MA, 2004), p. 27. The influence of John Sarrazin is particularly notable in any consideration of later-medieval Dionysian thinking because his translation of De coelesti hierarchia (c. 1167) would be the reference text of Aquinas, Albertus Magnus and Gallus: M. Edwards, ‘John Sarracenus and His Influence’, in The Oxford Handbook of Dionysius the Areopagite, ed. M. Edwards et al., Oxford Handbooks (Oxford, 2022), pp. 328–41, at pp. 328, 330–8.

⁷⁴ Rorem, Pseudo-Dionysius, p. 77.

⁷⁵ P. Lombard, Sentences, 2.9, in P. Lombard, On Creation, The Sentences, 2, ed. and trans. G. Silano (Toronto, 2008), pp. 38–9; Gregory the Great, Homily on the Gospels, 34.7, in Gregory the Great, Forty Gospel Homilies, ed. and trans. D. D. Hurst, Cistercian Studies, 123 (Kalamazoo, MI, 1990). The order of the virtues and principalities is the reverse of those in pseudo-Dionysius.

⁷⁶ Rorem, Pseudo-Dionysius, p. 20.

⁷⁷ As B. McGinn, The Flowering of Mysticism: Men and Women in the New Mysticism, 1200–1350, The Presence of God, 3 (New York, 1998), p. 83, has illustrated, Dionysius was regarded as the primary authority of both negative and angelic theology in the later Middle Ages.

⁷⁸ Aristotle, Metaphysics, 1026a19: Aristotle, Metaphysica Libri I–XIV: Recensio et translatio Guillelmi de Moerbeka, trans. William of Moerbeke, ed. G. Vuillemin-Diem, Aristoteles Latinus, 25, 3.2 (Leiden, New York, and Cologne, 1995), p. 127; Aristotle, Metaphysica, Libri I–X, XII– XIV: Translatio anonyma sive ‘Media’, trans. Anonymous, ed. G. Vuillemin-Diem, Aristoteles Latinus, 25, 2 (Leiden, 1976), p. 118. The connotation of the term scientia, ‘science’ or ‘knowledge’, greatly differed in the Middle Ages from its present-day sense as a field of research based on the empirical method. As J. A. Weisheipl, ‘Classification of the Sciences in Medieval Thought’, in Nature and Motion in the Middle Ages, ed. J. A. Weisheipl and W. C. Carroll (Washington, DC, 2018), p. 203, notes, science in its medieval sense describes ‘every field of intellectual endeavor in which true causal explanations could be discovered’. The threefold classification of the sciences into the natural (pertaining to the physical world), the mathematical (encompassing the mathematical arts including music) and theology was discussed notably by Boethius in his commentaries on the Isagoge of Porphyry. His work is heavily influenced by Platonist thought but also incorporates Aristotelian elements: see J. Dyer, ‘The Place of Musica in Medieval Classifications of Knowledge’, The Journal of Musicology, 24 (2007), pp. 3–70, at pp. 8–14; Weisheipl, ‘Classification of the Sciences in Medieval Thought’, pp. 207–11. Vetulus does not exhibit knowledge of the subtleties of Boethius’s idiosyncratic approach to the division of the sciences as discussed by Hicks, Composing the World, pp. 70–7.

⁷⁹ Later Christian texts that discuss the classification of the sciences include Cassiodorus (c. 490–585), Isidore of Seville (570–636), Hugh of St Victor, William of Conches (c. 1080–1154), Robert Kilwardby (c. 1215–1279) and Roger Bacon (c. 1220–1292): see Hicks, Composing the World, pp. 67–109; E. Grant, A History of Natural Philosophy (Cambridge, 2006), pp. 165–70.

⁸⁰ Dyer, ‘The Place of Musica’, p. 12.

⁸¹ The four notes of the tetrachord are also compared to the four elements in the fourth Berkeley treatise, written c. 1375. The anonymous author states that the four elements are analogous to the ‘harmony of the world’, referring, presumably, to the fact that the notes D, E, F and G are the finals of the church modes: Anonymous, The Berkeley Manuscript: University of California Music Library, Ms. 744 (Olim Phillipps 4450), ed. and trans. O. B. Ellsworth, Greek and Latin Music Theory (Lincoln, NE, and London, 1984), pp. 190–1.

⁸² ‘Et per arbores praedictas fit ascensus per totam musicam tam planam quam mensuratam usque ad atomum, similiter et reductio. Sed quaeritur quare per has arbores prius ascenditur quam descendatur, quod totum contrarium facit philosophus quando ostendit dialectico ordinationem et constitutionem naturae. Respondetur: Quia natura multum distat ab hac scientia. Nam in natura omne superius constituit suum inferius et maius est eo. Sed in hac scientia quae ad dei laudem inventa est, ut pluries dictum, nullus laudans est maior deo immo minor, et non constituit deum immo ascendit ad dei laudem ut constituatur ab eo.’ LDM, 44.2–6, p. 63.

⁸³ Desmond, Music and the Moderni, pp. 191–6, discusses the use of trees of Porphyry in 14th-century music theory, including that of Vetulus.

⁸⁴ I. Hacking, ‘Trees of Logic, Trees of Porphyry’, in Advancements of Learning: Essays in Honour of Paolo Rossi, ed. J. L. Heilbron, Biblioteca di Nuncius, Studi e Testi, 62 (Florence, 2007), pp. 219–61, at p. 227. A. R. Verboon, ‘The Medieval Tree of Porphyry: An Organic Structure of Logic’, in The Tree: Symbol, Allegory, and Mnemonic Device in Medieval Art and Thought, ed. P. Salonius and A. Worm, International Medieval Research, 20 (Turnhout, 2014), pp. 95–113, provides an overview of trees of Porphyry.

⁸⁵ C. M. Balensuela, Introduction to Ars cantus mensurabilis mensurata per modos iuris, ed. and trans. idem, Greek and Latin Music Theory, 10 (Lincoln, NE, 1994), p. 82.

⁸⁶ See Desmond, Music and the Moderni, p. 195.

⁸⁷ This tree was probably modelled on the arbor consanguinitatis, a type of tree of Porphyry that represents blood relationships and legal issues relating to marriage and inheritance; see Balensuela, Introduction to Ars cantus mensurabilis mensurata per modos iuris, p. 66.

⁸⁸ As Hicks, Composing the World, pp. 38–44, has shown, the creative agency of nature had been discussed in the 12th century, at which time there was a paradigm shift resulting in the reconfiguration of nature as ‘the efficient cause of material creation’. This impulse allowed nature to take on, through God’s will, the more immediate role of causation that had previously been attributed to God alone.

⁸⁹ ‘Potest enim tempus praefatum 8^e maioris prolationis praedictae dividi per binarium numerum. Et quilibet numerus … 4^e maioris prolationis vocatur, quod dividitur et reducitur per modum imperfectum.’ LDM, 37.8–9, p. 45. Later in the treatise Vetulus employs the term reduction repeatedly to describe the groupings of disparate parts of perfections during syncopation.

⁹⁰ On reduction as a return, see McGinn, The Flowering of Mysticism, p. 91.

⁹¹ P. Rorem, ‘Hugh of St Victor and Dionysius’, in The Oxford Handbook of Dionysius the Areopagite, ed. Edwards et al., pp. 367–78, at p. 371; Rorem, Pseudo-Dionysius, pp. 51, 53.

⁹² ‘Finis ad quem tendit est tota laus dei.’ LDM, 1.7, p. 26. ‘The end to which it [music] strives is the complete praise of God.’

⁹³ This text is analysed definitively in Proclus, The Elements of Theology, ed. and trans. Dodds.

⁹⁴ ‘Every effect remains in its cause, proceeds from it, and converts to it.’ Proclus, The Elements of Theology, prop. 35, trans. Dodds, p. 39.

⁹⁵ Rorem, Pseudo-Dionysius, p. 52.

⁹⁶ E. D. Perl, Theophany: The Neoplatonic Philosophy of Dionysius the Areopagite (Albany, NY, 2007), p. 35.

⁹⁷ Translated into Latin in the 12th century, this treatise was transmitted in a large body of manuscripts, both in translation and through commentaries, in the later Middle Ages. For an edition of the Liber de causis, and a list of late-medieval commentaries, see A. Pattin, ‘Le Liber de causis: Édition établie à l’aide de 90 manuscrits avec introduction et notes’, Tijdschrift voor Filosofie, 28 (1966), pp. 90–203. The terminology of ‘ascent’ and ‘descent’ that pervades Vetulus’s text is absent in the Liber de causis, whereas it is omnipresent in the Dionysian corpus. The style and contents of the Liber de causis is also far more learned than that of Vetulus’s work, making this an unlikely origin for the model encountered in Liber de musica.

⁹⁸ For an overview of Dionysian influence in the works of Aquinas and Albertus Magnus, see W. J. Hankey, ‘Dionysius in Albertus Magnus and his Student Thomas Aquinas’, in The Oxford Handbook of Pseudo-Dionysius the Areopagite, ed. Edwards et al. pp. 394–409. B. T. Coolman, Knowledge, Love, and Ecstasy in the Theology of Thomas Gallus (Oxford, 2017), pp. 57–73, discusses Gallus’s adaptation of the doctrine of procession and return.

⁹⁹ Bonaventure was the most prominent Franciscan thinker of the later Middle Ages and was influenced by the thought of Augustine of Hippo (354–430), pseudo-Dionysius and the Victorine school of Chartres. See Bonaventure, De reductione artium ad theologiam, ed. The Fathers of the Collegii S. Bonaventura, Doctoris Seraphici S. Bonaventurae Opera omnia, 5 (Turnhout, 2010).

¹⁰⁰ Reduction is a central precept of Bonaventure’s argumentative process that distinguishes his work from that of contemporaneous thinkers such as Aquinas: see A. Gerken, ‘Identity and Freedom: Bonaventure’s Position and Method’, trans. Myles Parsons, Greyfriars Review, 4, no. 3 (1990), pp. 91–105, at p. 93. As McGinn, The Flowering of Mysticism, p. 91, notes: ‘reduction … forms the heart of the Franciscan’s mode of argumentation, which, in its ceaseless piling up of ternary formulations often may seem arbitrary to those who have not grasped the basic structure of the Bonaventurean system’. De reductione artium ad theologiam is not one of Bonaventure’s most disseminated texts, but this compact treatise is believed to have been a principium – a sermon read at the inauguration of a university course, in this instance at the University of Paris – and as such exercised pedagogical influence. The treatise is transmitted in around 34 manuscripts: see J. C. Benson, ‘Identifying the Literary Genre of the De reductione artium ad theologiam: Bonaventure’s Inaugural Lecture at Paris’, Franciscan Studies, 67 (2009), pp. 149–78, at p. 158. A discussion of Bonaventure’s theory of reduction in texts including De reductione artium ad theologiam, Itinerarium mentis in deum and his commentary on the Sentences of Peter Lombard is set out in G.-H. Allard, ‘La technique de la “reductio” chez Bonaventure’, S. Bonaventura, 1274–1974, 5 vols. (Rome, 1973–4), II, pp. 395–416; A. Speer, ‘Bonaventure and the Question of a Medieval Philosophy’, Medieval Philosophy and Theology, 6 (1997), pp. 25–46, at pp. 40–2.

¹⁰¹ Allard, ‘La technique de la “reductio”’, p. 403.

¹⁰² Allard, ‘La technique de la “reductio”’, pp. 413–14.

¹⁰³ For a summary of Bonaventure’s text, see Z. Hayes, Introduction to Bonaventure, St. Bonaventure’s On the Reduction of the Arts to Theology: Translation with Introduction and Commentary, ed. and trans. idem, Works of Saint Bonaventure, 1 (St Bonaventure, NY, 1996), pp. 1–10.

¹⁰⁴ See A. Speer, ‘Metaphysica reducens: Metaphysik als erste Wissenschaft im Verständnis Bonaventuras’, Recherches de Théologie Ancienne et Médiévale, 57 (1990), pp. 142–82, at pp. 159–60. Under this system, the arts are viewed as a preparatory stage for the study of theology. This stance is also taken by Boethius, whose fusion of Aristotelian and Platonic philosophy advocates for the study of the Liberal Arts as a preparatory stage on the journey to the study of theology: see Weisheipl, ‘Classification of the Sciences in Medieval Thought’, pp. 210–11, 219–20.

¹⁰⁵ P. Rorem, ‘Dionysian Uplifting (Anagogy) in Bonaventure’s “Reductio”’, Franciscan Studies, 70 (2012), pp. 183–8, at p. 187. Along with the literal, tropological and allegorical, anagogy is one of the four basic modes of reading spiritual texts, and originated in the works of early Christian scholars including Origen (c. 185–253), Gregory of Nyssa (c. 335–394), Didymus the Blind (c. 313–398) and Jerome (c. 347–420). See H. de Lubac, Medieval Exegesis: The Four Senses of Scripture, trans. E. M. Macierowski, 3 vols. (Grand Rapids, MI, 1998), II, pp. 179–216; B. McGinn, The Presence of God: A History of Western Christian Mysticism, I: The Foundations of Mysticism: Origins to the Fifth Century (London, 1991), pp. 157–82.

¹⁰⁶ This translation was later reiterated by Hugh of St Victor in his commentary on De coelesti hierarchia, which would ultimately exercise significant influence upon Bonaventure. For a discussion of Eriugena’s reduction into unity as theorised in his Periphyseon, see R. van Nieuwenhove, An Introduction to Medieval Theology (Cambridge, 2012), pp. 64–6.

¹⁰⁷ C. M. Cullen, Bonaventure (Oxford, 2005), pp. 29–30, 89–90.

¹⁰⁸ This model is also encountered in the Didascalion of Hugh of St Victor, which would ultimately influence Bonaventure’s text: R. Martello, ‘St. Bonaventure as a Disciple of Hugh of Saint-Victor: The Influence of the Didascalicon on the Reduction of the Arts to Theology’, Il Santo, 58 (2018), pp. 137–82.

¹⁰⁹ Bonaventure, Itinerarium mentis in Deum, ed. and trans. Z. Hayes, Works of Saint Bonaventure, 2 (Saint Bonaventure, NY, 2002), 2.10, pp. 74–7.

¹¹⁰ Bonaventure, Itinerarium mentis in Deum, 4.2, pp. 96–9; Allard, ‘La technique de la “reductio”’, p. 412.

¹¹¹ This pattern resonates with the distinction that Bonaventure draws between philosophy, a discipline that concerns natural things, and theology, a discipline that provides a mirror to the divine and that through this facilitates greater understanding thereof: see Speer, ‘Metaphysica reducens’, pp. 155–9.

¹¹² Vetulus does not mention theology, but this is implied. As a whole, Liber de musica reads as a devotional text that is actualised through a remarkably intricate music-theoretical system.

¹¹³ This position is advocated in the context of 15th-century music theory in R. C. Wegman, ‘“Musical Understanding” in the 15th Century’, Early Music, 30 (2002), pp. 47–66.