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Published online by Cambridge University Press:  23 February 2022

John N. Crossley*
Monash University
Constant J. Mews*
Monash University
Carol J. Williams*
Monash University


The Musica speculativa of Jean des Murs played a key role in renewing interest in the teaching of Boethius in the fourteenth century. We argue that this treatise is much more than a summary of the Boethian De institutione musica in presenting its core teachings as fully consistent within an Aristotelian theory of knowledge. Two versions of its prologue (1323 and 1325 respectively) are examined together with their relationship to Jean’s Notitia artis musicae (1321) and the innovative significance of its mathematical-style presentation of the teaching of Boethius about proportions with its appeal to clear diagrams. We aim to guide the modern reader through the thought patterns and diagrams of Jean des Murs, demonstrating why the Musica speculativa was so widely studied in the later Middle Ages. The two different prologues are presented in English translation for the first time.

Research Article
© The Author(s), 2022. Published by Cambridge University Press

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The following abbreviations are used:

BnF Bibliothèque nationale de France

SM Jacobus, Speculum musicae


1 There have been many variants of the vernacular version of the name of Johannes de Muris, including Jean de Murs, used in New Grove and John of Murs in Dictionary of Scientific Biography, ed. C. C. Gillispie (New York, 1973), pp. 128–33, article by E. Poulle. Biographical details are given by L. Gushee in ‘New Sources for the Biography of Johannes de Muris’, Journal of the American Musicological Society, 22 (1969), pp. 3–26, and G. L’Huillier, ‘Aspects nouveaux de la biographie de Jehan des Murs’, Archives d’histoire doctrinale et littéraire du moyen âge, 54 (1979), pp. 272–6.

2 The Musica speculativa has been edited by C. Falkenroth, Die Musica speculativa des Johannes de Muris: Kommentar zur Überlieferung und kritische Edition, Beihefte zum Archiv für Musikwissenschaft, 34 (Stuttgart, 1992); E. Witkowska-Zaremba: Musica Muris i nurt spekulatywny w muzykografii średniowiecznej/Musica Muris and Speculative Trend in the Medieval Musicography, Studia Copernicana, 32 (Warsaw, 1992); S. Fast, Johannis de Muris Musica <speculativa>, Institute of Mediaeval Music, Musicological Studies, 61 (Ottawa, 1994), online at, based on her PhD dissertation, submitted under the name of S. Scea, ‘A Critical Edition of Johannes de Muris’s “Musica (speculativa)”’ (University of Iowa, 1990); her choice for the principal text is the manuscript Milan, Biblioteca Ambrosiana, H. 165 inf. The edition and translation of C. Meyer, Jean de Murs, Écrits sur la musique: Traduction et commentaire (Paris, 2000), pp. 134−95, is based on that of Witkowska-Zaremba. In his edition Meyer says: ‘nous a semblé offrir le texte le plus sûr’ (p. 55). The text as edited by Falkenroth is also available in TML (Thesaurus Musicarum Latinarum) for Version A at, and for Version B at In this study, we refer to the editions of the Musica speculativa simply by the editor’s name and within the work by the numbers in the text provided by Witkowska-Zaremba (which are also used by Meyer) and the page numbers in Witkowska-Zaremba and Meyer.

3 Joseph Dyer refers to the renewal of interest in musica speculativa in the early fourteenth century, but claims ‘Muris’s Musica speculativa was intended to rectify the neglect of Boethius by providing a summary of the first two books of De institutione musica for the benefit of those lacking advanced mathematical skills’ in his ‘Speculative “Musica” and the Medieval University of Paris’, Music & Letters, 90 (2009), pp. 177–204, at pp. 179–80. Gilles Rico offers a detailed exposition of interest in Boethius in a university context in his DPhil thesis, ‘Music in the Arts Faculty of Paris in the Thirteenth and Early Fourteenth Centuries’ (Oxford University, 2005), especially pp. 76–148 on glosses on Boethius. Christian Meyer describes the treatise as ‘un résumé axiomatisé’ of the teaching of Boethius in ‘Un abrégé universitaire des deux premiers livres du “de institutione musica” de Boèce: Édition et commentaire’, Archives d’histoire littéraire et doctrinale du moyen âge, 65 (1998), pp. 91–121, at p. 96.

4 From now on all references in the form ‘Boethius II.1’ will refer to De institutione musica unless otherwise indicated.

5 Dyer lays out the evidence of these manuals in ‘Speculative “Musica”’, pp. 189–92. See also Rico, ‘Music in the Arts Faculty of Paris’, p. 30.

6 See the introduction to Codices Boethiani: A Conspectus of Manuscripts of the Works of Boethius, i, ed. M. T. Gibson and L. Smith, Warburg Institute Surveys and Texts, 25 (London, 1995), pp. 26–7; C. Bower, ‘Boethius’ De institutione musica: A Handlist of Manuscripts’, Scriptorium, 42 (1988), pp. 205–51; and M. Bernhard, ‘Glosses on Boethius’ De institutione musica’, in A. Barbera (ed.), Music Theory and its Sources (Notre Dame, Ind., 1990), pp. 136–49. An abridgement of the first two books produced no later than 1273/4 has been edited by Meyer, Jean de Murs.

7 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), online at Johannes de Grocheio, Ars musice 5.5, ed. C. J. Mews, J. N. Crossley, C. Jeffreys, L. McKinnon, and C. J. Williams, TEAMS (Kalamazoo, 2011), pp. 56–8.

8 Grocheio, Ars musice 1.2–4.18, ibid ., pp. 43–56, largely taken from Book 1 of Boethius, De institutione musica. On the date of Grocheio’s treatise, see the editors’ discussion in the Introduction, pp. 10–12.

9 Margaret Bent has discovered an early fifteenth-century inventory recording that Jacques, traditionally identified as ‘of Liège’, was referred to as Magister Jacobus de Hispania, leading her to propose that he could possibly be identified with an Oxford-educated master James of Spain, related to the royal family of Spain; M. Bent, Magister Jacobus de Ispania, Author of the Speculum musicae, Royal Musical Association Monographs, 28 (Farnham, 2015), pp. 6–7 and 82–91. Rob Wegman suggested that Hispania instead could refer to Hesbaye, an archdeaconry of Liège (R. C. Wegman, ‘Jacobus de Ispania and Liège’, Journal of the Alamire Foundation, 8 (2016), pp. 253–74, at p. 253). See Jacques de Liège, Speculum musicae, ed. Roger Bragard, Corpus Scriptorum de Musica, 3 (Rome, 1961), Book II, ch. 56: ‘Timens autem ne tacta Boethii Musica mihi concessa tolleretur a me, ut de ea memoriale <aliquid> mihi retinerem, ut amplius in ea proficerem, ut confidentius illa uti possem, qui de duobus primis libris, quos Parisius audieram, aliqua extraxeram, plura coepi et de illis et de aliis excerpere, in aliquibus locis textum Boethii quem habebam nudum, sine scriptis, sine glossis abbreviare, in aliquibus locis qui mihi difficiliores videbantur, ut occurrebat, exponere in textu et figuris.’ References to this edition will be in the form ‘SM II.56’.

10 SM II.56.

11 SM III.1, 3, p. 11: ‘Hic venerabilis Boethii Musica, quae nunc magna ex parte derelicta videbatur, ad memoriam revocatur. Nam, etsi aliqui in suis musicae tractatibus de illa sumant aliqua, illa sunt pauca et de assumptis quaedam dicunt quae minime ad intentionem vadunt Boethii, ut patebit infra.’

12 SM VII.6.

13 There had been a Dominican convent at Moray (Moravia) since 1233/34. The vitality of Moray is evident from the database People of Medieval Scotland (although Jerome of Moray is not mentioned there), at It is quite possible that Jerome’s interest in all five books of Boethius may have been stimulated by comments of John of Garland in his Musica plana as preserved in Paris, BnF lat. 18514, fols. 85r–94r, ed. C. Meyer, Musica Plana Johannis de Garlandia (Baden-Baden, 1998), pp. 3–21, quoted by Jerome at the outset of his Tractatus. Speaking about chromatic, enharmonic and diatonic scales, John comments (ed. Meyer, p. 4): ‘De hiis tribus supra Boecius 4o que 5. Libro diffusius est prosequtus.’ John seems to refer to the glosses on all five books on Boethius that precede the Musica plana in this manuscript (fols. 1r–85r); the authorship of these glosses deserves further investigation.

14 See the Index fontium in Hieronymus de Moravia, Tractatus de musica, ed. G. Lobrichon and C. Meyer, Corpus Christianorum Continuatio Mediaeualis, 250 (Turnhout, 2012), pp. 272–5. Jerome refers to Book V. 2–3, 8, 9, 11 at the opening and close of his Tractatus. On his reading of Boethius, see C. Meyer, ‘Lecture(s) de Jérôme de Moravie, lecteur de Boèce’, in Michel Huglo and Marcel Peres (eds), Jérôme de Moravie. Un théoricien de la musique dans le milieu intellectuel parisien du XIIIe siècle (Paris, 1992), pp. 55–74.

15 MS Paris, BnF lat. 7378a, fol. 60vb. The Notitia artis musicae is edited by U. Michels in Johannes de Muris, Notitia artis musicae et Compendium musicae practicae. Petrus de Sancto Dionysio: Tractatus de musica, Corpus scriptorum de musica, 17 (Rome, 1972), and edited and translated by Meyer in Jean de Murs, Écrits sur la musique, pp. 58–111. It is available in TML at Karen Desmond argues that the date 1319 given in MS Paris, BnF lat. 7378A, fol. 60vb implies only that the Notitia was written between 1319 and 1321; K. Desmond, Music and the Moderni, 1300–1350 (Cambridge, 2019), p. 28, but the date 1321 comes from Murs himself; see Notitia, ed. Michels, 9.

16 Notitia, Prologue, ed. Meyer, p. 58. The innovative character of the Notitia is emphasised by Dorit Tanay, ‘The Transition from the “Ars Antiqua” to the “Ars Nova”: Evolution or Revolution?’, Musica Disciplina, 46 (1992), 79–104.

17 SM VII.1: ‘Currunt enim, ut aiunt, et ab Aristotele in libro metheorum sumunt opiniones et scientiae revolutiones nam et, ubi nunc est arida, prius fuit aqua.’ Notitia II.15, ed. Meyer, p. 110: ‘Currunt enim opiniones et scientiae revolutiones ad circulum revertentes, quamdiu summae placuerit voluntati eius, qui non necessitatus omnia condidit in hoc mundo et omnia voluntarie segregabit.’

18 SM VII.1. Aristoteles Latinus, Liber Meteorum X.2.II, ed G. Vuillemin-Diem (Brussels, 2008), p. 85: ‘Accidit enim sepe in talibus repulsa prima parte fluentis corporis propter non subcedere, aut propter artitudinem aut propter repercutere, circulum et reuolutionem fieri spiritus. Hoc quidem enim in anterius prohibet procedere, hoc autem posterius impellit, quare compellitur in latus, qua non prohibetur, ferri, et sic semper habitum, donec utique unum fiat, hoc autem est circulus: cuius enim una latio figure, hoc necesse circulum esse.’

19 Witkowska-Zaremba provides an English language summary of the manuscripts of Version A on pp. 148–66. Using her sigla, the closing rubric to the Musica speculativa in (O) Oxford, Bodleian Library, Bodley, 77, fol. 99va (1450–1500), describes the work as an abbreviation of the De institutione musica of Boethius: ‘Explicit musica Boecii abbreviata a magistro Johanne de Muris anno domini 1323 mense iunii parisius in sarbona [sic].’ By contrast, (S) St Paul in Lavantthal, Benediktinerstift, 264/4 (c. 1400) offers the title by which it is more often known: ‘Explicit musica speculativa secundum Boetium per magistrum Johannem de Muris abbreviata Parisius in Sorbona anno domini 1323.’ This copy concludes with a diagram of the monochordal instrument of des Murs and explanatory notes. While questions about the dates offered by U. Michels, Die Musiktraktate des Johannes de Muris, Beihefte zum Archiv für Musikwissenschaft, 8 (Wiesbaden, 1970), are raised by Desmond in Music and the Moderni, p. 100, the fact that des Murs was very attentive to chronology in his astronomical writings suggests that these rubrics deserve respect.

20 Aristotle, Politics VIII.3, ed. F. Susemihl (Leipzig, 1872), p. 339: ‘propter quod quidem Homerus ita poetizavit: “sed est quidem velut ad epulas <vocare> congaudere” et ita dicens alteros quosdam “qui vocant cantitaturam naturam tamquam delectantem omnes”.’

21 Musica speculativa, I.10 W-Z (hereafter I.10 W-Z), p. 172; Meyer, p. 136: ‘Verum quia istis diebus libri antiquorum philosophorum nedum de musica, sed et de ceteris mathematicorum pluribus non leguntur, et ob hoc accidit eos tamquam inintelligibiles au nimis difficiles abhorreri, visum est mihi bonum, ut ex Musica Boetii, quam secundum vires mihi a Deo datas perstudui eamque favente Deo aliqualiter intellexi, tractatum brevem elicerem, in quo conclusiones pulchriores et essentialiores ad ipsam artem musicae pertinentes cum sermonis claritate et evidentia sententiae manifestare conabor.’

22 I.11−14 W-Z, p. 173; Meyer, p. 136:

‘Omnem doctrinam et omnem disciplinam ex praeexistenti cognitione fieri.

‘Ante cognitionem sensitivam non aliam inveniri.

‘Experientiae multiplici ut in termino status acquiescere.

‘Experientiam circa res sensibiles artem facere.’

We have followed the reading of Milan, Biblioteca Ambrosiana, H. 165 inf, fol. 1r.

23 Notitia II.14, in Meyer, Jean de Murs, p. 110: ‘Nemo tamen dicat nos statum musicae et finem eius immutabilem tetigisse.’

24 The four propositions are in I.15−77 W-Z, pp. 173−6; Meyer, pp. 138−42: ‘Pythagoram nobis artem tradidisse sonorum; … Propter symphoniam subiungere vim numerorum; … Iam tres harmonias perfectas esse sonantes; … Has tres melodias numeros dare clarificantes.’

25 Concerning the two versions see F. Hentschel, Sinnlichkeit und Vernunft in der mittelalterlichen Musiktheorie: Strategien der Konsonanzwertung und der Gegenstand der Musica Sonora um 1300 (Stuttgart, 2000), pp. 314–22.

26 Michels, Die Musiktraktate, p. 21, n. 21 acknowledges the assistance of Frobenius in understanding the explicit: ‘Hic liber expletur, si <sic ms.> quid nimis est, resecetur, si minus <nimis ms.> addatur, et sic ars vera paratur. Summe decem cubice, duplate, terque quadrate et semis: armoniae sunt hec <hee ms.> sic ab<b>reviate.’ (‘Here ends the book, which, if it is too much, can be shortened, if too little, added to, and thus the true art is obtained. Take ten cubed, a double and three squares and a half: these Harmonies are thus abbreviated. Summe decem cubice … may be interpreted as ‘ten cubed [plus] a double [of 10] and three times the square, and a half’, that is to say 103 + 2 × 10 + 3 × 102 + 10/2, or 1325. Michels interprets this as saying that this 1325 version was a shortening of the 1323 version, which was no longer being read in the Sorbonne.

27 On the decree, see Helmut Hucke, ‘Das Dekret “Docta Sanctorum Patrum”’, Musica Disciplina, 38 (1984), 119–31.

28 This version is transcribed in TML, accessed 27 August 2020. It was labelled as Version A/B by Falkenroth, Die Musica speculativa des Johannes de Muris, pp. 13–15. More comprehensive textual comparison is needed to identify precisely how the Version B, preserved in several copies, including Ambrosiana H. 165 inf., fols. 126r–132v, relates to the ‘mixed’ (possibly transitional) version preserved in three MSS: Milan, Bibl. Ambrosiana C. 241 inf. (copied in Paris in 1401); Berlin, Staatsbibliothek Preußischer Kulturbesitz, lat. fol. 600 and Paris, BnF lat. 7295; see Michels, Notitia artis musicae, p. 16, and Hentschel, Sinnlichkeit und Vernunft, pp. 317 and 319.

29 Milan, Biblioteca Ambrosiana, H. 165 inf., fol. 1r: ‘QVoniam musica est de sono relato ad numeros adequatio necesse est utrumque numerum, scilicet, et sonum simul consyderare.’

30 Gushee, ‘New Sources’, p. 6. As noted in J. N. Crossley, ‘The Writings of Boethius and the Cogitations of Jacobus de Ispania on Musical Proportions’, Early Music History, 36 (2017), pp. 1–30, at p. 26, n. 127, the numbers in the diagrams seem to have been copied by someone other than the main scribe.

31 The texts in the editions of the Musica speculativa of Jean des Murs are exemplary insofar as the editors have consulted a huge range of manuscripts and noted variant readings. The diagrams, however, fare much less well. Witkowska-Zaremba’s, which Meyer uses, is almost error free. The editions by Fast and Falkenroth, however, have numerous errors.

32 Al-Khwārizmī’s Arithmetic was the basis for algorism, the methods of calculating with Arabic numerals. Jean’s Quadripartitum numerorum contains an extended exposition of al-Khwārizmī’s Algebra, though it must be admitted this work only dates from 1343, twenty years or so after the Musica speculativa; G. L’Huillier, Le “Quadripartitum numerorum” de Jean de Murs: Introd. et éd. critique (Geneva and Paris, 1990).

33 See Poulle, ‘John of Murs’.

34 I.85 W-Z, p. 177; Meyer, p. 144: conclusio prima: ‘patet in hoc figura’; I.154 W-Z, p. 181; Meyer, p. 152: conclusio sexta: ‘In figuris haec omnia declarantur’; I.176 W-Z, p. 182; Meyer, p. 154: conclusio septima: ‘Et hoc patet in figura’; I.217 W-Z, p. 185; Meyer, p. 158: conclusio duodecima : ‘Quae autem nunc dicta sunt, praesens figura declarat; I.249 W-Z, p. 188; Meyer, p. 164: conclusio quarta decima: ‘ut videri potest in hac figura’.

35 The designations ‘Figura D’, etc., are taken from Witkowska-Zaremba. The calculations for this figure will be discussed below; see Table 2.

36 See, for example, Boethius III.9, in Anicii Manlii Torquati Severini Boeti:. De institutione arithmetica libri duo. De institutione musica libri quinque, ed. G. Friedlein (Leipzig, 1867), p. 279.

37 See also below, n. 42 on Part II where Jean des Murs uses frequencies as well as string lengths.

38 Boethius, De institutione arithmetica, II.54; this is the last figure of the De institutione arithmetica and may be found at the very end, p. 173, in Friedlein’s edition. Note that the orientation of the numbers varies; see n. 42 below.

39 This sentence does not occur in Friedlein’s edition and may be a later addition, but does in Anicius Manlius Severinus Boethius, De institutione musica, Patrologia cursus completus, series latina, ed. J. P. Migne, 221 vols. (Paris, 1844–1904), 63, col. 1178B.

40 In Witkowska-Zaremba, p. 176, there are errors in the text of the proportions in the diagram, but the correct proportion names are to be found in Paris, BnF lat. 7207, fol. 294v, though the numbers are reversed from left to right; on the variation in orientation see n. 42 below. Meyer’s version is correct (p. 142).

41 See Crossley, ‘Cogitations’, p. 10.

42 The orientation of the figures with the numbers increasing from left to right or vice versa seems not to have troubled medieval readers. Cf. Boethius, Fundamentals of Music, trans., with Introduction and notes by C. M. Bower, ed. C. V. Palisca (New Haven and London, 1989), p. 98, n. 19, discussing the diagrams in Boethian manuscripts, specifically those for Boethius III.3: ‘The diagrams … reflect a stage of musical thought in which direction (left or right) had no implication with respect to pitch. In modern terms one can think of the numbers as representing the lengths of a string in one direction and the frequency of the pitches for the other direction: a string twice as long creates a note an octave lower, while a note of twice the frequency is an octave higher.’ However, it is not until Part II, Propositio sexta, that Jean des Murs notes the inverse connection between frequency and string length (II.91–3 W-Z, p. 201, Meyer, p. 189). ‘Et quoniam vult Boetius, quod longior chorda plures partes et maius spatium obtinet breviori, ergo sibi maiorem numerum attribuit et sonum gravem, breviori chordae minorem numerum et sonum acutum, licet fieri aeque bene posset e converso: longiori, eo quod pauciores motus continet breviori, sibi minorem numerum dare, breviori vero plures numeros, eo quod plures motus habet, ut ipse innuit libro suo; quocumque modo fiat, numquam proportio variatur.’ (But he does have a related remark in propositio seconda, II.28–31 W-Z, p. 182: Meyer, p. 177.)

43 Equivalently (n + 1) : n. Jean des Murs is unusual in that he happily uses fractions in some of his work, though the aim was always to present proportions by ratios reduced to their lowest terms where the numbers involved were whole numbers. For example, a sesquitertian proportion of 8 : 6 would be presented using the ratio 4 : 3.

44 I.77 W-Z, p. 176; Meyer, p. 142: ‘iam tempus est huius figurae misteria et inclusa mirabilia extrahere sigillatim’.

45 I.71 W-Z, p. 176; Meyer, p. 142: ‘Sed haec figura quasi unum chaos, in quo latitant plures formae, potest satis rationabiliter appellari.’ Chaos is the Latin version of the Arabic قوس (qaws), which means ‘arch’ or ‘bow’ or ‘arc of a circle’ and is also the name of the constellation Sagittarius. (Thanks to Charles Burnett for this elucidation in London and email of 26 June 2018.)

46 Boethius, De institutione arithmetica, II.54, p. 173 in Friedlein’s edition.

47 Bellissima reports that the ancient Greeks used the sequence fourth, tone, fourth. See F. Bellissima, ‘Propositions VIII.4–5 of Euclid’s Elements and the Compounding of Ratios on the Monochord’, BSHM Bulletin: Journal of the British Society for the History of Mathematics, 30 (2015), pp. 183–99, at p. 193.

48 I.79 W-Z, p. 177; Meyer, p. 144: ‘Omnis proportio superparticularis minor est proportione multiplici, quoniam multiplex continet minorem integre, ut bis vel ter vel quater et sic deinceps; sed superparticularis numerus numquam continet bis minorem, sed semel cum aliqua sui parte, ut media, tertia, quarta et cetera, semper augendo partes et minuendo proportiones.’ More than one whole plus one aliquot part is a multi-superparticular proportion and if there are more parts it is a multi-superpartient proportion. Murs only deals with one of these latter (in conclusio quinta decima near the end of his treatise in its first form; see below). However, other redactions do include definitions of these terms; see e.g. Musica speculativa, ed. Falkenroth, p. 87.

49 At the end of conclusio septima (I.175 W-Z, p. 182; Meyer, p. 154) he does simply say: ‘Et hoc patet in figura’ (‘And this is clear in the figure’).

50 See Crossley, ‘Cogitations’, p. 18.

51 See Bellissima, ‘Propositions’, pp. 192–3.

52 As mentioned earlier, when dealing with proportions medieval writers did not necessarily specify the order of the numbers; a sesquialtera proportion simply meant that one number was one and a half times the other.

53 In the text here, following the discussion in the seventh conclusion about dividing the tone, he repeats the Boethian comment that a semitone is called such because it is imperfect, not because it is half a tone. Figura E is, in part, on fol. 295v of Paris, BnF lat. 7207, but does not have all the arcs and only contains the top line of the table.

54 For a discussion of the terminology, see Crossley, ‘Cogitations’, p. 13.

55 I.168 W-Z, p. 182; Meyer, p. 154: ‘Et est ars inveniendi tales harmonias talis.’

56 The fifth number, 273⅜, corresponding to three whole tones, he has calculated earlier, before he gives the explanation (I.164 W-Z, p. 182; Meyer, p. 154).

57 We do not know why 682⅔ is omitted; perhaps because of the fraction two-thirds.

58 Des Murs uses the word pasodia, which has caused much discussion but simply seems to be an amalgam of πασῶν and διά with the order reversed from the normal diapason. It also occurs in the fifteenth-century MS Ghent, Rijksuniversiteit 70 (71), fol; 103r; see Cuiusdam Cartusiensis Monachi Tractatus de Musica Plana, ed. S. Lebedev (Tutzing, 2000), p. 50.

59 The Greek names are only introduced in Figura F; see below.

60 I.217 W-Z, p. 185; Meyer, p. 158: ‘Quae autem nunc dicta sunt, praesens figura declarat.’ See also below regarding the differentia.

61 A full discussion of the method may be found in Crossley, ‘Cogitations’, pp. 20–2.

62 Figura H W-Z, p. 188; Meyer, p. 162.

63 I.244 W-Z, p. 187; Meyer, p. 162: ‘Propter hoc vide figuram subscriptam.’

64 I.249 W-Z, p. 188; Meyer, p. 164: ‘ut videri potest in hac figura’.

65 Boethius uses them in his Book III.

66 They would represent different fractions of a tone.

67 I.249 W-Z, p. 188; Meyer, p. 164: ‘ut videri potest in hac figura’.

68 I.270 W-Z, p. 189; Meyer, p. 166: ‘verum semitonium in rerum natura non existere’.

69 More than one whole plus one aliquot part to one is a multi-superparticular proportion, e.g. 1⅓ : 1, and if there are more parts it is a multi-superpartient proportion, e.g. 2⅔ : 1.

70 I.271 W-Z, p. 189; Meyer, p. 166, read: ‘comma in numero nullo esse, sed maius 75 et minus 74’. Clearly the first words contradict the previous statement that the comma is expressible by the ratio 531441 : 524288 (or 1 and 7153/524288 : 1), but then the rest seems to be taken from Boethius III.12: ‘In qua numerorum proportione sit comma, et quoniam in ea quae major sit quam 75 ad 74, minor quam 74 ad 73.’ (There are many different versions in the various manuscripts of the Musica speculativa.)

71 A major semitone is greater than 16 : 15 and less than 15 : 14. This is correctly presented in a comment in MS Munich, Universitätsbibliothek 4º Cod. ms. 743, and this is the only correct manuscript version cited by Witkowska-Zaremba, p. 231.

72 I.154 W-Z, p. 181; Meyer, p. 152: ‘In figuris haec omnia declarantur.’

73 Boethius, Fundamentals of Music, trans. Bower, p. xxxi.

74 I.304 W-Z, p. 191; Meyer, p. 170: ‘Nec Boetius in sua Musica nec alii musici, quos viderim, hanc quaestionem determinant.’

75 Cf. Boethius VII.6: ‘Utrum diatessaron ante diapente sit consonantia.’

76 I.305 W-Z, p. 191, Meyer, p. 170: ‘non est dubium, quin supra diapente optima sit et dulcis’.

77 See Witkowska-Zaremba, p. 192; Meyer, p. 173. The diagram is not included here.

78 I.316 W-Z; Meyer, p. 172: ‘Et sic diatessaron non ex se est, sed ex duabus consonantiis actu, scilicet diatessaron et diapente, quibus duobus positis impossibile est illam non poni, immo forte non nisi secundum quid differt ab ambabus.’ Note that Meyer corrects Witkowska-Zaremba, who interchanges diatessaron and diapason. This accords with Paris, BnF lat. 7207, fol. 297v.

79 II.1 W-Z, p. 193, Meyer, p. 174: ‘in quo omnes consonantiae et earum partes et partes partium denotantur, ac inde per consequens de compositione variorum instrumentorum et cognitione ignotorum, inventioneque novorum maximam praestat fidem’. Further, although he only goes as far as two octaves plus a fifth, he is explicit that the methods can be extended further (see II.127 W-Z, p. 203; Meyer, p. 150).

80 M. Husson, ‘Coping with the Tiny: Measures, Computations, and Reasoning with Small Amounts in Jean des Murs’s Quadrivial Works’, Erudition and the Republic of Letters, 4 (2019), pp. 146−65, at p. 153.

81 Paris, BnF lat. 7207 does not include this figure.

82 Jordanus de Nemore, De Elementis Arithmetice Artis: A Medieval Treatise on Number Theory, ed. H. L. L. Busard, 2 vols. (Stuttgart, 1991).

83 A simple version, akin to our version of Figura L, may be found in Milan, Biblioteca Ambrosiana, H. 165 inf, fol. 11v.

84 Meyer, p. 175, n. 63, notes that understanding this caused difficulties and, in certain manuscripts, phrases had been added to clarify matters.

85 II.14 W-Z, p. 194; Meyer, pp. 174: ‘ut auris habeat consonantiam iudicare, quam prius ignorabas’.

86 For an interesting discussion of a fourteenth-century explanation of this tetrachord see J. Bates and S. McCoy, ‘Mercury’s Tetrachord’, Early Music, 10 (1982), 213–15.

87 II.14 W-Z, p. 194; Meyer, p. 174: ‘et informatione intellectus et sensus, miraberis circa sonorum consonantias apparentes, tunc mirabiles consonantias naturales iudicabis tunc mirabiles consonantias naturales iudicabis’.

88 Boethius, ed. Friedlein, p. 279. Cf. n. 66 above.

89 In Paris, BnF lat. 7207, fol. 298v, the numbers have been inserted: 192 for b, 288 for c and 216 for d, which correspond to the relevant proportions.

90 Boethius, ed. Friedlein, pp. 283–5.

91 Ibid., p. 281.

92 The diagram, Figura P, in Witkowska-Zaremba (p. 198, repeated in Meyer, p. 182) is at best confusing since the semicircles are identical, which means that the high and the low commas both occur at the left, rather than one on the left and one on the right as in Paris, BnF lat. 7207, fol. 298v.

93 II.48 W-Z, p. 198; Meyer, p. 182: ‘His expeditis monochordum scire velitis.’

94 II.50 W-Z, p. 198; Meyer, p. 182: ‘de earum reductione ad sensibiles figuras, quae multum placent mathematicis, quoniam veritas, quae est in intellectu, per eas ad iudicium visus et auditus conformiter reducta est’.

95 II.51 W-Z, p. 198; Meyer, p. 182: ‘[cum] … omnis consonantia composita sit ex tonis et semitoniis maioribus vel minoribus aut eorum partibus, ut visum est, infertur iam conveniens esse ad divisionem monochordi accedere’.

96 II.54 W-Z, p. 198; Meyer, p. 182: ‘Omnis divisio monochordi, quod in se continet implicite et virtualiter omnia genera instrumentorum, vadit per tetrachorda.’ Italics have been added for later reference. Grocheio says virtually the same as this but about the vielle in De musica, [12.2]: ‘Ita viella in se uirtualiter alia continet instrumenta.’

97 II.55 W-Z, p. 198; Meyer, p. 182: ‘Omne autem tetrachordum vocant musici spatium duorum tonorum cum minori semitonio: et hoc est diatessaron.’

98 II.56 W-Z, p. 198; Meyer, p. 182: ‘nam ut in praecedentibus est ostensum, bis diapason ad semel diapason reducitur, diapason autem ad diapente cum diatessaron, diapente autem diatessaron praesupponit; et haec omnia prius sunt manifesta. Diatessaron autem nullam consonantiam praesupponit, sed tonum et semitonium.’

99 See n. 78 above.

100 II.67 W-Z, p. 199; Meyer, p. 184: ‘Notandum est, quod in duobus tetrachordis coniunctis sunt septem chordae, sed in disiunctis octo.’

101 II.78 W-Z, p. 200; Meyer, p. 186: ‘ubi viget religio catholica fidem in orbe terrarum’.

102 This is the unlabelled figure after II.77 W-Z, p. 202.

103 Jean’s use of the word ‘monochord’ is confusing. Throughout Book II he discusses the division of a string or strings but concludes with a diagram of a nineteen-stringed instrument that covers over two octaves, which he still seems to refer to as a monochord. This could, however, be interpreted in terms of pitches produced by moving a bridge on a single string.

104 II.99–101 W-Z, p. 202, Meyer, p. 190: ‘quoniam illa maneries canendi quae suo viguit tempore, super numeros, tempore nostro sic est solum accidentaliter variata, quod nostra placentior est auditui quam antiqua. Subtiliataque multum est musica per exercitium modernorum non solum litteratorum hominum in hac arte studentium auxilio vel inventione, sed et vulgus commune et specialiter iuvenes ac etiam mulieres ad hoc moventur …’.

105 II.101–2 W-Z, p. 202, Meyer, p. 190: ‘nescio qua forte nisi naturali industria nunc a superiori circulo regulata. Mutantur enim haec continue et illa revoluto forte circulo aliquo redibunt et erunt sicut prius.’

106 II.127 W-Z, p. 203, Meyer, p. 192: ‘Continet autem hoc instrumentum 19 chordas, scilicet bis diapason cum diapente, licet sit possibile ulterius augmentari. Et est in figura trianguli orthogoni, quantum ad duo sui latera; sed tertium latus non sub una linea cadere potest, sed maxime ad circumferentiam accedit, super tria puncta descripta.’ Witkowska-Zaremba includes a diagram on p. 204 showing nineteen strings. This diagram is slightly different from that in Paris, BnF lat. 7207, fol. 300r, which does not show the strings.

107 Elżbieta Witkowska-Zaremba informed us (email of 10 March 2018) that this diagram was drawn in a similar way to that ‘in Pr3 but is also transmitted in MSS Pr1, Pr2, K1, K4. Transmissions S and W differ a little bit in shape.’ (The sigla are those of Witkowska-Zaremba, pp. 148 ff.)

108 The strings are drawn at right angles to those in the figure in Witkowska-Zaremba, p. 204, and Meyer, p. 193, and it is difficult to see the ends of them, but their lengths appear to correspond to the pitches immediately above them.

109 II.130 W-Z, p. 203, Meyer, 192; cf. Grocheio, Ars musice 12.2, ed. Mews etal., p. 72: ‘Ita viella in se virtualiter alia continet instrumenta.’ Cf. n. 97 above.

110 II.130 W-Z, p. 203, Meyer, p. 192: ‘Diversas tamen figuras, ad quas potest hoc monochordum transferri, exempli causa tibi describere volo. Quorum figurae sunt in hoc ordine consequentes.’

111 Cf. also Lawrence Gushee, review of Falkenroth (see n. 2 above), Music & Letters, 76 (1995), pp. 275–80, at p. 279.

112 SM VII.6.

113 Aristotle, Nicomachean Ethics, 1095b 20 = Aristoteles latinus, XXVI.1−3 Fasc. 3: ‘Multi quidem igitur omnino bestiales videntur esse, pecudum vitam eligentes.’

114 Aristotle, Nicomachean Ethics, 1119a 15 = Aristotelis latinus XXVI.1−3 Fasc. 3: ‘Quecumque autem ad sanitatem sunt vel ad bonam habitudinem delectabilia existencia, haec appetet mensurate, et ut oportet, etalia delectabilia non impedimenta hiis existencia, vel preter bonum, vel super substanciam.’

115 Aristotle, Metaphysics, 980a 27 = Aristotelis latinus XXV, 2: ‘Causa autem est quod hic maxime sensuum cognoscere nos facit et multas differentias demonstrat.’

116 Aristotle, Politics, 1338a 28 (= ps.-Thomas Aquinas [Peter of Auvergne], In Politicorum, L. VIII.1. I 1115): ‘Quapropter Homerus in poemate suo ita inquit. Sed quale est vocare ad mensam opipare paratam. Deinde nominas quosdam alios, subdit et citharoedum, qui omnes demulceat. etalio loco ait Ulyxes, optimam esse degendi rationem, quando laetis hominibus convivae audiunt citharoedum sedentes per ordinem.’

117 Cf. Boethius I.1 (ed. Friedlein, p. 178): ‘Musica naturaliter nobis esse conjunctam, et mores vel honestare vel evertere.’

118 Cf. Boethius I.1 (ed. Friedlein, p. 180): ‘Unde Plato etiam maxime cavendum existimat, ne de bene morata musica aliquid permutetur.’

119 See Boethius I.3, who takes this classification from Nicomachus. The only ‘equal’ comparison is of equal numbers, e.g. 1 : 1, 4 : 4, etc.; all other comparisons are called ‘unequal’.

120 Thus 15 is greater as a superparticular of 12, since it is one whole (12) plus one-fourth (3) and conversely 12 is less as a superparticular of 15, since one has to add one part (a fourth of 12) to 12 to get 15. Cf. the first sentence of n. 42 above.

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