¹ Walden, D. K. S., ‘Charting Boethius: Music and the Diagrammatic Tree in the Cambridge University Library’s De Institutione Arithmetica, Ms Ii. 3. 12’, Early Music History, 34 (2015), pp. 207–228 Google Scholar.

² Until recently Jacobus has been known as Leodiensis, but Margaret Bent has proposed that the true name is Jacobus de Ispania. See Bent, M., Magister Jacobus de Ispania, Author of the Speculum musicae (Royal Musical Association Monographs, 28; Farnham, 2015)Google Scholar for considerations of his biography and, for discussion of his name, Wegman, R. C., ‘Jacobus de Ispania and Liège’, Journal of the Alamire Foundation, 8 (2016), pp. 253–274 Google Scholar.

³ Re the dating see Bent, Jacobus, pp. 53 ff. There is only one complete manuscript, Paris, Bibliothèque nationale de France (hereafter BnF) lat. 7207, though ‘thought by its editor to be Florentine, can now be shown on the basis of its miniatures by Cristoforo Cortese to be from the Veneto, datable c. 1434–40’ (Bent, Jacobus, p. i); there is also a second, partial, one likewise in Paris (BnF lat. 7207A) that contains the first five and part of the sixth book, and a third fifteenth-century one in Florence: Biblioteca Medicea Laurentiana, Plut. XXIX. 16, which contains excerpts (Bent, Jacobus, p. 4).

⁴ Although we know the manuscripts we have are from the fifteenth century there seems no evidence of the extent to which the Speculum was known, let alone read, before that time. It was, however, regarded as worth preserving or it would not have been copied.

⁵ See, for example, K. Desmond, ‘Behind the Mirror: Revealing the Contexts of Jacobus’s Speculum musicae’ (Ph.D. diss., New York University, 2009), which discusses Books III and VII at length. Much interest has focused on this last book and the dispute between Jacobus and Jean des Murs. (See Bent, Jacobus, pp. 2 and 42, and Desmond, ‘Mirror’, pp. 328 ff.)

⁶ de Liège, Jacques [Jacobus de Ispania], Speculum musicae, ed. R. Bragard (Corpus Scriptorum de Musica, 3; Rome, 1961)Google Scholar, Book I, chapter 1, ‘multis cogitationibus est explendum’ (many cogitations had to be expended). (Future references will simply be in the form Jacobus, I. 2, i.e. Bk. I, ch. 2.) The text is available at http://boethius.music.indiana.edu/tml/14th/JACSP1A, accessed 23 January 2017.

⁷ There are two accepted approaches to such understanding, which may be roughly characterised as (1) to attempt to think in the way the ancient writers did and (2) to present the results that they achieved in modern notation and terminology. The best authors use both, though I tend to favour the first.

⁸ Boèce, , Institution arithmétique, ed. and trans. J.-Y. Guillaumin (Paris, 1995 Google Scholar); Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, De institutione musica libri quinque, ed. G. Friedlein (Leipzig, 1867). References will generally be to the English translations: Boethius, , Fundamentals of Music, trans., with Introduction and notes by C. M. Bower, ed. C. V. Palisca (New Haven and London, 1989)Google Scholar and Masi, M., Boethian Number Theory: A Translation of the De Institutione Arithmetica (Studies in Classical Antiquity, 6; Amsterdam, 1983)CrossRefGoogle Scholar.(References to the De musica for the Latin will be given in the form I. 2 (i.e. Bk. I, ch. 2).)

⁹ See Jacobus, I. 2 and many other places.

¹⁰ See Dyer, J., ‘Speculative “Musica” and the Medieval University of Paris’, Music & Letters, 90 (2009), pp. 177–204 Google Scholar, at 191. In fact Boethius had knowledge of much more of Euclid, including Book VIII, though apparently Jacobus did not (see n. 63 below).

¹¹ Jacobus, I. 24: ‘Excerptae sunt autem illae maiore ex parte de Iordani Arithmetica et Euclidis Geometria.’ ( de Nemore, Jordanus, De elementis arithmetice artis: A Medieval Treatise on Number Theory, ed. H. H. L. Busard, 2 vols. (Stuttgart, 1991 Google Scholar). In VII. 1 he explicitly, but briefly, refers to Euclid Book I.

¹² See p. 14 of Folkerts, M., ‘Euclid in Medieval Europe’, §III in The Development of Mathematics in Medieval Europe (Aldershot, 2006)Google Scholar.

¹³ In I. 28 he explicitly refers to Euclid Book VII and Jordanus Book III.

¹⁴ I shall generally refer to these as ‘Arabic numerals’, though strictly speaking they should be called Hindu-Arabic since the numerals originated in India and were subsequently transmitted through the Arab world, the Maghreb and then into western Europe. Chrisomalis, S., Numerical Notation: A Comparative History (Cambridge, 2010)Google Scholar, uses ‘Western system’. Jordanus’s De elementis does not treat algorism but he wrote one work on algorism (see ibid., p. 9) and it is possible he also wrote another Demonstratio Jordani de algorismo. (See ‘Jordanus de Nemore’, in Dictionary of Scientific Biography, ed. C. C. Gillispie (New York, 1973), vii, pp. 176 and 178.)

¹⁵ There is one exception to this in Boethius, Fundamentals, II. 28, pp. 82–3, which is echoed by Jacobus, I. 41; see below.

¹⁶ The reason there are fifty-one is that this is the maximum extent of the monochord; see Jacobus, II. 126: ‘Et, secundum illam monochordi ordinationem, altissima consonantia in se contineret ter diapason cum tono.’ (And according to the arrangement of the monochord, the largest consonance contained in it is the triple diapason plus a tone.)

¹⁷ See Beaujouan, G., ‘The Transformation of the Quadrivium’, in R. L. Benson and G. Constable with C. D. Lanham (eds.), Renaissance and Renewal in the Twelfth Century (Cambridge, Mass., 1982), pp. 463–487 Google Scholar, at 467.

¹⁸ L. Weber, ‘Intellectual Currents in Thirteenth-Century Paris: A Translation and Commentary on Jerome of Moravia’s Tractatus de musica’ (Ph.D. thesis, Yale University, 2009), at p. 2.

¹⁹ It is not clear when Jacobus studied the Arithmetica, as opposed to the Musica, of Boethius. Bent, Jacobus, pp. 144 ff., has noted the influence of Kilwardby and if Jacobus did indeed study in Oxford as well as Paris he could certainly have learnt algorism (which is discussed below) there since English translators of Arabic into Latin were the first to bring algorism to general notice. See e.g. The Earliest Arithmetics in English, ed. R. R. Steele (Early English Text Society, es 118; London, 1922) (repr. Millwood, NY, 1975).

²⁰ Jacobus, VI. 62 and VII. 17.

²¹ See Bent, Jacobus, pp. 9–10 and G. Rico, ‘Music in the Arts Faculty of Paris in the Thirteenth and Fourteenth Centuries’ (D.Phil. thesis, Oxford University, 2005), p. 33. Also Bent, Jacobus, p. 60 says: ‘The late 1320s seems right for the composition of at least the final book of the Speculum, in his self-confessed old age. It is only at the very end of his treatise that Jacobus declares that it was his primary intent to defend the art of the ancients, and a secondary purpose to outline speculative theory and chant.’

²² Jacobus refers to Boethius V in I. 2 and elsewhere. In II. 56, Jacobus says: ‘Fearing that these things I had learned from Boethius’s Musica might someday escape me, and so that I might retain them in my memory, and so that I might use them with more confidence, I excerpted some passages from the two first books, which I had heard in Paris, and I began to excerpt these and other passages.’ (Translation from Desmond, ‘Mirror’, pp. 97–8.) (‘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.’) Dyer (‘Speculative’, p. 191) points out that there is documentary evidence that only the first two books were lectured on and on p. 179 explicitly says: ‘Jacques de Liège (? 1260–after 1329) reported that he heard lectures on the first two books of Boethius’s De institutione musica at Paris, most likely in the 1270s, when he would have been a student there’, citing Jacobus, II. 136.

²³ Jacobus, II. 56. See Bent, Jacobus, p. 9, where she argues that at least one copy of the Musica would have been available ‘in Paris around the rue St-Jacques in the late thirteenth century’.

²⁴ Jacobus, I. 1: ‘Opus autem quod assumpsi, laboriosum et longum mihi est et multis cogitationibus est explendum, tardeque nimis est inceptum.’

²⁵ See Desmond, ‘Mirror’, p. 98. Jacobus, II. 56: ‘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’.

²⁶ Such numbers are described in detail in Nicomachus of Gerasa, Introduction to Arithmetic, trans. M. L. D’Ooge, with studies in Greek arithmetic by F. E. Robbins and L. C. Karpinski (New York, 1926; repr. 1972), Book II. 8 ff. The numbers are those of the quantity of dots making up a square, such as ::, which is the square number 4.

²⁷ (Two of these are to comments on Jordanus.) Jacobus relies heavily on propositions from Jordanus, De elementis arithmetice artis, in his Book III where he is attempting to show that the tone is not divisible into two equal parts.

²⁸ Jacobus only mentions Euclid’s De arte geometria from Boethius’s translation, according to Bent, Jacobus, p. 11. (See Jacobus, I. 9, III. 24, 28, IV. 12 and VII. 15.)

²⁹ Algorism and Jacobus’s debt to it will be treated below.

³⁰ See, for example, G. C. Cifoletti, ‘Mathematics and Rhetoric: Peletier and Gosselin and the Making of the French Algebraic Tradition’ (Ph.D. diss., Princeton University, 1992).

³¹ See Crossley, J. N. and Henry, A. S., ‘Thus Spake al-Khwārizmī’, Historia Mathematica, 17 (1990), pp. 103–131 Google Scholar, from Cambridge, Cambridge University Library, MS. Ii. vi. 5, and M. Folkerts, Die älteste lateinische Schrift über das indische Rechnen nach al-Hwarizmi, Edition, Übersetzung und Kommentar (Philosophisch-historische Klasse; Abhandlungen der Bayerischen Akademie der Wissenschaften, 113; Munich, 1997), from New York, Hispanic Society of America, MS HC 397/726, and for Fibonacci, see B. Boncompagni, Il Liber Abbaci di Leonardo Pisano (Rome, 1857); a new edition by André Allard is reputed to be nearing completion. See also n. 122 below.

³² The translations from Greek were rarely used; see Folkerts, ‘Euclid’, p. 5.

³³ See, inter alia, Dyer, ‘Speculative’, p. 191 and Rico, ‘Music’, p. 30, paraphrasing Lafleur, C., ‘La réglementation “curriculaire” (“de forma”) dans les introductions à la philosophie et les guides de l’étudiant de la Faculté des Arts de Paris au XIII^e siècle: Une mise en contexte’, in C. Lafleur and J. Carrier (eds.), L’Enseignement de la philosophie au XIII ^e siècle : Autour du ‘Guide de l’Etudiant’ du ms. Ripoll 109 (Studia Artistarum; Turnhout, 1997), pp. 521–548 CrossRefGoogle Scholar.

³⁴ Nicomachus, Introduction.

³⁵ Jordanus, De elementis.

³⁶ Desmond, ‘Mirror’, p. 74.

³⁷ See Beaujouan, G., ‘Motives and Opportunities for Science in the Medieval Universities’, in A. Crombie (ed.), Scientific Change (London, 1961), pp. 219–236 Google Scholar, and Rico, ‘Music’, p. 21. It is not clear that this restriction was very significant as there were so many feast days in the Church’s year.

³⁸ Beaujouan, ‘Motives’, p. 222.

³⁹ See Dyer, ‘Speculative’, p. 182.

⁴⁰ See ibid., p. 185, quoting Chartularium Universitatis Parisiensis sub auspiciis consilii generalis facultatum Parisiensum ex diversis bibliothecis tabulariisque collegit et cum authenticis chartis contulit, ed. H. Denifle (Paris, 1899; repr. Brussels, 1964) i, p. 278 (no. 246) and Thorndike, L., University Records and Life in the Middle Ages (New York, 1944), pp. 53–54 Google Scholar.

⁴¹ ‘aliquos libros mathematicos audiverit’. See Dyer, ‘Speculative’, p. 185, quoting Chartularium, ed. Denifle, iii. 145 (no. 1319); Thorndike, Records, p. 246.

⁴² See Rico, ‘Music’.

⁴³ For a thorough discussion of how Euclid treated numbers and magnitudes see Grattan-Guinness, I., ‘Numbers, Magnitudes, Ratios, and Proportions in Euclid’s Elements: How Did He Handle Them?’, Historia Mathematica, 23 (1996), pp. 355–375 Google Scholar. Because of our adeptness in associating numbers (particularly decimal numbers) with magnitudes we have lost sight of this. Also we tend to interpret Euclid in terms of algebra, a subject that had only just begun to penetrate western Europe in the Middle Ages and was understood by few. See also Crossley and Henry, ‘Thus Spake’, from Cambridge, Cambridge University Library, MS. Ii. vi. 5, and Folkerts, Die älteste lateinische Schrift, from New York, Hispanic Society of America, MS HC 397/726.

⁴⁴ Jacobus, I. 1, referring explicitly to Boethius, V. 2, says: ‘because [Boethius] does not consider music theory as sound in itself, but as numbered sound’ (‘quia [Boethius] non considerat musica sonum per se sumptum, sed sonum numeratum’).

⁴⁵ Guy, Tractatus de tonis, in the collection of music manuscripts in London, British Library, MS Harley 281, fol. 60^r, ‘de numero relato ad sonum’. (See C. J. Mews, C. J. Williams, J. N. Crossley and C. Jeffreys, Guy of Saint-Denis: Tractatus de tonis (Kalamazoo, 2017.) Cf. Boethius, I. 7: ‘Illud tamen esse cognitum debet, quod omnis musicae consonantiae aut in duplici aut in triplici aut in quadrupla aut in sesqualtera aut in sesquitertia proportione consistant; et vocabitur quidem, quae in numeris sesquitertia, diatessaron in sonis . . .’. See also Grocheio, Ars musice, in Harley 281, fol. 41^v [5. 1], pp. 56–7: ‘Dicentes eam esse de numero relato ad sonos’; John of Garland, who was alive c. 1270–1320 (see Johannes de Grocheio, Ars musice, ed. and trans. C. J. Mews, J. N. Crossley, C. Jeffreys, L. McKinnon and C. J. Williams (Medieval Institute Publications; Kalamazoo, 2011), wrote in his Musica plana (see Musica plana Johannis de Garlandia, ed. C. Meyer (Baden-Baden, 1998), reportatio 1. 8, p. 3): ‘Alia est relata de qua est musica in qua determinatur de numeris relatis ad sonos.’

⁴⁶ ‘vel de sono armonice numerato’. Cf. Grocheio, Ars musice, fol. 41^v [5. 3], pp. 56–7: ‘musica est ars vel scientia de sono numerato armonice’; cf. Robert Kilwardby, De ortu scientiarum, ed. A. G. Judy (Auctores Britannici Medii Aevi, 4; London and Toronto, 1976), pp. 51–2, paragraph 128: ‘et ideo posuerunt recte musicam audibilem esse de sono harmonice numerato vel de numero sonorum harmonico’.

⁴⁷ ‘sicut arismetica . . . absolute considerato’. Cf. John of Garland, Musica plana, reportatio 1. 8, p. 3: ‘quia alia est absoluta de qua est Arismetica, in qua determinatur de numeris absolute’.

⁴⁸ Guy of Saint-Denis, Tractatus de tonis, fol. 60^r [1. 1. 4]: ‘Cum enim musica secundum boecium sit de numero relato ad sonum, vel de sono armonice numerato, sicut arismetica est de numero in se et absolute considerato, quod est proportio in ipsis numeris hoc esse videtur concordantia in sonis musicis.’

⁴⁹ Crossley, J. N., ‘A Sense of Proportion: Jacobus Extending Boethius around 1300’, to appear in C. Monagle (ed.), ‘Making Known Worlds New’: Essays in Honour of Constant J. Mews (forthcoming Amsterdam University Press, 2018)Google Scholar.

⁵⁰ It is notorious that American and British systems are different – and confusing.

⁵¹ Interestingly, numbers are included among magnitudes because all numbers can be measured by other numbers, namely their divisors; thus 21 can be measured by 3 since 7 times 3 gives 21. The converse is not true; not only are magnitudes not numbers, they cannot always be measured. (Note that measuring involves comparison and one needs a unit of measure.) Jacobus only needed to consider lengths – that of the string on the monochord – but other writers of that time had great difficulty with ratios of other kinds of magnitude; see Rommevaux, S., ‘La proportionnalité numérique dans le livre VII des Éléments de Campanus’, Revue d’Histoire des Mathématiques, 5 (1999), pp. 83–126 Google Scholar.

⁵² To put it another way, √2 is irrational, i.e. cannot be expressed in the form m/n with whole numbers m and n.

⁵³ Jacobus, I. 1, perhaps referring to Boethius, V. 2, says: ‘because he does not consider music theory as sound in itself, but as numbered sound’ (‘quia non considerat musica sonum per se sumptum, sed sonum numeratum’). Boethius, V. 2, says inter alia: ‘Sensus namque confusum quiddam ac proxime tale, . . . accipit vero ratione integritatem atque unas persequitur differentias.’ (‘So sense finds something confused, yet close to the truth, but it receives the whole by reason. . . . or: Sense conceives of nothing of the whole, but only gets as far as an approximation. Reason makes the judgement and searches out differences.’)

⁵⁴ See Boethius, I. 9: ‘Non omne judicium dandum esse sensibus sed amplius rationi esse credendum. . . . Postrema ergo perfectio, agnitionisque vis in ratione consistit, quae certis regulis sese tenens nullo unquam errore prolabitur.’

⁵⁵ See e.g. Jacobus, I. 29 and also in I. 8, where he says, referring to Aristotle’s Prior Analytics: ‘Vel dicitur arithmetica esse de multitudine absolute sumpta, ut distinguitur a multitudine ad aliquid dicta.’

⁵⁶ There are anomalies in English for the first few numbers, parts usually starting from one third. Further, American English uses ‘one fourth’ whereas British English prefers ‘one quarter’.

⁵⁷ M. Bent, personal conversation, June 2016 and email, 1 December 2016. Johannes Ciconia (c. 1370–1412), Nova musica, Book I, ch. 27, and Book II, ch. 56, uses ratio in the modern sense: I, 27: ‘Hieronymus: Diatessaron symphonia est que constat ex ratione epitrita et fit ex sonitibus quatuor unde et nomen accepit. Fulgentius: Symphonia diatessaron est cuius sonus in arithmeticis epitritus vocatur, ut 3 ad 4’, and II. 56: ‘Cum enim omnis musica ratione numerorum constet, tamen propter privilegium rationalis creature, musica humane vocis rithmus, id est numerus arithmeticorum, appellari meruit, ut est octonarius, senarius, quaternarius, id est sesquitertius, sesqualter, duplex.’

⁵⁸ See Nicomachus, Introduction, p. 264, n. 2, where D’Ooge says that ἀναλογία ‘may be translated “proportion,” and Nicomachus points out that an ἀναλογία is, strictly speaking, a combination of ratios [sic]’. D’Ooge almost exclusively uses the English word ‘ratio’ in the footnotes, perhaps because Nicomachus concerns himself with particular, named, proportions (sesquialter, sesquitertian, etc.) and does not need to use it. (Bower in Boethius, Fundamentals, p. 65, n. 34, says: ‘A verbal relationship between the words proportio and ratio in Latin and Greek is unfortunately lost in English; proportio (λόγος) must be translated “ratio,” and proportionalitas (ἀναλογία) “proportion”.’ (Emphasis added.) However, on p. 7, n. 30, he had already said: ‘“Ratio” is a translation of the Latin proportio, which Boethius translated from the Greek λόγος.’ Rommevaux, S., ‘A Treatise on Proportion in the Tradition of Thomas Bradwardine’, Historia Mathematica, 40 (2013), pp. 164–182 Google Scholar, at 165, n. 3 is in between, using ‘ratio’ and ‘proportionality’. Significantly, Walden, ‘Charting’, uses ‘proportion’; see p. 223 in particular.

⁵⁹ Desmond, ‘Mirror’, p. 88 quotes Odington’s succinct description: ‘Ratio is the relationship of quantity. Proportion is the relationship of ratio.’ Walter Odington, Summa de speculatione musicae, ed. F. Hammond (Corpus scriptorum de musica, 14; n.p., 1970), p. 48.

⁶⁰ Hawkins, J. M. and Allen, R. (eds.), The Oxford Encyclopedic Dictionary (Oxford, 1991)Google Scholar, entry for ‘ratio’.

⁶¹ These two ratios represent the same proportion or, alternatively, we may say they ‘are in the same proportion’. However, as is current practice, I shall sometimes use the terminology ‘proportion a : b’.

⁶² Nicomachus, Introduction, p. 265, but see there n. 1, where D’Ooge describes the definition given by Nicomachus as ‘a poor one’. See also ibid., p. 264, n. 2.

⁶³ For his theory of continued proportions Boethius apparently used Euclid, Book VIII, especially Propositions 2 and 4. However, Jacobus would not have had access to this book directly.

⁶⁴ So all numbers are of the form 2 ^m 3 ⁿ . A certain number of subtractions are needed to state the exact proportions in the way that Boethius and Jacobus chose to.

⁶⁵ Technically this is the definition of prime number but Jacobus did not refer to Euclid, Book VII, which is where the extended study of prime numbers really begins.

⁶⁶ See for example Masi, Boethian, p. 188 and Fig. 2, p. 7, of R. M. Thomson: ‘John Dunstable and his Books’, Musical Times, 150, no. 1909 (Winter 2009), pp. 3–16.

⁶⁷ The authors often invert the order of the numbers in a proportion (as has been done here), but their terminology (see below) usually makes the meaning clear. Cf. p. 701 of Rommevaux, S., ‘The Transmission of the Elements to the Latin West: Three Case Studies’, in E. Robson and J. Stedall (eds.), The Oxford Handbook of the History of Mathematics (Oxford, 2009), pp. 687–706 Google Scholar.

⁶⁸ I use the symbol ⊗ to indicate it is not a standard kind of multiplication.

⁶⁹ The modern algebraic way of thinking of this is to say that in adding two tones together the new proportion is obtained by multiplying: we say 9/8 times 9/8 = (9 × 9)/(8 × 8), so the proportion is 81 : 64.

⁷⁰ The fact that a given proportion can always be represented in its lowest terms requires the fact that one cannot go on dividing by a number infinitely many times. Rommevaux, ‘La proportionnalité’, p. 100, points out that Euclid VII. 1 (the Euclidean algorithm) does not explicitly invoke this, but Campanus did, and so did Euclid in VII. 31. She also remarks that Campanus credits many of his ideas to Jordanus (p. 88 and annexe).

⁷¹ In Boethius, Fundamentals, pp. 26–7, n. 88, Bower translates primi numeri as ‘smallest integers’, which is equivalent to the above definition, since if we have a : b and c (≠ 1) divides both a and b, then the proportion is also represented by a/c : b/c, and vice versa, and these two numbers a/c and b/c are clearly smaller than a and b.

⁷² If there is a number m that divides both a and b, then a : b reduces to a/m : b/m. If there is another number that divides both a/m : b/m, repeat the procedure. Eventually this process terminates (cf. n. 70 above). Thus we may define a proportion as a class (or set or collection) of ratios where two ratios a : b and c : d belong to the same class if they reduce to the same lowest terms.

⁷³ See Murdoch, J., ‘The Medieval Language of Proportions: Elements of the Interaction with Greek Foundations and the Development of New Mathematical Techniques’, in Crombie (ed.), Scientific Change, pp. 237–271 Google Scholar. Even the simple pre-Eudoxian version discussed by Murdoch would have been difficult for Jacobus to follow.

⁷⁴ Though he does use both Roman and Arabic numerals.

⁷⁵ The word ‘octave’ would perhaps be better replaced by ‘diapason’, the word Jacobus uses, which gives the sense of going through (dia) all [the notes of the scale] without the preconception of there being eight notes. The words ‘diatessaron’ and ‘diapente’, however, do give the numerical indication of four and five tones respectively.

⁷⁶ See Walden, ‘Charting’, pp. 223 and 227 and A. Moyer, E., ‘The Quadrivium and the Decline of Boethian Influence’, in N. H. Kaylor, Jr. and P. E. Phillips (eds.), A Companion to Boethius in the Middle Ages (Leiden, 2012), pp. 479–517 CrossRefGoogle Scholar, at 482.

⁷⁷ See Masi, Boethian, I. 32, p. 114. ‘Demonstratio quemadmodum omnis inaequalitas ab aequalitate processerit’ (‘Demonstration that every inequality comes from equality’).

⁷⁸ Nicomachus, Introduction, pp. 222–9. D’Ooge points out that Theon of Smyrna had already defined these proportions.

⁷⁹ Modern English still preserves sesqui in ‘sesquiannual’ – every eighteen months. The terminology, found in Nicomachus, Introduction, chapters XVIII–XXIII, is remarkable in that it gives a unique designation to each proportion (provided the first term is greater than or equal to the second). Given such a proportion a : b in its lowest terms with a ≥ b, then a = qb + r, with r < a, the designation of the proportion is ‘q multiple super-r-partient (with b parts)’. (In the mid-thirteenth century Campanus introduced the term ‘denominatio’ for the (partly fractional) number, as opposed to ratio, q + r/b; see Rommevaux, ‘La proportionnalité’, §5, pp. 104 ff., but the word is not found in Euclid or Boethius.)

⁸⁰ Jacobus often omits the number of parts; see, for example, all the ratios in the Appendix below.

⁸¹ ‘Et illa est super quadraginta septem partiens octuagesimas primas’ (‘And that is a super-47-partient with 81st [parts]’).

⁸² Jacobus, II. 97: ‘Hexatonus est inaequalium vocum consonantia sex tonos in se continens in dupla super 7153 partiente consistens proportione.’ (‘The hexatone is an inequaltiy of a consonance of pitches of six tones, which is in duplesuper-7153-partient proportion.’) The parts are 262,144ths. Cf. Bower, Fundamentals, p. 95, where the size of the parts is clear from the diagram but not in the text.

⁸³ See Nicomachus, Introduction, p. 54, for D’Ooge’s modern general algebraic formulation of Nicomachus’s constructions.

⁸⁴ See Nicomachus, Introduction, p. 233. Nicomachus usually uses ‘sesquialter’ but Boethius, with a few exceptions, uses ‘sesqualter’.

⁸⁵ 1 is included because it is the source of all number.

⁸⁶ From Nicomachus, Introduction, p. 232, footnotes omitted.

⁸⁷ See p. 147 of Guillaumin, J.-Y., ‘Boethius’s De institutione arithmetica, and its Influence on Posterity’, in Kaylor and Philips (eds.), A Companion, pp. 135–161 Google Scholar.

⁸⁸ Square numbers are treated in chapter IX, p. 242, of Nicomachus, Introduction.

⁸⁹ Boethius, Fundamentals, pp. 82–3.

⁹⁰ Observe how this is different from our calculation for this interval above.

⁹¹ Jacobus gives Boethius’s explanation in I. 57, explicitly referring to Boethius, II. 28.

⁹² The fact that this is written the other way round does not concern Boethius; cf. n. 67 above.

⁹³ 192 : 216=24×8 : 24×9 and 216 : 243=27×8 : 27×9.

⁹⁴ Because he only multiplies when necessary to avoid fractions, this guarantees the resulting numbers are relatively prime and the only numbers used for such multiplications are powers of 2 and/or 3. (256 is only divisible by 2 (eight times) and 243 is only divisible by 3 (four times).)

⁹⁵ This proportion seems to be as far as a number of teachers of music theory reached in the thirteenth century. See, for example, Grocheio, Ars musice, fol. 41^r [4.13], and Guy of Saint-Denis, Tractatus de tonis, fol. 61^r [1.1.8].

⁹⁶ The largest number in Boethius is 663,552 in III. 14; see p. 110 of Boethius, Fundamentals. Jacobus’s use of algorism for large calculations will be discussed below.

⁹⁷ A list of the fifty-one musical intervals he considers will be found near the end of Jacobus II. 126. The first twenty culminate in the octave (diapason) and the remaining ones involve intervals greater than an octave and these are mainly multiplex multisuperpartient proportions (cf. the Appendix below).

⁹⁸ Bent, Jacobus, p. 145, says: ‘In Book II, Jacobus defines an unprecedented total of fifty-one intervals, using a gamut with an upward extension, also defined in Book V. 4 for notes above ee up to aaa.’

⁹⁹ See Boethius, Fundamentals, p. 86. Jacobus gives Boethius’s explanation in I. 57, explicitly referring to Boethius, II. 28 (chapter 31 in Friedlein’s edition).

¹⁰⁰ See The Euclidean Division of the Canon: Greek and Latin Sources, ed. A. Barbera (Lincoln, Nebr. and London, 1991), Proposition 9, p. 147. (Barbera translates ‘λόγος’ as ‘ratio’.) Bower, C., ‘Boethius and Nicomachus: An Essay Concerning the Sources of De institutione musica ’, Vivarium, 16 (1978), pp. 1–45 Google Scholar, at 12 definitively says: ‘Chapters 1–2 of Book IV are certainly drawn from Sectio canonis.’

¹⁰¹ Boethius merely says ‘diapason sex tonis non constet’ (‘the diapason (octave) does not consist of six tones’). However, in the Sectio canonis, a work from which Boethius derives much, does say ‘Sex proportiones sequioctavae maiores sunt uno duplici intervallo’ (‘Six sesquioctave proportions are greater than one duple interval’) and proceeds to list the numbers as in Figure 4, penultimate column. No reason is given there for starting with 262,144 but Barbera reports that, in a Greek version, Isaac Argyros says: ‘We certainly learned to put together ratios similar to one another, and through this A [the starting number] becomes 262,144’; see The Euclidean Division, ed. Barbera, Proposition 9, pp. 215, n. 23 and 254.

¹⁰² This list is not in the Sectio canonis.

¹⁰³ Obviously Boethius does not include the final column I have inserted here.

¹⁰⁴ Although the techniques of working with fractions were known at the time (from the work of al-Khwārizmī) it is clearly easier to work with whole numbers.

¹⁰⁵ At most this needs Euclid, Book VII, Propositions 24 and 26, and in fact Jacobus does refer specifically to this book in his III. 27 while also referring to Jordanus de Nemore. Then Jacobus III. 28 is titled ‘Quod numeri quilibet in sua proportione minimi numerant quoslibet alios in eadem sumptos proportione’ (‘That arbitrary numbers in their minimum proportion measure (numerant) any others in the same proportion’).

¹⁰⁶ Seven, because there are six tone intervals between the seven notes.

¹⁰⁷ Boethius, III. 14, labels his diagram: ‘Differentia. 7153, 524288, 531441, comma seu spatium quo majores sunt sex toni diapason’. Since the difference between the two large numbers is 7153, this is a superpartient proportion of the form one whole and 7153 parts each of size 1/524,288. Boethius argued that this is ‘ultimate interval heard which can really be perceived’ (see Boethius, Fundamentals of Music, pp. 96–7). The adjective ‘Pythagorean’ distinguishes this from the syntonic or Didymus’s comma, which has the ratio 81 : 80 (see P. Pesic, ‘Hearing the Irrational: Music and the Development of the Modern Concept of Number’, Isis, 101 (2010), pp. 501–30 at 521 and 506, n. 16).

¹⁰⁸ I. 44: ‘Comma est inaequalium sonorum minima sensu perceptibilis consonantia.’

¹⁰⁹ In perhaps plainer English this can be restated as: a comma is the difference between a tone and a minor tone (two minor semitones) or by how much 9:8 exceeds two minor semitones or six tones exceed an octave. ‘Comma, ut est dictum, est illa particula vel spatium qua vel quo superat tonus perfectus tonum minorem sive sesquioctava proportio proportiones duorum minimorum semitoniorum, vel hexatonus diapason.’

¹¹⁰ In modern terms one may think of 531441 : 262144 ⊗ 1 : 2 = 531441 : 524288, since division (subtraction) of proportions is equivalent to multiplying by the inverse proportion.

¹¹¹ Immediately following the preceding quotation he says: ‘Hoc autem est in superpartiente proportione, sicut ex terminis ostenditur sequentibus: 531441 524288 497664 472392.’ (‘But this is in superpartient proportion just as is shown by the following numbers.’) Boethius uses ‘terminus’ for number also in II. 12 and elsewhere.

¹¹² ‘Inter primum horum terminorum et quartum est sesquioctava proportio in qua fundatur tonus perfectus; inter secundum et <quartum> est proportio duorum minimorum semitoniorum in qua tonus fundatur minor.’

¹¹³ Indeed the definition of a minor tone is that it is two minor semitones: ‘inter secundum et <quartum> est proportio duorum minimorum semitoniorum in qua tonus fundatur minor’.

¹¹⁴ Jacobus, II, 44: ‘Ergo, inter terminos illos, est proportio cuiusdam inaequalitatis non multiplicitatis’ (‘So between those numbers there is a proportion of inequality [see n. 78] that you cannot multiply’).

¹¹⁵ Jacobus, II. 64.

¹¹⁶ 4,782,969/4194304=1+588,665/4,194,304.

¹¹⁷ Another way of looking at this (a modern way) is to compare 588,665/4,194,304 and 1/9. If we cross-multiply this is the same as comparing 588,665×9 and 4,194,304×1, i.e. 5,297,985 and 4,194,304. The former is greater, so 588,665/4,194,304 is larger than 1/9. Likewise with 588,665 × 8 and 4,194,304×1 we compare 4,709,320 and 4,194,304, so 588,665/4,194,304 is less than 1/8. Finally we could simply compare the decimal fractions for 1/9, 588,665/4,194,304 and 1/8. As decimal fractions these are 0·111…, 0·1403… and 0·125–but decimal fractions were not introduced until the sixteenth century (by Stevin in 1585; see e.g. Crossley, J. N., The Emergence of Number, 2nd edn (Singapore, 1987), p. 140)CrossRefGoogle Scholar.

¹¹⁸ Interestingly Jacobus does not point out that a major tone is exactly an octave less than seven tones; a fact that comes from the ratio 2187×2187 : 2048×2048, which can be written as 9⁷ : 2×8⁷ or {(9/8)⁷ ÷ 2} : 1.

¹¹⁹ See Jordanus, De elementis, p. 9.

¹²⁰ There were many copies of Alexandre de Villedieu, Carmen de algorismo, and John Sacrobosco, De arte numerandi, circulating in the thirteenth century and they were even rendered into English very early. See e.g. Arithmetics, ed. Steele. Sacrobosco’s text was last edited in de Dacia, Petrus and de S. Audomaro, Petrus, Opera quadrivialia, ed. F. S. Pedersen, 2 vols. (Copenhagen, 1983–4), i, pp. 174–201 Google Scholar.

¹²¹ Jacobus, I. 2: ‘Nam, de numero, licet sit scientia speculativa, ad praxim tamen extenditur in algorismo, et numerus, in secundo Arithmeticae, ad figuras extenditur geometricas’ (‘For concerning number, especially for theoretical science, it is extended in practice however by algorism, and number, in the second book of [Boethius’s] Arithmetica, it is extended to geometrical figures’). ‘Jacobus cites the De ortu scientiarum of Kilwardby (‘hic Robertus’) five times early in Book I (chs. 2, 7, 8), where he follows Kilwardby’s classification of music, distinguishing it from Boethius and Isidore [of Seville]. Music is placed among the speculative sciences’ (Bent, Jacobus, p. 145). See also n. 19 above.

¹²² Jacobus, II. 78: ‘O commendabilis arithmetica! O utilis algorismi scientia!’ If ‘science’ seems anachronistic, ‘knowledge’ or even ‘know-how’ will do equally well here.

¹²³ Crossley, J. N., ‘Old-fashioned versus Newfangled: Reading and Writing Numbers, 1200–1500’, Studies in Medieval and Renaissance History, 3rd ser., 10 (2013), pp. 79–109 Google Scholar.

¹²⁴ For a modern algorithmic approach to computing with Roman numerals see Detlefsen, M., D. K. Erlandson, J. C. Heston, and C. M. Young, ‘Computation with Roman Numerals’, Archive for History of Exact Sciences, 15/2 (1976), pp. 141–148 CrossRefGoogle Scholar.

¹²⁵ For an account of how complicated multiplication using Roman numerals was see Murray, A., Reason and Society in the Middle Ages (Oxford, 1978), p. 156 Google Scholar.

¹²⁶ See e.g. Paris 7207, fol. 67^r and also Folkerts, Die älteste. In general in Paris 7207, only small numbers, say, those below one hundred, are not written in Arabic numerals.

¹²⁷ As happens elsewhere, e.g. in the Cambridge MS of al-Khwārizmī, the Arabic numerals in diagrams seem to have been written by someone other than the main scribe (see Crossley and Henry, ‘Thus Spake’).

¹²⁸ See J. N. Crossley and C. Williams, ‘Snow on the Mountains: Number in Grocheio in the Thirteenth Century’, unpublished technical report, available at http://www.csse.monash.edu.au/publications/2005/tr-2005-186-full.doc (accessed 20 January 2017) and Burnett, C., ‘The Semantics of Indian Numerals in Arabic, Greek and Latin’, Journal of Indian Philosophy, 34 (2006), pp. 15–30 CrossRefGoogle Scholar.

¹²⁹ See, for example, Rico, ‘Music’, p. 10. It is interesting that as late as the sixteenth century the same books were being used as basic mathematics books as in Paris around 1300. Martín de Rada, OSA, who was renowned for his knowledge of mathematics, wrote, when he was in the Philippines, in a letter to Fr Alonso de la Veracruz on 3 June 1576: ‘we have nothing more than Euclid and Archimedes on geometry, Ptolemy and Copernicus on astronomy, Vitellius on perspective, [and] Hali ben Zagel on astrology. I also have a book on triangles and the directions of mote regio [Regiomontanus] and the Ephemerides of Cipriano Leonistio and the Alfonsine and Prusenic tables.’ (‘Al muy reverendo padre nuestro el maestro fray Alonso de la Vera Cruz provincial de los agustinos en la nueva España. Mi padre . . .’; translation by the present author from the transcription from Bibliothèque nationale de France, fonds espagnol 325. 7 (M F 13184), fols. 35–6 by D. Folch Fornesa; previously available at http://www. upf. edu/asia/projectes/che/s16/rada7. htm, viewed 6 April 2011.)

¹³⁰ See Dyer, ‘Speculative’, p. 179, quoting from Die Musica speculativa des Johannes de Muris: Kommentar zur Überlieferung und kritische Edition, ed. C. Falkenroth (Beihefte zum Archiv für Musikwissenschaft, 34; Stuttgart, 1992), p. 74 (prologue of ‘version A’): ‘these days the books of the older philosophers, not only those concerning music, but also those of the other mathematicals, are not read; and this happens because [students] shrink from them as unintelligible or excessively difficult’ (‘Verum, quia istis diebus libri antiquorum philosophorum nedum de musica, sed et de ceteris mathematicis non leguntur, et ob hoc accidit quod eos tamquam inintelligibiles aut nimis difficiles abhorreri’).

¹³¹ See Dyer, ‘Speculative’, p. 191. Rico, ‘Music’, p. 95 says: ‘Indeed, it is only in the late fourteenth-century copies of Johannes de Muris’[s] Musica speculativa that canonical problems of musica were to be treated with the use of Arabic numerals.’ I find this curious since de Muris was surely more mathematically adept than Grocheio or Jacobus and yet both seem to have used Arabic numerals (see n. 130 above).

¹³² Jacobus, III. 56: ‘si defeci [est] quia non sufficio ad plene capiendam tam arduam materiam quae multas requirit cogitationes, multas numerorum collationes. . . . Si amatores musicae sint theoriae, delectentur. Ludant hi in numerorum proportionibus, in variis et stupendis numerorum comparationibus.’

¹³³ Jacobus, I. 1.

¹³⁴ Ibid.

¹³⁵ He mentions the monochord over four hundred times.

¹³⁶ Boethius, II. 12: speaking of the terms in a proportion he says: ‘But I say “terms” meaning whole numbers’ (‘terminos autem voco numerorum summas’).

¹³⁷ As noted earlier the numbers would always be powers of 2 and 3 (and their products) only.

¹³⁸ One problem remains. When Jacobus finally establishes a proportion, stated in the terminology of Boethius, which is a superparticular or superpartient proportion, he does not mention the size of the parts. In the simple example given earlier the proportion for two tones is a super-17-partient proportion, namely 1+17/64 : 1. He says: ‘The ditone is an interval enclosing two tones having a super-17-partient proportion.’ Why is the 64 not mentioned?