1 The term ‘harmonic progression’ is used here to refer to any progression between chords of two or more notes each. The expression ‘directed harmonic progression’ (or ‘directed progression’ for short) is used to refer to any such progression which appears to proceed according to an aesthetic effect of inevitability, meaning in part that the second chord is directly implied by the first.

2 F. Hammond, ‘Jacques de Liège’, in New Grove, ix, pp. 453–5. The reference to ‘cadence’ remains in the revised version of the article in the dictionary's more recent edition: F. Hammond and O. B. Ellsworth, ‘Jacobus of Liège’, in New Grove II, xii, pp. 734–6.

3 S. Fuller, ‘Tendencies and Resolutions: The Directed Progression in Ars Nova Music’, Journal of Music Theory, 36 (1992), pp. 229–58, at 230; J. Bain, ‘Fourteenth-Century French Secular Polyphony and the Problem of Tonal Structure’ (Ph.D. diss., New York University, 2001), pp. 73–4; D. Cohen, ‘“The Imperfect Seeks its Perfection”: Harmonic Progression, Directed Motion, and Aristotelian Physics’, Music Theory Spectrum, 23 (2001), pp. 139–69, at 161 and 164. Margaret Bent has sought to distance Jacques's term cadentia from the phrase ‘directed progression’: M. Bent, Counterpoint, Composition and Musica Ficta (New York and London, 2002), pp. 13–15.

4 Jacques's distinction between consonance (consonantia) and concord (concordia), discussed below, is maintained throughout this article when discussing his theory.

5 See in particular the third paragraph of the chapter. The chapter is given with a translation in App. 1 for reference during this discussion. Letters have been added in the translation of paragraph 4 to aid the discussion of App. 2. The text and all other citations from the treatise are taken from Jacques de Liège, Speculum musicae, ed. R. Bragard, 7 vols. (CSM 3; Rome, 1955–73). Citations of this work give first the book and chapter numbers of the text then the volume and page numbers of the edition. Translations of this and all other cited texts are by the present author except where otherwise stated.

6 See e.g. passages cited in Sarah Fuller, ‘On Sonority in Fourteenth-Century Polyphony: Some Preliminary Reflections’, Journal of Music Theory, 30 (1986), pp. 35–70, at 44.

7 Cohen, ‘“The Imperfect Seeks its Perfection”’, 146.

8 It would follow logically from this that imperfect consonances ought to be named as imperfect forms of the perfect consonance to which they pertain: thus a major sixth would be an imperfect octave. No doubt the weight of tradition would have lain heavily against any move to reform theoretical terminology in this way, had such been at any stage contemplated.

9 Cohen, ‘“The Imperfect Seeks its Perfection”’, 146–8.

10 ‘Natura, quod imperfectum est et incompletum, ut perfectam habeat formam ad id tendens quo deficit moveri compellitur, cumque consonantiae imperfectae seu dissonantiae praedictae consonantium comparatione imperfectae sint, et ipsarum perfectionem non habeant unaquaeque ut inesse consonantis perfectionis constituatur, eam natura gliscit adire. Expertis theoricae peritis haec indubie nota sunt, quia si vel in tertia, sexta vel decima sit vocum positio non fit quies, sed quaelibet ut suae copuletur perfectioni ad eam coacta movetur. Et hinc est quod cantus quilibet mensura ordinatus in perfecta consonantia finem habet, quamvis ante finem ultimum quidam in imperfectis consonantiis seu dissonantiis interdum habeat terminari in quo quia audientis bene disposita auris non quiescit, ultimus consonans finis addicitur.’ Ugolino of Orvieto, Declaratio musicae disciplinae, ed. A. Seay, 3 vols. (CSM 7; Rome, 1955–61), ii, p. 12.

11 The instability of imperfection is raised again towards the end of the treatise as part of the criticism of the new style. SM VII. xlviii; vol. vii, pp. 93–5.

12 ‘For a thing naturally desires its perfection, it tends naturally from the imperfect to the perfect; for all things desire what is good, and flee what is bad and imperfect.’ (‘Appetit enim res naturaliter suam perfectionem, tendit naturaliter de imperfecto ad perfectum; omnia enim bonum appetunt, malum et imperfectionem fugiunt.’) SM VII. xxxviii; vol. vii, p. 77.

13 ‘Since therefore a discant by its nature should concord with the tenor or tenors, it will be so much more perfect when it uses more perfect concords. However as long as it has been properly composed, because not always, not too often is a good, better or even best concord used, but by turns now one, now another, according to a way suitable for its melody. And the discant should not always ascend when the tenor ascends or descend when it descends, but rather ascend when it descends and descend when it ascends; it is allowed now and then in appropriate places, where the beauty of the melody permits, that the discant may descend with the tenor or ascend with the same. For much attention should be paid so that the beauty of its melody is served … [NB. Similar motion is specifically allowed here.]

But I say this in general that, just as form gives being, and good and perfect form gives good and perfect being to appropriate matter, so a good arrangement of voices concording with each other makes good, effective discant, so that he is an expert and discerning discantor or composer of discants who knows the natures of consonances and in what way they should be ordered in discants, which ones are more appropriate at the beginning, which in the middle, and which at the end, and in what way they should be varied alternating between lower and upper voice parts. He should consider this and many other things sought for good discant and observe the end, that there is melody and entertainment for the ears coming forth and arising from discant.’

(‘Cum igitur discantus de natura sua concordare debeat cum tenore vel tenoribus, tanto perfectior erit quanto perfectioribus utitur concordiis. Dum tamen fuerit convenienter compositus, quia non semper, non nimis saepe utendum est una concordia bona, meliore vel etiam optima sed alternatim nunc una, nunc altera secundum modum ipsius cantus aptum. Nec debet semper discantus ascendere cum tenor ascendit vel descendere cum descendit, sed potius ascendere cum descendit et descendere cum ascendit, licet interdum in locis aptis, ubi permittit pulchritudo cantus, possit discantus cum tenore descendere vel cum eodem ascendere. Multum enim est attendendum ut ipsius cantus servetur pulchritudo. … Sed hoc in generali dico quod, sicut forma dat esse, et bona et perfecta forma supra debitam materiam bonum et perfectum esse, sic bona ordinatio vocum concordiam inter se habentium bonum facit discantum efficientem, ut est peritus et discretus discantator vel discantuum compositor qui novit consonantiarum naturas et qualiter in discantibus debeant ordinari, quae aptiores sint in principio, quae in medio, quae in fine, qualiter alternatim sub et supra variari debeant. Haec et alia multa ad bonum discantum requisita consideret et observet finalem, ut est ex discantu proveniens et consurgens melodia auriumque recreatio.’) SM VII. v; vol. vii, pp. 12–13.

14 The language for describing ‘falling’ in Latin texts of the period was predictably straightforward, as in Roger Bacon: ‘it is said that by falling a thing is moved to a place and rests in the place …’ (‘dicitur quod per cadentiam movetur res ad locum et quiescit in loco …’). Opera hactena inedita Roger Baconi, fasc. 13, Questiones supra libros octo physicorum Aristotelis, ed. F. M. Delorme and R. Stelle (London, 1935), p. 409. Jacques's chapter offers nothing that corresponds to or is dependent upon an idea of this sort.

15 The absence of such a formulation in the chapter itself is the more telling given that a suitable expression of the idea can be found in the chapter on motion in the first book: ‘Therefore motion is a certain imperfect state of the mobile thing itself according to which the mobile thing is in potential and tends towards a more perfect state, whether in regard to natural or artificial things.’ (‘Est igitur motus actus quidam ipsius mobilis imperfectus secundum quem mobilem est in potentia et tendit ad actum perfectiorem sive in naturalibus, sive in artificialibus.’) SM I. xxiiii; vol. i, p. 73.

16 Lexicon musicum Latinum medii aevi, ed. M. Bernhard, fasc. 3 (authenticus–canto) (Munich, 1997), p. 281. The later applications of the word referring to closure (mentioned in n. 25 below) seem like specific limitations of this more general use.

17 ‘Musica … est motus vocum consistitque in quantitate, quia in numero vocum simul iunctarum.’ SM VI. lxviiii; vol. vi, p. 191. Compare J. Smits van Waesberghe, ‘De commentator anonymus in Micrologum Guidonis Aretini en zijn verhouding tot de Luiksche muziekschool’, in Muziekgeschiedenis de Middeleeuwen (Tilburg, 1936–46), i, p. 154.

18 SM I. xxiv; vol. i, pp. 74–5.

19 Such is congruent with the general theory of motion in the Middle Ages; see J. A. Weisheipl, ‘The Interpretation of Aristotle's Physics and the Science of Motion’, in N. Kretzmann, A. Kenny and J. Pinborg (eds.), The Cambridge History of Later Medieval Philosophy (Cambridge, 1982), pp. 521–36.

20 ‘Nonne processus ab una voce ad aliam motus quidam est? Nonne ascensus et descensus quidam motus sunt? Et hi ad inaequales pertinent voces, cum quis ascendit de re in mi, vel e converso descendit; et consimiliter in omnium inaequalium vocum coniunctionibus ascensus est vel descensus; et cum possit esse motus qui nec ascensus, nec descensus est, ut ille qui est in spatio aequali, non est omnino expers motus processus qui inter voces est aequales.’ SM VI. lxviiii; vol. vi, p. 191.

21 He makes two further allusions to the problem of the unison without resolving it. The first is near the end of the chapter: ‘And so in singing carefully, one produces often and more often the varied melodies of notes simple and mixed according to arsis and thesis that have been touched on, but he should not neglect to sing appropriately in their places restruck or same-sounding notes and for the first time in the joints of the [Guidonian] hand in which boys are initially trained.’ (‘Diligenter igitur cantando, saepe et saepius proferat quis tactas varias vocum simplicium et mixtarum secundum arsim et thesim modulationes, nec ignoret voces repercussas seu unisonantes suis in locis convenienter decantare et primitus in manus iuncturis in quibus pueri primo instruuntur.’) SM VI. lxviiii; vol. vi, pp. 198–9. The other occurs in the next chapter: ‘And song can thus be described more generally: “Song is the appropriate performance of distinct notes; singing is joining together distinct notes appropriately.” And, according to this, the unison, which requires distinct, though equal, notes, not inflected by arsis and thesis, is not excluded from song, when someone sings re re re re in crying aloud. Surely birds, certain fish and crickets sing? They seem to sound the unison for the most part. And a tenor, which is made purely from repercussions, that is with equal notes sounding the same, is called a song. So the unison should not be excluded from song, just as it is not segregated from the number of the consonances, as was seen above in the second book.’ (‘Potest autem cantus generalius sic describi: “Cantus est apta vocum distinctarum prolatio; cantare est voces distinctas apte coniungere.” Et, secundum hoc, unisonus qui voces requirit distinctas, aequales tamen, non per arsim et thesim inflexas, a cantu non excluditur, cum quis vociferando dicit re re re re. Nonne aves, pisces quidam et cicada cantant? Et tamen unisonare saltem maiore ex parte videntur. Et tenor, qui ex puris confectus est repercussis idest vocibus aequalibus et unisonantibus, cantus quidam dicitur. Non igitur unisonus excludatur a cantu, sicut nec a numero consonantiarum, sicut supra libro secundo visum est, secluditur.’) SM VI. lxx; vol. vi, p. 200.

22 Modern philosophy explains it through a concept of ‘double intentionality’, a distinction between the sounds that are literally heard (which do not move) and what is heard in them (a musical movement). R. Scruton, The Aesthetics of Music (Oxford, 1997), pp. 1–96; id., ‘Musical Movement: A Reply to Budd’, British Journal of Aesthetics, 44 (2004), pp. 184–7.

23 A. Seay, ‘The “Liber Musices” of Florentius de Faxolis’, in Musik und Geschichte: Leo Schrade zum sechzigsten Geburtstag (Cologne, 1963), pp. 71–95, at 88–90. E. E. Lowinsky, ‘Ascanio Sforza's Life: A Key to Josquin's Biography and an Aid to the Chronology of his Works’, in E. E. Lowinsky and B. J. Blackburn (eds.), Josquin des Prez (London, 1976), pp. 31–75, at 48. The date has now been refined to 1485–92 in Florentius de Faxolis, Book on Music, ed. B. J. Blackburn and L. Holford-Strevens (The I Tatti Renaissance Library, 43; Cambridge, Mass. and London, 2010), p. x.

24 Florentius de Faxolis, Book on Music, ed. Blackburn and Holford-Strevens, pp. 156–9 and 285–6, nn. 104–5. Cadentia has already been fleetingly mentioned in the treatise when dealing with counterpoint (Book II, ch. 14), ibid., 144.

25 The linguistic register of the word cadentia and its musicological uses are illuminatingly discussed by L. Holford-Strevens, ‘Humanism and the Language of Music Treatises’, Renaissance Studies, 15 (2001), pp. 415–49, at 430. Faxolis's use of the term should be set against a more or less contemporaneous one in Jean Le Munerat's De moderatione et concordia grammatice et musice (1490), ed. in Don Harran, In Defence of Music: The Case for Music as Argued by a Singer and Scholar of the Late Fifteenth Century (Lincoln, Nebr. and London, 1989), p. 98. The word appears just once there in a discussion of the relationship between words and music in chant, seeming to mean melodic cadence. Instances of the verb cado to mean ‘finish’ (i.e. make a cadence in the modern sense) can be found in Lambertus's Tractatus de musica of the 1270s and Engelbert of Admont's De musica of the early fourteenth (Lexicon musicum Latinum medii aevi, ed. Bernhard, fasc. 3, pp. 281–2). Presumably Le Munerat was drawing on parlance of this kind.

26 P. Aaron, Libri tres de institutione harmonica (Bologna, 1516; repr. New York, 1978), sigs. Giiii^v–Hiiii^r.

27 ‘Cadentia igitur est, cuiuslibet partis cantus particula, in fine cuius, vel quies generalis, vel perfectio reperitur. Vel Cadentia est generalis, vel perfectio reperitur. Vel Cadentia est quaedam ipsius Cantilenae partis terminatio, perinde atque in orationis contextu Media distinctio, atque Distinctio finalis. Studentque periti Musici, ut Cadentiarum Meta fiat, ubi et orationis pars, seu membrum terminat.’ S. Vanneo, Recanetum de musica aurea (Rome, 1533; repr. Bologna, 1969), fols. 85^v–86^r. Vanneo's definition of cadentia is dependent on that given by Tinctoris for clausula: ‘Clausula est cuiuslibet partis cantus particula in fine cuius vel quies generalis vel perfectio reperitur.’ Johannes Tinctoris, Diffinitorium musice: Un dizionario di musica per Beatrice d'Aragona, ed. and trans. Cecilia Panti (Florence, 2004), p. 10.

28 Quotations from Roger Bacon for the first two senses and from Johannes de Garlandia for the third are given by Dictionary of Medieval Latin from British Sources, ed. R. E. Latham, fasc. II C (London, 1981). A quotation using the term in the first sense from Paulus Sanctinus Ducensis in a source of 1330 or 1340 is given in C. de Fresne, D. P. Carpenter and G. A. L. Henschel, Glossarium mediae et infimae Latinitatis, 10 vols. (Niort, 1883), ii, p. 14.

29 Anonymous II, Tractatus de discantu, ed. A. Seay (Colorado Springs, Colo., 1978), p. 32.

30 TML: ANOUPS2. See C. A. Moberg, ‘Om flerstämmig musik i Sverige under medeltiden’, Svensk Tidskrift för Musikforskning, 10 (1928), pp. 67–82, at 72.

31 Franchinus Gaffurius, Extractus parvus musice, ed. F. A. Gallo (Antiquae Musicae Italicae Scriptores, 4; Bologna, 1969), p. 130. Guillaume Guerson, Utillissime musicales Regule cunctis summopere necessarie plani cantus simplicis contrapuncti rerum factarum tonorum et artis accentuandi tam exemplariter quam practice per magistrum Guillermi Guersoni de Villalonga nouitter conpilate (Paris, c. 1495), sig. bvi^v. The requirement of good word setting is moved to third place in the list by Gaffurius, who eliminates the third given by the other authors (which is hard to distinguish from the second).

32 Such usage seems to be echoed by Jacques de Liège in the sixth book of Speculum musicae, where he discusses changes that a psalm tone may undergo to accommodate the words: ‘Again it occurs whenever a monosyllabic word or another which is accented by its nature falls at the end of a mediation, and in such a case not all the notes of the mediation are observed, as appears here: [he gives a music example.]’ (‘Item accidit quandoque ut in fine mediationis cadat dictio monosyllaba vel alia quae de natura sua acuitur, et in tali casu non omnes mediationis voces observantur, ut hic patet: Deus, iudicium tuum regi da etc.’) SM VI. lxxxvi; vol. vi, pp. 249–50.

33 From the surviving sources, Faxolis is the first writer to have used cadentia to refer unambiguously to a distinct musical object (a ‘cadentia’). Jacques always uses the word in the singular, referring (as the discussion below will argue) to an abstract idea. For him, ‘cadentia is said to be in consonances’, not to comprise them.

34 In fact, John of Garland made this same distinction; but although Jacques quotes from and refers to Franco of Cologne and Lambertus (whom he calls Aristotle), he makes no direct allusion to that author. Yet if the quotation in the 87th chapter of the second book (SM II. lxxxvii; vol. iia, p. 204) derives directly from Jerome of Moravia's Tractatus de musica, Jacques was presumably acquainted also with John of Garland's work, which is included in that treatise.

35 He includes the unison in this category. SM II. x; vol. iia, pp. 29–34.

36 ‘Concord is generally taken as a mixture of distinct sounds produced at the same time that is pleasing to the senses … For discord is a mixture of distinct sounds produced at the same time that is harsh and displeasing to the senses.’ (‘Concordia generaliter sumpta … est sonorum distinctorum simul tempore productorum placens apud sensum permixtio … Est enim discordia sonorum distinctorum simul tempore productorum apud sensum dura displicensque permixtio.’) SM IV. xxxi; vol. iv, p. 92. See also R. Crocker, ‘Discant, Counterpoint and Harmony’, Journal of the American Musicological Society, 15 (1962), pp. 1–21; S. Fuller, ‘“Delectabatur in hoc auris”: Some Fourteenth-Century Perspectives on Aural Perception’, Musical Quarterly, 82 (1998), pp. 466–81, at 469–73.

37 ‘Again, since the mixture is the union of other miscible elements, it more properly occurs between concordant notes than between discordant ones. For miscible entities do not remain in their own forms under combination but are changed; and a third [entity] arises from them which is called the mixture, as appears to be so with material elements. And in its way this occurs between concording distinct sounds coming to the ear at the same time.’ (‘Ideo cum mixtio sit miscibilium alteratorum unio, magis proprie locum habet in vocibus concordantibus quam in discordantibus. Miscibilia enim in mixto non manent sub propriis formis sed alteruntur et fit unum tertium ex illis quod mixtum dicitur, ut patet in materialibus elementis. Suo autem modo sic fit in concordantibus simul venientibus ad auditum distinctis sonis.’) SM IV. xxxi; vol. iv, p. 92.

38 The scheme has admirable clarity in Garland's De mensurabili musica: Johannes de Garlandia, De mensurabili musica, kritische Edition mit Kommentar und Interpretation der Notationslehre, ed. E. Reimer (Beihefte zum Archiv für Musikwissenschaft, 10–11; Wiesbaden, 1972), i, pp. 67–72. Lambertus's formulation is difficult to construe because of corruptions in the text. The octave, fifth and fourth, together with their octave compounds, constitute respectively the perfect, middle and imperfect concords. The sixths are imperfect discords, the thirds are middle and the tone, semitone and tritone are perfect. No mention is made of the sevenths. Scriptores de musica medii aevi, ed. E. de Coussemaker, 4 vols. (Paris, 1864–76), i, p. 260; see also S. Fuller, ‘Organum – discantus – contrapunctus in the Middle Ages’, in T. Christensen (ed.), The Cambridge History of Western Music Theory (Cambridge, 2002), pp. 477–502, at 485–7.

39 Jacques's scheme is more cumbersome than Garland's, in part because it does not reduce octave compounds of simple consonances to the same categories as the simple consonances themselves. Even allowing for a subtle sensibility at work here, it is unclear why, for example, the thirds should be middle and the tenths imperfect when the tone is imperfect but the ninth middle. In Garland's scheme, thirds and tenths are all imperfect. The tone and ninth are both discords; but provision is made for situations in which they may be regarded as middle, as is discussed below.

40 ‘Concordia media est cum voces simul prolatae, differe multum ab auditu dinoscuntur; placent tamen sibi … Imperfecta concordia tunc fieri dicitur cum voces, simul prolatae, multum ab auditu differe noscuntur, aliqualiter tamen concordant …’. SM IV. xxxvi and xxxvii; vol. iv, pp. 99 and 100.

41 Mention must be made also of a passage with strikingly similar language from the second book of the treatise: ‘Again, if one of two [singers] holding the extreme notes of a fifth descends and the other ascends through some [note] following, so that they are at a third, they fall into a unison, towards which they tend. They always perform a minor third, not a major one: imagine if one sings la sol in descending, the other re mi in ascending, so that they sound together and concord perfectly in fa, because the imperfect concord tends to the perfect; or if someone sings sol fa, and the other ut re, so that they fall into mi at the unison; these, in the first case and in the second, use the minor third which is between re and fa and between mi and sol.’ (‘Item si duorum diapente voces tenentium extremas per aliquam secundam unus descendat et alius ascendat, ut in tertia, in unisonum, ad quem tendunt, cadunt. Semper semiditono funguntur, non ditono, puta si quis descendendo dicat la sol, alius ascendendo re mi, ut in fa unisonent et perfecte concordent, quia imperfecta concordia tendit ad perfectam, vel si quis dicat sol fa, et alter ut re, ut in mi cadant in unisonum, illi, hic et ibi, semiditono utuntur qui est inter re et fa et inter mi et sol.’) SM II. lxxx; vol. iia, p. 191. Here the minor third is described as tending to the unison; and the singers themselves (metonymically for the music) are described as falling from a third into a unison. Moreover, it is the minor rather than the major third that is ‘always’ performed in singing by contrary stepwise motion from fifth to unison, suggesting that the progression is specific and directed.

While it is clear that cado is used here in a way directly connected with a harmonic progression, it is no more obvious than in the chapter on cadentia what is thought to be ‘falling’ in writing thus; and cadentia itself is at least associated with harmonic progression. The occurrences of tendo seem less ambiguous; but habits of thinking about directed musical motion that are now current did not exist in Jacques's day, so the significance of the locution must not be assumed. The word is used elsewhere in the treatise in connection with melodic progression to a modal final, or simply to some other specified note, without conveying any sense that the musical effect is one of fulfilling a continuation implicit in a given start. The following from the thirty-ninth chapter of the sixth book (‘Concerning the final notes of the modes’; De modorum vocibus finalibus) is typical: ‘And so it should be said that scales have final notes in the defined keys of the monochord. Otherwise, there would be a great confusion in modes or songs if they were not concluded by fixed ends, but could wander to indefinite final notes. And so in the same way as an end is ordained by nature for all consistent things, so in songs and scales there are defined and final notes and keys of the monochord to which all other notes of the melody itself are ordered and tend.’ (‘Dicendum igitur quod toni voces habent finales in determinatis monochordi clavibus. Alias, in modis vel cantibus esset magna confusio si ad certos fines determinati non essent, sed vagari possent in indeterminatis vocibus finalibus. Sicut igitur omnium natura constantium terminus positus est, sic in cantibus et tonis sunt determinatae voces et claves monochordi finales ad quas ceterae omnes ipsius cantus ordinantur atque tendunt voces.’) SM VI. xxxviiii; vol. vi, p. 95. See also vol. vi, pp. 87, 300, 302 and 328.

It seems that tendo indicates here an ordered melodic progression to the modal final; but there is no indication that the motion as such is predictable (i.e. directed) from any given note of the melody: such would be contrary to the nature of chant melody, and contrary to Guido d'Arezzo's account of melodic retrospection (cited below). If the sense of tendo here is carried over to the passage above, then, it is not so much that a particular imperfect concord necessarily implies the following perfect one, but that in practice some perfect concord always follows an imperfect one – a categorial order rather than directed motion. It may not be a very large step from here to assert that the minor third implies a particular unison following by contrary motion, but this is not one that Jacques actually takes.

In any case, the passage can be understood without reference to the concept of the directed progression. It is part of a discussion affirming the greater concordance of the minor third than the major one. To begin with, this is affirmed on the grounds that the minor third includes a tone and a minor semitone rather than two tones (‘there is no complete melody from pure tones’; ‘ex puris tonis nulla est integra melodia’). The progression of the minor third to a unison following a fifth is discussed, and this contributes to the argument by showing that it is the minor rather than the major third that leads to the unison. The language here is descriptive not regulative. The passage argues that the minor third is more concordant than the major one because it is used in progressing to the unison. The unison confers concord on the minor third on account of their adjacency. As will become apparent in the discussion below, the use of cado in this way is congruent with Jacques's concept of cadentia and need not be aligned with any implication of ‘directedness’ in the progression.

42 There seems to be no reason why Jacques should not have suggested that the tone might progress to the fifth, as it often does in Ars antiqua motets and as is exemplified by John of Garland (discussed below).

43 SM I. xxviiii; vol. i, pp. 86–90.

44 ‘Quamvis autem sensus iudicet aliquos sonos concordare, aliquos non, et, quantum ad concordantes, aliquos magis, aliquos minus concordare dicat, et similiter in discordantibus, non est tamen sensus causa concordiae in concordantibus, nec discordiae in discordantibus, sed provenit hoc naturali ex proportione miscibilium vocum vel in proportione ex partibus principalibus talium consonantiarum, seu causis aliis essentialibus vel accidentalibus, quas etsi non noscat sensus, percipere potest eas intellectus.’ SM IV. xxxi; vol. iv, p. 93.

45 Jacques distinguishes two orders of consonances: that according to the way of perfection, which concerns form, and that according to the way of imperfection, which concerns material (and progresses by ascending order of the size of the pitch interval). He states the principle of the way of perfection thus: ‘According to the way of perfection, those [consonances] are former of which the mixtures and proportions are more perfect, so that those whose sounds concord more, are united more and endured by the hearing.’ (‘Secundum viam perfectionis, illae sunt priores quarum mixtiones et proportiones sunt perfectiores, ut sunt illae quarum soni magis concordant, magis uniuntur et se compatiuntur apud auditum.’) SM II. viiii; vol. iia, p. 27. The matter is taken up again closer to the chapter on cadentia in SM IV. xxiiii–xxvi; vol. iv, pp. 61–5.

46 See the discussion in K. Falconer, ‘Consonance, Mode, and Theories of Musica Ficta’ in U. Günther, L. Finscher and J. Dean (eds.), Modality in the Music of the Fourteenth and Fifteenth Centuries/Modalität in der Musik des 14. und 15. Jahrhunderts (Musicological Studies and Documents, 49; Neuhausen-Stuttgart, 1996), pp. 11–29, at 15–18.

47 The falling intended is that conveyed in a passage discussed below from the seventh book: ‘no imperfection falls back or could fall truly into that highest Trinity’ (‘in summa illa Trinitate nulla prorsus cadat vel cadere possit imperfectio’). SM VII. xxx; vol. vi, p. 61.

48 The sleight of hand is noteworthy because of the precision with which concepts of perfection and imperfection are handled in Book VII and the rigour with which they are distinguished.

49 It may be tempting to understand ‘close’ here as a requirement for voice-leading proximity, an idea central to Marchetto of Padua's theory (discussed below). The examples of cadentia (discussed in App. 2) suggest that this cannot be so. For Jacques, the principles of good melodic writing within the individual voices of the polyphony ensured the continuity between one consonance and another (see n. 13 above).

50 In fact, at this point in his reasoning, Jacques suggests that in the proportion ‘g:i’ of two minor semitones, it is ‘g’ that lacks a unit. The reason for this is that he sets the numbers within the frame of a whole tone represented by the proportion 66429:59048 (9:8 × 7381:7381). As ‘i’ is already given, it is ‘g’ that he is inclined to see as deficient to complete the proportion. Later on he acknowledges that ‘i’ is the number lacking in respect of the proportion of two minor semitones (SM II. xlviiii; vol. iia, p. 121). The apotome, or major semitone, is also inaccurately realised in this numerical series: ‘f:h’ (66429:62208) as compared with 531441:497664 ≃ 66430:62208, its accurate calculation. Jacques claims that ‘f:h’ represents the apotome as it includes a minor semitone and the comma, without acknowledging that this will be only an approximation, as the value of the comma itself is not exact.

51 ‘Ea autem, quae sunt prope, sunt quasi idem et illi termini sunt minimi numeri proportionis toni minoris, ut patebit infra.’ SM II. xlviiii; vol. iia, p. 119.

52 ‘Nec ea, quae sunt prope, sunt idem, sed quasi. Non sequitur: “Iste est prope villam, ergo est in villa, vel ad villam.” Haec enim dictio “prope” quamdam vim exceptivam vel exclusivam importare videtur.’ SM II. xlviiii; vol. iia, p. 121. The use of prope for approximation in this passage may be compared with the following from Aquinas: ‘But islands that are close to land are more-or-less parts of the land’ (‘Set insule que sunt prope terram sunt quasi partes terre’). Kevin White, ‘Three Previously Unpublished Chapters from St. Thomas Aquinas's Commentary on Aristotle's Meteora: Sentencia super meteora 2.13–15’, Mediaeval Studies, 54 (1992), pp. 49–93, at 93). Perhaps the idiom was common in speech.

53 ‘Nec putandum est ex uno tactu chordae in aere expansae sonum unum generari, sed plures, tot scilicet quot vicibus chorda tremula percutit aerem sufficienter ad causandum sonum. Sed, quia ea quae prope sunt, quasi nihil differre videntur, illi autem soni vicini propinqui sunt et similes. Non discernens inter illos, sensus iudicat ibi sonum esse unum continuatum, velut color rubeus vel viridis in aliqua parte sumitatis trochi positus, cum trochus velociter rotatur, videtur circulariter totam illius trochi sumitatem afficere, et idem patet in aliquibus iridis coloribus.’ SM I. xxvi; vol. i, p. 80. This passage is echoed in the second chapter of the second book, where the mechanics of sound are recapitulated as a prelude to the discussion of consonances: ‘For the string of a psaltery struck once makes several sounds before it ceases, just as trembling itself it strikes the air many times. These [sounds] do not, however, make consonance, because their distinct note is uncertain, unstable and confused and not well sensed. For sense judges the sound there to be one and continuous, on account of the equality and proximity [of the sounds]. For one [sound], as if suddenly and imperceptibly, snatches the other in regard to weight [i.e prominence or intensity?] or speed, because things that are near seem more or less not to differ at all.’ (‘Percussa enim semel psalterii chorda plures, antequam quiescat, facit sonos, sicut pluries ipsa tremula aerem verberat. Illi tamen consonantiam non faciunt quia incerta est, instabilis et confusa et non bene sensui nota illorum distinctio. Iudicat enim ibi sonum esse sensus unum et continuum propter aequalitatem et vicinitatem. Unus enim, quasi subito et imperceptibiliter, in gravitate vel in velocitate rapit alium, quia, quae propre [recte: prope] sunt, quasi nihil differre videntur.’) SM II. ii; vol. iia, p. 10.

54 Here, as so often in Speculum musicae, Jacques is indebted to Boethius, whose account of a spinning top (turbo) he borrows and adapts. De institutione musica, ed. G. Friedlein (Leipzig, 1867), I. iii (p. 190).

55 Ars nova music, by contrast, permitted strings of imperfect consonances in succession before arriving at perfect consonances. The characterisation of Ars antiqua polyphony as being based on the succession of contrasting sonorities has been illuminatingly discussed in C. Dahlhaus, Untersuchungen über die Entstehung der harmonischen Tonalität (Saabrücker Studien zur Musikwissenschaft, 1968), trans. R. O. Gjerdingen, Studies on the Origin of Harmonic Tonality (Princeton, 1990), pp. 71–83.

56 ‘Concord is given more by the high notes than by the low; and experience teaches, and sense too, that it is so. It contributes to the confirmation of this that the high voice seems more perfect than the low, because the low is closer to silence than the high, just as low and slow movement is calmer than fast. And similarly when a man proceeds in ascending from low notes to high, the spirit seems to emit more in a high [note] than in a low one, to make more and stronger movements.’ (‘Concordia autem amplius datur acutis vocibus quam gravibus et, quod ita sit, experientia docet et sensus. Ad cuius confirmationem facit quod vox acuta perfectior videtur quam gravis, quia gravis magis appropinquat silentio quam acuta, sicut motus gravis et lentus quieti quam velox. Et cum idem homo procedit ascendendo a gravibus vocibus ad acutas, in acutis plures emittere videtur spiritus quam in gravibus, plures et fortiores facere motus’.) SM VII. viii; vol. vii, pp. 20–1.

57 ‘Quodsi concordia imperfecta ante perfectam perficiatur propter cadentiam in ipsam, rationabile videtur ut concordia aliqua, puta diatessaron, augeatur ex perfectiore si communicet cum illa.’ SM VII. viii; vol. vii, pp. 21–2. Presumably ipsam refers to perfectam [concordiam] here.

58 Mention should be made here of the statement heading the third chapter of the Compendium de musica: ‘The reason for the goodness of consonance is perfected cadentia in the genre of inequality.’ (‘Ratio bonitatis consonantiae est perfecta cadentia in genere inaequalitatis.’) Jacques de Liège, Tractatus de consonantiis musicalibus, Tractatus de intonatione tonorum, Compendium de musica, ed. J. Smits van Waesberghe, E. Vetter and E. Visser (Divitiae musicae artis, A/IXa; Buren, 1988), pp. 88–122, at 93. The work has been attributed to Jacques de Liège (ibid., pp. 8–9), and if this identification is correct, then the statement is of evident relevance to an assessment of his concept of cadentia. The doctrine of the Compendium differs from that of the Speculum musicae in various respects (the distinction between consonance and concord is not maintained in the same way, and the unison is not included among the consonances, to cite just two important differences), so the understanding of cadentia may differ also.

The term cadentia is not commented on or explained in what is a highly compressed text, but several occurrences of the verb cado in the second and third chapters may relate to it. The verb appears three times through the two chapters to indicate consonances ‘falling’ into particular ‘genera of inequality’ (i.e., types of proportion, multiple or superparticular, for the interval kinds identified as consonances in the second chapter: fourth, fifth, octave, twelfth and double octave). Perfected cadentia may then simply refer to the fact that all intervals deemed to be consonances fall within these fundamental types of interval proportion. If this is so, the expression seems odd, but it is nonetheless clear that cadentia is regarded as an agency of perfection, in a way different from that of the Speculum musicae but congruous with it. However, the ‘falling into genera’ of these subsequent statements may simply be a standard idiom for the alignment of things with general types and have no direct relationship to cadentia in the specific sense implied by the opening sentence.

The second chapter introduces grades of consonance, similar to the grades of perfect concord introduced in the fourth book of Speculum musicae: ‘Of the consonances, the fourth is good, the fifth better and the octave best. This is therefore because the parts of the definition of the consonance meet in the same [parts] following more and less [i.e., proceeding from greater to smaller: 4:3, 3:2, 2:1].’ (‘Consonantiarum autem bona est diatessaron, diapente melior, diapason vero optima. Hoc ideo quia partes diffinitionis consonantiae eisdem secundum magis et minus competunt’; p. 92.) There is a clear numerical falling in this series, and if the cadentia referred to at the beginning of the second chapter relates to this, then it is like the descent through the order of concords that is indicated in Speculum musicae.

Following the introduction of grades of consonance, the verb cado occurs again, this time in reference to the different auditory effects of the consonances: ‘For the fifth falls more sweetly in the ear than the fourth, and the octave than the fifth.’ (‘Dulcius enim diapente in aure cadit quam diatessaron, et diapason quam diapente’; p. 92.) This suggests an empirical correlate for the rational ‘falling’ through proportions and seems like an adumbration of the more developed theory of contrasting degrees of concordant blend in the cadentia of Speculum musicae. In this tentative form, there is a hierarchy of perfection proceeding from superparticular proportions (fourth and fifth) to the simpler multiple proportion of the octave, and this is matched by the greater sweetness of the octave in comparison with the fourth and fifth.

The precise significance of the Compendium's statement on cadentia is unclear, but the relationship in which it sets the concept to the idea of perfection is unmistakable and tallies with the theory of the Speculum musicae.

59 ‘Haec consonantia, propter summam unionem quam eius important voces, alias de se malas, pro quanto cadunt in eam, reddit meliores … Etiam propter eius bonitatem consonantias de se imperfectas, ut unam sextam vel septimam, ut immediate nituntur cadere in eam, meliores reddit, in hoc cum unisono conveniens.’ SM II. xi and xv; vol. iia, pp. 36 and 45).

60 The sloppy reasoning of Ars nova theorists also roused his ire. D. Tanay, Noting Music, Marking Culture: The Intellectual Context of Rhythmic Notation 1250–1400 (Neuhausen-Stuttgart, 1999), pp. 146–81.

61 Jacques was blind, like his Ars antiqua predecessors, to the duple groupings inherent in the third, fourth and fifth of John of Garland's six rhythmic modes (though following Franco he referred to just five modes). SM VII. xviiii; vol. vii, p. 40.

62 ‘Discant is called “the consonance of distinct songs” because, just as consonance requires distinct notes combined at the same time, so discant [requires] distinct songs combined at the same time; and, just as whatever sounds combined simultaneously do not create a mixture that presents itself smoothly and sweetly to the auditor, so does not every distinct song combined at the same time make discant, but those which concord mutually so that it seems as if there is one song from their good concord although they are several, just as from their distinct notes the octave or fifth make it seem as if there is one sound on account of good concord … Therefore he who discords with the other does not sing discant … What therefore is discant unless it is as if one song comes from two or more distinct songs on account of good concord! To discant is to make it seem as if one song arises from two or more distinct songs through sweet concord.’ (‘Dicitur discantus “consonantia distinctorum cantuum” quia, sicut consonantia requirit distinctas voces simul mixtas, sic discantus distinctos cantus simul mixtos et, sicut non quicumque soni simul mixti faciunt mixtionem suaviter dulciterque auditui se facientem, sic nec omnes distincti cantus simul mixti discantum faciunt, sed illi qui invicem concordant ut per bonam illorum concordiam ex illis fiat quasi cantus unus cum sint plures, sicut ex distinctis vocibus ipsius diapason vel diapente propter bonam concordiam efficitur quasi sonus unus … Qui ergo cum alio discordat, non discantat … Quid est igitur discantus nisi duorum cantuum vel plurium distinctorum propter bonam concordiam quasi cantus unus! Discantare est de duobus vel pluribus distinctis cantibus propter suavem concordiam quasi cantum unum facere.’) SM VII. iv; vol. vii, pp. 10–11.

63 SM VII. xxx; vol. vii, pp. 60–2. In this, Jacques follows the Thomist tradition of analogia entis, found in the treatises of Lambertus and Franco of Cologne. Tanay, Noting Music, Marking Culture, pp. 42–4.

64 ‘[1] For it is said that, before God took up human flesh, the essential salvation of God, song could be from imperfect [time values]. [2] Again, in the same way that God is three in persons, as he is one in substance, natural song should no more be drawn into the divine Trinity than into unity. [3] Again, whether [song] is sung from perfect or from imperfect [time values], God is neither more nor less three and one.’ (‘Dicit enim quod, antequam Deus carnem assumeret humanam, salva Dei essentia, ex imperfectis cantus esse poterat. Item cum similiter sic Deus trinus est in personis, sicut unus in substantia, non plus debet cantus naturalis referri in Trinitatem divinam quam in unitatem. Item sive cantetur ex perfectis sive ex imperfectis, neque plus neque minus Deus est trinus et unus.’) SM VII. xxx; vol. vii, p. 60.

65 ‘Omnis perfectio a summa et prima procedit perfectione et per consequens omnis ternarius, trinitas vel ternareitas ratione perfectionis quam importat iure in primam summam et perfectissimam reducitur Trinitatem. … Et cum cantus ex perfectis ratione ternarii in summam primam et perfectissimam reducatur Trinitatem, non sic cantus ex imperfectis cum in summa illa Trinitate nulla prorsus cadat vel cadere possit imperfectio.’ SM VII. xxx; vol. vii, pp. 60–1.

66 ‘“<Cum similiter sic> Deus trinus est in personis, sicut unus est in substantia, non plus debet cantus naturalis ex perfectis confectus referri in divinam Trinitatem quam in ipsius unitatem.” Dicendum quod cantus naturalis ex perfectis compositus refertur et in divinam Trinitatem et in ipsius unitatem: in Trinitatem ratione perfectionis et distinctionis quam importat ternarius, in ipsam vero unitatem ratione concordiae quae in huius modi cantibus requiritur, nam concordia dicitur distinctarum vocum vel distinctorum cantuum in unum <redacta> concordia.’ SM VII. xxx; vol. vii, p. 61.

67 The ideology of unity underpinning Western polyphonic theory at its earliest stage is examined by D. Cohen, ‘Metaphysics, Ideology, Discipline: Consonance, Dissonance, and the Foundations of Western Polyphony’, Theoria, 7 (1993), pp. 1–85.

68 SM IV. xli; vol. iv, pp. 106–7.

69 ‘But now other discantors or composers of discant observe no requirement of these conditions, they do not make discant with good concord, they make wild cantus, wrongly pleasing, difficult, complicated and more or less unmeasurable, so that they badly heed [two of] the four causes that are needed in discants: the material [cause], that the voices concord when performed together; the formal [cause], that there is a harmonious arrangement, disposition, combination of them.’ (‘Nunc autem aliqui discantores vel discantuum compositores nullam harum conditionum observant proprietatem, discantus faciunt non bonae concordiae, cantus faciunt silvestres, male placentes, difficiles, intricatos et quasi immensurabiles, sicque male custodiunt quattuor causas quae in discantibus requiruntur: materialem, ut sunt voces concordantes simul prolatae; formalem, ut est conveniens ordinatio, dispositio, compositio illarum.’) SM VII. v; vol. vii, p. 13.

70 The style is discussed in S. Fuller, ‘Discant and the Theory of Fifthing’, Acta musicologica, 50 (1978), pp. 241–75.

71 ‘Placet enim multis una tertia, si dulciter in loco suo, scilicet ante unisonum vel diapente, decantetur.’ SM IV. xxxvi; vol. iv, p. 99.

72 See Cohen, ‘Metaphysics, Ideology, Discipline’.

73 ‘Sciendum est, quod omnis discordantia ante perfectam concordantiam sive mediam aequipollet concordantiae mediae, et hoc proprie sumitur ante unisonum vel diapason: Ante unisonum tonu<s>. Tonus ante diapason. et sic de singulis. Et improprie sumitur ante mediam. Sed multum invenitur in multis partibus organi, ut tonus ante diapente, ut in hoc exemplo: Semiton<ium> ante diapente. Tonus ante diapente. Tonus ante diatessaron. Et sciendum, quod numquam ponitur discordantia ante imperfectam concordantiam, nisi sit causa coloris sive pulchritudinis musicae.’ Johannes de Garlandia, De mensurabili musica, ed. Reimer, i, p. 74. The explanation of Garland's hierarchy of discords given by Carl Dahlhaus (Studies on the Origin, p. 80) is at odds with the example given by Garland of the semitone progressing to a fifth.

74 ‘omnis imperfecta discordantia immediate ante concordantiam bene concordat’. Franco of Cologne, Ars cantus mensurabilis, ed. G. Reaney and A. Gilles (CSM 18; [Rome], 1974), p. 68.

75 ‘Et sic patet, quod vilis discordantia sive taediosa, quae est sexta, et refutabilis ab omnibus in maiori parte, et ipsa est paenultima ante perfectam concordantiam, quae est diapason, optima concordantia fit sub tali ordinatione et positione punctorum sive sonorum, ut praedictum est.’ F. Reckow, Der Musiktraktat des Anonymus 4, 2 vols. (Wiesbaden, 1967), i, p. 80.

76 ‘Diaphonia est concors discordia inferiorum vocum cum superioribus, sic dicta quia non per totum proceditur per concordias, sed quia concordia sequens tollit offensionem discordiae prioris, et haec organum communiter appelatur.’ Walter Odington, Summa de speculatione musicae, ed. F. Hammond (CSM 14; [Rome], 1970), p. 127.

77 Richard Taruskin argues that the Guidonian idea ‘discordia concors’ was a central aesthetic tenet of motet composition in the thirteenth and fourteenth centuries, a view that lends Odington's theoretical perspective a special pertinence even though, as an Englishman, he would have been at the periphery of the main developments. R. Taruskin, The Oxford History of Western Music, 6 vols. (New York, 2006), i, pp. 148, 226, 229, 236, 255, 261, 286 and 424.

78 ‘Which are the concording discords: There are six concording discords: semiditone, ditone, diapente with tone, diapason with semiditone, diapason with ditone, diapason with diatessaron. About these enough has been said. Therefore they are called concording discords for the reason that although they themselves discord, put before others [these] bestow a sweeter concord on them.’ (‘Quae sunt concordes discordiae. Concordes discordiae sunt sex: semiditonus, ditonus, diapente cum tono, diapason et semiditonus, diapason et ditonus, diapason et diatessaron. De his satis dictum est. Ideo dictae sunt concordes discordiae eo quod ipsae etsi discorderent aliis propositae suaviorem illis tribuunt concordiam.’) Odington, Summa de speculatione, ed. Hammond, p. 75.

79 ‘Consonantiarum tres sunt per se et perfectae, scilicet: unisonus, diapason et diapente. Tres sunt per accidens, scilicet: semiditonus, ditonus in ordine ad diapente vel unisonum, vel tonus cum diapente in ordine ad diapason; una est perfecta, et non perfecta per accidens, scilicet diatesseron.’ Anonymus, De musica libellus (Ms. Paris, Bibl. Nat., lat. 6286); Anonymus, Tractatus de discantu (Ms. Saint-Dié, Bibl. Municipale, 42); Pseudo-Franco de Colonia, Compendium discantus (Ms. Oxford, Bodl. Libr., Bodley 842); Anonymus, Traitié de deschant (Ms. Paris, Bibl. Nat., lat. 15139); Anonymus, Traitié de deschant (Ms. Paris, Bibl. Nat., lat. 14741), ed. G. Reaney (CSM 36; Neuhausen-Stuttgart, 1996), pp. 50–1.

80 ‘duoroum sonorum sibimet permixtorum ad aurem veniens aspera atque iniocunda permixtio’. The Lucidarium of Marchetto of Padua. A Critical Edition, Translation, and Commentary, ed. and trans. J. W. Herlinger (Chicago and London, 1985), pp. 200–1. All references to Marchetto's Lucidarium cite both text and translation of Herlinger's edition.

81 Ibid.

82 ‘Quanto enim dissonantia minus distat a consonantia, tanto minus distat a sua perfectione et magis assimilatur eidem, et ideo magis amicabilis est auditui, tamquam plus habens de natura consonantie.’ Ibid., pp. 208–9. His theory is not strong, as on its account diminished thirds and augmented sixths should be more ‘endurable by the ear and mind’ than minor thirds or major sixths, being closer than these to the unison and octave respectively.

83 ‘quod uterque sonus ad locum ire cupit ubi est permixtio iocunda, amicabilis, et suavis, hoc est consonantia’. Ibid., pp. 200–1.

84 ‘Oportet enim quod quando due voces sunt in dissonantia que compatitur ab auditu quod ipsarum quelibet requirens consonantiam moveatur ita videlicet, ut si una in sursum tendit, reliqua in deorsum.’ Ibid., pp. 202–3.

85 See the fortieth chapter of the sixth book, ‘That the modes are known principally from their final notes’ (‘Quod a vocibus finalibus principaliter modi cognoscuntur’). SM VI. xl; vol. vi, p. 98. See also n. 41 above for another instance where Jacques's aesthetics of polyphony compares with that of monophony.

86 ‘Cum autem quilibet cantus omnibus vocibus et modis fiat, vox tamen quae cantum terminat, obtinet principatum; ea enim et diutius et morosius sonat. Et praemissae voces, quod tantum exercitatis patet, ita ad eam aptantur, ut mirum in modum quandam ab ea coloris faciem ducere videantur.’ Guido of Arezzo, Micrologus, ed. J. Smits van Waesberghe (CSM 4; [Rome], 1955), 139–40.

87 The idea is particularly clear in a passage from Johannes Boen's Musica: ‘Moderns have increased the similarity between the third and the sixth on account of reciprocal influence, which they have mutually, so that just as three thirds may follow one another, so also may three sixths; and they maintain this for this reason, that a song, which is rated imperfect on account of thirds and sixths, not however being inharmonious draws and attracts the ears, so that the perfection of the song, which is to follow through a fifth or octave, of which the thirds and sixths are heralds and maidservants, they [the moderns] declare sweeter being longer awaited, as here [he gives a music example]; but they do not maintain this for fifths and octaves, in case the ear ceases from its attention, thinking that motion ceases with the end having been achieved.’ (‘Inter tertiam et sextam propter vicissitudinem, quam habent adinvicem, moderni similitudinem adauxerunt, quod sicut tres tertias se invicem sequi licet, ita et tres sextas, hoc ideo statuentes, ut cantus ille, qui per tertias et sextas imperfectus censetur, non tamen discors aures trahat et alliciat, ut perfectionem cantus, qui per quintam sequetur vel octavam, quarum tertie et sexte sunt nuntie et ancille, exspectatam diutius indicent dulciorem, ut hic non autem hoc statuentes in quintis vel octavis, ne auris cesset ab advertentia, putans, quod habito fine cesset motus.’) W. Frobenius, Johannes Boens Musica und seine Konsonanzenlehre (Stuttgart, 1971), p. 70.

88 Jacques himself refers to a motet (Non pepercit Deus nato proprio) that employs intervals larger than an octave in its melodic progression: SM II. xiiii; vol. iia, p. 42.

89 Fuller, ‘Tendencies and Resolutions’, 230. Cohen's translation is similar, though he does not suggest that contrary motion is intended. Cohen, ‘“The Imperfect Seeks its Perfection”’, p. 164.

90 This interpretation renders the meaning of ascendendo and descendendo congruent with the following, less ambiguous passage from the De discantu et consonantiis attributed to Johannes de Muris: ‘And other intervals, namely the semiditone and ditone, the tone with fifth make imperfect consonance, because they tend to ascend or descend into the aforementioned perfect intervals, that is to say, the semiditone into the unison, the ditone into the diapente, the tone with fifth into the octave, by ascending or descending in turn.’ (‘Et aliae species videlicet semiditonus et ditonus, tonus cum diapente faciunt consonantiam imperfectam, quia tendunt ascendere vel descendere in speciebus praedictis perfectis scilicet semiditonus in unisono, ditonus in diapente, tonus cum diapente in diapason, ascendendo vel descendendo seriatim.’) Scriptores ecclesiastici de musica sacra potissimum, ed. M. Gerbert (St Blaise, 1784), iii, p. 306.

It seems here that the minor third is thought to ascend into the unison, and the major third and major sixth to descend respectively into the fifth and octave, following the melodic movement of the lower voice. This is, then, the same principle of description as Jacques's but following the lower rather than the upper voice.

91 Fuller, ‘“Delectabatur in hoc auris”’, 471–2.

92 His examples may be compared with the interval progressions given in the Discantus positio vulgaris, to demonstrate the continuity of Jacques's musical thinking with that of the thirteenth-century Hieronymus de Moravia, Tractatus de musica, ed. S. M. Cserba, 2 vols. (Regensburg, 1935), ii, pp. 191–2. See also Fuller, ‘Organum – discantus – contrapunctus’, pp. 487–8.