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THE WRITINGS OF BOETHIUS AND THE COGITATIONS OF JACOBUS DE ISPANIA ON MUSICAL PROPORTIONS

Published online by Cambridge University Press:  12 September 2017

John N. Crossley
John N. Crossley*
Affiliation:
Monash University, Australia
*
Email: John.Crossley@Monash.edu
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Abstract

Thirteenth-century music theory, which followed the ideas of Boethius, was very largely concerned with the numerical proportions associated with musical intervals. Numbers provided an intellectual foundation that did not suffer from the vagaries of the senses. In general neither Boethius nor his greatest exponent, Jacobus (writing c. 1320), explained how they obtained the numbers they used. In this essay I attempt to reconstruct their methods and show how they developed ideas from the first-century Nicomachus to achieve their aims. Jacobus is explicit in saying that the use of the relatively newly introduced methods of algorism – calculating with Arabic numerals – made his cogitations easier. I shall argue that the manuscripts we have of his Speculum musicae show that Jacobus did indeed use algorism in his work.

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Type
Research Article
Information
Early Music History , Volume 36 , October 2017, pp. 1 - 30
Copyright
© Cambridge University Press 2017 

In 2015 this yearbook published an excellent presentation, by Daniel Walden, of the overall strategy of Boethius’s approach to the theory of music, musica speculativa, as described in the first book of his Arithmetica and presented in a twelfth-century manuscript.Footnote 1 There was increasing interest in the music theory of Boethius during the first century of the University of Paris, culminating in the encyclopedic Speculum musicae of Jacobus de Ispania (otherwise known as Jacobus Leodiensis).Footnote 2 This six-hundred page encyclopedia of music theory was written largely in Paris in the period 1290 to the 1320s.Footnote 3 Over the centuries interest in the work of Jacobus has been fragmentaryFootnote 4 and usually centred on his last book and his dispute with Jean des Murs, but I shall only be concerned with Books I and II.Footnote 5

In writing his enormous Speculum Jacobus lamented it had required ‘many cogitations’.Footnote 6 Understanding and plausibly reconstructing some of these cogitations is the aim of this article.Footnote 7 In writing his first two books Jacobus was clearly following in the footsteps of Boethius and his approach was Pythagorean. He refers to both the De institutione musica and the De institutione arithmetica,Footnote 8 and frequently mentions the newly rediscovered ideas of Aristotle that were increasingly current when he was writing. In his first two books, although he mentions the Philosopher, his concerns with musical proportions are purely intellectual and do not involve the senses.Footnote 9 Both Boethius and Jacobus had access to the first few books of Euclid.Footnote 10 It is not clear which version (or versions) of Euclid Jacobus was familiar with. In III. 24 he simply says: ‘But the excerpts are in the main from part of Jordanus’s Arithmetic and Euclid’s Geometry.’Footnote 11 Robert of Chester’s version, however, was widely used at that time.Footnote 12 In fact relatively little Euclidean knowledge is required and much of what is needed was already in the texts of Boethius (though Boethius does not explicitly mention Euclid) or else via Jordanus. Indeed, apart from his explicit reference to Euclid’s Book VII there is no indication that he obtained his knowledge of Euclid other than via Boethius and Jordanus’s Arithmetic.Footnote 13

The great advantage of Jacobus is his knowledge of algorism – the use of Hindu-Arabic numerals and calculations with them, especially their facility for calculations with large numbers.Footnote 14 Using algorism Jacobus extends the techniques of Boethius and takes them to new heights so that his Speculum marked the zenith of Boethian ideas about the theory of music.

In order to reconstruct his cogitations it seems necessary to revisit the works of Boethius and his inspiration, Nicomachus of Gerasa. Some of these ideas appear to have been lost in translations into English, so they will be discussed first. Next, the techniques of Nicomachus will be discussed, though these are not always explicitly described by him. This remains a problem even in modern mathematics since mathematicians from Euclid on seem to work backwards: they work out how the numbers, formulae and axioms, etc., should be and then present the work going from the last-discovered, but basic, elements; Boethius and Jacobus exhibit the same trait.Footnote 15 Nevertheless there is strong circumstantial evidence provided from their works. After that I turn to Jacobus’s own advances in pushing the ideas and techniques of Boethius even further. Boethius was concerned to identify musical intervals with numerical proportions and, at the end of his second book (II, [126]), Jacobus lists fifty-one such intervals, the largest extending to three octaves plus a tone.Footnote 16 How Jacobus managed the necessary computations is my final contribution.

The influence of Boethius on music theory was felt until at least the nineteenth century. In the thirteenth and fourteenth centuries his influence was significantly revived seven hundred years after his death and eventually all five books of his De institutione musica were studied ‘around 1320 . . . after a 200-year eclipse’.Footnote 17 Key figures in this revival were Jerome of Moray (fl. 1272) and Jacobus. The former revived interest in Boethian music theory and considered all five books of De musica, rather than just the first two, and the latter specifically extended the work of Boethius.Footnote 18

It was in Paris that Jacobus studied the De institutione musica of Boethius in detail.Footnote 19 He himself mentions being in Paris twiceFootnote 20 and it seems generally agreed that he was there sometime between the 1290s (certainly after 1277) and 1310.Footnote 21

Jacobus refers to works of Boethius 620 times: De institutione musica, De institutione arithmetica and the Consolation of Philosophy. Jacobus himself records that he heard [lectures on] the first two books of the De Musica in Paris. It is clear, however, that he ultimately read all five books,Footnote 22 and this was made possible because a ‘certain worthy man’ (quodam valente) made a copy of the Musica available to him.Footnote 23

Despite his persistence, Jacobus found the mathematics of Boethius hard to understand and he states this quite firmly at the very beginning: ‘But the work that I took on was laborious and long for me, and many cogitations had to be expended, and it was begun too late [in my life].’Footnote 24 He is explicit in his treatment of Boethius: ‘In some places of Boethius’s text I added no notes, nor abbreviated with glosses, and in other places which seemed to me more difficult, wherever this occurred, I endeavoured to explain in text and figures.’Footnote 25

At that time there were two traditions in mathematics: one in academia, the other in commerce. Jacobus seems to have taken advantage of both. In academia the focus was on theoretical mathematics (mathematica speculativa) and was not concerned with calculation; rather it focused on the abstract properties of numbers together with relations between numbers. This academic tradition considered, for example, the notions of square, triangular and even pentagonal numbers.Footnote 26 De institutione arithmetica is a principal source for Jacobus and he refers to it more than two hundred times. He also refers to Jordanus Nemorarius fifteen times,Footnote 27 but Euclid only five times, though he cites these two, together with Boethius’s inspiration, Nicomachus, as his mathematical authorities.Footnote 28

Jacobus was clearly familiar with algorism,Footnote 29 although calculation was the primary concern of those engaged in commerce and the union of the two traditions – academic and commercial – only really came about in the sixteenth century.Footnote 30 The methods of algorism not only included the four basic operations of addition, subtraction, multiplication and division, but also fractions and the extraction of square and even cube roots. These were known increasingly widely because algorism had originally reached western Europe in the preceding century through the Maghreb in works such as those of al-Khwārizmī and Fibonacci (Leonard of Pisa).Footnote 31

While it is true that Latin translations from both Greek and Arabic of essentially the whole of Euclid were available at the time, only a fraction of the work of Euclid was known at all widely.Footnote 32 In the University of Paris, the syllabus centred on just the first six books.Footnote 33 It appears from what he writes that Jacobus did not use, and perhaps did not even understand, material from the books beyond Book VI (except for one reference to Book VII with Jordanus in III. 28). These earlier books had already been available to Boethius though he had developed most of his Arithmetic from Nicomachus, who lived c. 100 ce.Footnote 34 The influence of Jordanus, in particular his De elementis arithmetice artis, however, should not be underestimated.Footnote 35 According to Karen Desmond, his ‘four-hundred proposition De elementis arithmetice artis was his best-known text and became the standard source of theoretical arithmetic in the Middle Ages’.Footnote 36

Before proceeding to the techniques involved it may be useful to recall the attitude to the teaching of mathematics and number. Instruction in Paris in the late twelfth and early thirteenth centuries concentrated on the Trivium: Logic, Grammar and Rhetoric. The Quadrivium of mathematical sciences: Arithmetic, Geometry, Astronomy and Music, was very much relegated to a lower role. Guy Beaujouan pointed out long ago that these subjects were only taught on feast days, of which there were, it must be said, very many (but not on very solemn feasts, though there were about sixty supplementary teaching days).Footnote 37 The Quadrivium was not part of the regular cursus studiorum.Footnote 38 Indeed the newly felt influence of Aristotle drove people away from mathematics.Footnote 39 As early as 1255 a detailed statute stipulating the syllabus conspicuously excludes the Quadrivium,Footnote 40 though a century later a statute of 1366 decreed that the ‘scholars at Paris were obligated to attend lectures on “some mathematical books”’.Footnote 41 Gilles Rico, however, has recently shown that there was much more teaching of, and interest in, mathematics than has previously been noted.Footnote 42

The Quadrivium, the mathematical part of the seven arts, was traditionally divided into two parts: those concerned with number, namely Arithmetic and Music, and those concerned with magnitudes: Geometry and Astronomy. The distinction between number and magnitude goes back at least to Euclid.Footnote 43 Magnitudes – length, area, weight, etc. – have an obvious connection with geometry and astronomy; quite literally we see (or feel) the relation. The division between number and magnitude had been absolutely fundamental in the thinking of Euclid and dominated Boethius also. Once this connection had been established between music and numbers, the medieval mind, when thinking about music, only thought of numbers – and not magnitude.

Boethius defined music as ‘numbered sound’.Footnote 44 Guy of Saint-Denis, writing about the time Jacobus was in Paris, said: ‘For since music is, according to Boethius, about number related to sound,Footnote 45 or about sound numbered “harmonically”,Footnote 46 just as arithmetic is about number in itself and considered absolutely,Footnote 47 because this proportion is in the numbers themselves this seems to be concord in musical sounds.’Footnote 48 So if music is numbered sound, as Boethius and others called it, then it has nothing to do with magnitudes. This mind-set was handed down to Jacobus and was the source of his problems with the equal division of the tone, which I have discussed elsewhere.Footnote 49

Nowadays it is hard to appreciate this difference; for us lines, areas, volumes all have numerical sizes and we compare them willy-nilly by using those numbers. It was not so in the Middle Ages, and certainly not for Jacobus. In order to understand some of the difficulties consider how we compare sizes of clothing even within one system.Footnote 50 Baby clothes, although there are sizes 1, 2, 3, also have 0 and 00 because the number series does not fit. Shoes come in sizes with, sometimes, an additional letter indicating the fit. To Euclid the connection between magnitude and number was through the idea of measuring.Footnote 51 If we have a given length, and we regard that as a unit, then when we try to measure another length by counting the number of times it takes putting our unit measure end to end along the line we sometimes get a whole number. This is not always the case and the most famous counter-example is the diagonal of a unit square. This cannot be represented as a fraction, so there is no fraction of the unit that divides the length of the diagonal exactly.Footnote 52

When musical intervals were compared discrepancies were immediately apparent: discrepancies that would not have occurred if they had been treated in terms of magnitudes. However, for this we shall need to consider proportions and their use, in particular the composition of proportions, but before that the relation between arithmetic and music, which is more philosophical, must first be considered.

Boethius’s abhorrence of basing ideas on the senses is well known and recorded at several places in his work. Jacobus followed Boethius and Pythagoras in not trusting the senses but requiring the theory of music to be in the domain of rules and reason.Footnote 53 In this he follows the authority of Boethius, Book I, chapter 9: ‘Not all judgement ought to be given to the senses but reason ought more to be trusted. . . . The ultimate perfection and the faculty of recognition consists in reason, which, holding itself to fixed rules, does not fall into any error.’Footnote 54 So music was put on a firm intellectual foundation, rather than one based on the senses, and this view persisted. Indeed, as will be seen below, for Jacobus number had a much more valid place than magnitude.Footnote 55

From at least the time of Aristotle 1 was not regarded as a number but the source of numbers. The other numbers were the usual, so-called ‘natural numbers’, otherwise known as cardinal numbers. Ordinal numbers 1st, 2nd, 3rd, etc. were used in two ways: first as numbering items listed in order, as the name suggests, but also as names of parts or fragments, e.g. one third, one eighth.Footnote 56 Fractions are not used as such but parts of a whole play a central role in the theory of proportions, since all three authors, Nicomachus, Boethius and Jacobus, are particularly concerned with proportions that involve one or several such parts.

‘Proportion’ seems the most appropriate term and the best translation of λόγος as used by Nicomachus and proportio as used by Boethius and Jacobus. As Margaret Bent pointed out to me, the Latin word ratio was rarely used in the sense of the modern English word ‘ratio’ by medieval writers on music theory,Footnote 57 who generally reserved it for the sense of ‘reason’, yet seems to have largely been used by many translators in translating proportio.Footnote 58 Medieval writers seem to have been more circumspect.

It therefore seems important to be more precise and distinguish between ratios and proportions, both of which are nowadays normally written a : b. A ratio is given by two numbers and we write a : b. Proportions are represented by ratios.Footnote 59 Nevertheless even the Oxford Encyclopedic Dictionary (but not the Oxford English Dictionary) says ‘the ratios 1:5 and 20:100 are the same’.Footnote 60 One wonders what a medieval writer would have made of this! Clearly the expressions 1 : 5 and 20 : 100 are different. The natural question for a medieval (or even a modern) person then is, ‘What is the name of this ratio?’ Amending the quotation to ‘the ratios 1:5 and 20:100 are in the same proportion’, resolves the problem and then the name of the proportion is clear: ‘one fifth’.Footnote 61 Nicomachus’s definition of λόγος: ‘a proportion is a relation between two numbers’ is then also clear.Footnote 62 Nicomachus also uses ’αναλογία for a combination of proportions. Boethius translates ’αναλογία as proportionalitas, which Jacobus also uses. The Latin preserves both the sense and the grammar of the Greek using the abstract noun. Proportionalitas is a sequence of numbers such that each is in the same proportion to its predecessor, for example, 9 12 16, where successively we have the proportion of 3 : 4. The modern term is ‘continued proportion’ which retains the meaning of the ’ανά as well as the λογός. Boethius, II. 12 says: Proportionalitas est aequarum proportionum collectio (A continued proportion is a collection [sequence] of equal proportions).Footnote 63 On occasion Boethius and Jacobus present a sequence of numbers not in the same proportion, and this will be my first example.

Pythagoras is reputed to have determined that the musical intervals of an octave (diapason), fourth and fifth corresponded to the proportions 2 : 1, 4 : 3 and 3 : 2; these were the basic intervals used by Boethius and Jacobus. All of these numbers can be obtained from just the numbers 1, 2, and 3 by multiplication; no other numbers, except products of these numbers, are needed for the calculations presented in this article.Footnote 64 The most important point is that 2 and 3 are numbers whose only divisors are 1 and themselves.Footnote 65

These numbers generate what is possibly the most notable example of a sequence of proportions, namely 6 8 9 12, which occurs at the end of Book II of the Arithmetica of Boethius.Footnote 66 This sequence demonstrates all the basic proportions of Pythagorean music theory: 8 : 6 is in the same proportion as 4 : 3, which is the musical fourth (diatessaron, for example, C–F); 9 : 8 is the tone (e.g. F–G), and 12 : 9 is in the same proportion as 3 : 2, which is the fifth (diapente, e.g. F–C) while 12 : 6, which is in the same proportion as 2 : 1 (taking the extreme terms), is the octave (diapason).Footnote 67 (The fifth is also represented by the proportion 9 : 6.)

Other musical intervals have proportions that can be calculated from these basic ones and such are the main concern of Jacobus in his Book II, as will be seen below. When we add two tones together on the musical scale, e.g. F–G and G–A we get the interval F–A: a third, or interval of two tones; similarly for other intervals. Because proportions are being composed together, instead of addition we have to ‘multiply’ the elements together, so I shall write ⊗ instead of+for the putting together (addition) of proportions.Footnote 68 When we consider the proportions for the two tones we are composing, namely 9 : 8 with 9 : 8, then the resulting proportion is 9 : 8 ⊗ 9 : 8, i.e. 81 : 64.Footnote 69 Conversely, when we subtract one interval from another we ‘divide’ proportions.

The simplest forms of numbers in a given proportion, a : b, are given when it is in its lowest terms, that is to say, there is no number (other than 1) that divides both a and b. Footnote 70 For example 3 : 2 or 9 : 8, but not 18 : 12 since 2 divides both numbers. Boethius and Jacobus usually present proportions in their lowest terms; the Boethian term is primi numeri but Jacobus also uses primi termini, which is to say that in a proportion a : b, there is no number (except 1) that divides both a and b. The obvious translation is ‘prime numbers’ but a better translation is ‘relatively prime numbers’ since although 9 and 8 are relatively prime, neither is a prime number.Footnote 71 Modern parlance also uses the term co-prime. For example 18 : 12 reduced to its lowest terms is 3 : 2, and the ratio 256 : 243, which will feature below, cannot be reduced because only powers of 2 (namely, 2, 22=4, 8, 16, 32, 128) divide 256 and only powers of 3 (3, 32=9, 81) divide 243 and these are the only divisors of all the numbers used below. Any ratio a : b can be reduced to a ratio in its lowest terms.Footnote 72 The fact that only 2 and 3 are used makes determining whether a proportion is expressed in its lowest terms easy since one only has to check whether both numbers are divisible by 2 or 3. The general property from Euclid Book VII, Proposition 30, that a prime number divides a product only if it divides one of the factors, is not needed since we are only concerned with the two small numbers 2 and 3, and Jacobus nowhere explains how he knows the numbers are co-prime.

Considerations of proportions occupy Books I and II of the Speculum musicae. Book I provides a thorough presentation of proportion as it was then conceived.Footnote 73 It relentlessly pursues Jacobus’s desire to set music on a comprehensive and firm mathematical, or more precisely numerical, footing. This means, in particular, that he needs to include material from Boethius’s De institutione arithmetica. The niceties of modern mathematical notation were not available to him, so everything had to be done in words (and numerals).Footnote 74 Moreover, early writers had no great facility with fractions (though they were used). These matters are reflected by the concern of the three authors with terminology. In particular they treat proportions expressed as so many wholes plus so many parts to so many wholes. However, almost all of the proportions lie between 1 : 1 and 1 : 2 or, the other way round, between 1 : 1 and 2 : 1; indeed, it is usually the latter order that is used. The octave (diapason) is struck on the monochord by halving (or doubling) the length of the struck string. So any discussion can usually be reduced to talking about an interval within an octave.Footnote 75

All of the musical proportions mentioned are of a small number of special kinds. Their logical relations are well illustrated in the Cambridge University MS Ii.3.12, fol. 22v discussed by Walden and succinctly described by Moyer.Footnote 76 Jacobus is only concerned with the major kinds of ‘inequalities’ as Boethius calls them: multiplex, superparticular, superpartient, multiplex superparticular and multiplex superpartient – the mention of ‘inequality’ comes from the fact that the two numbers representing a proportion are different.Footnote 77 These names came from Nicomachus and have a simple logic.Footnote 78 A multiplex proportion is one where the first number is a (whole number) multiple of the second, such as represented by 2 : 1 (or with some whole number larger than 2). ‘Super’ in superparticular and superpartient means the first number exceeds the second, ‘particular’ refers to one n-th and ‘partient’ refers to more than one n-th. Although the authors do not use fractional notation it is convenient to do so here. Thus a superparticular proportion is given by a ratio equivalent to 1+1/n : 1, though only whole numbers are used, for example, 3 : 2 is a superparticular proportion, while a superpartient proportion is given by a ratio equivalent to 1+m/n : 1 where m is larger than 1 and less than n, and the m is added to the name. For example, 7 : 4 is in supertripartient proportion: one whole plus three fourths to one (1¾ : 1).

Names were given to specific superparticular proportions, such as sesquialter (or sesqualter), sesquitertia, etc., sesqui indicating one plus one part and the general form corresponding to the Latin ordinal number altera, tertia, quarta, etc., reflecting the nth part; such proportions will often be named in the form ‘sesqui-nth-particular’ below.Footnote 79 For superpartient proportions the same principle applies. The forms are now of the form ‘super-m-partient’ where m indicates the number of parts, for example, superbipartient corresponds to a ratio of the form 1+2/m : 1. Note, however, that the number m may, or may not, be indicated.Footnote 80 For example, in Book II, chapter 87, Jacobus considers the proportion of 128 to 81.Footnote 81 This is a super-47-partient proportion of 81st parts (1+47/81 : 1). Multisuperpartient proportions correspond to ratios of the form k+m/n : 1 where k is larger than 1. In an example from both Boethius and Jacobus treated below, the size of the parts is often not stated explicitly by Jacobus.Footnote 82

The first thing needed here is a method for calculating sesqui-nth proportions. It was apparently by straight experimentation that Nicomachus had worked out a technique for adjusting numbers in a proportion in order to be able to take successive sesqui-nth proportions.Footnote 83 In his Book II, chapters III and IV, he generates tables for successive numbers in sesquialter and sesquiterita proportions (see Figure 1).Footnote 84 The principle he uses is ‘since doubles generate sesquialters corresponding to them in number, the first row of multiples will be 1, 2, 4, 8, 16, 32, 64.Footnote 85 Now since 2 is the first after unity, this will be the origin of one sesquialter only, 3,’ which is not divisible by 2. The next double, 4, is indeed divisible by 2 twice, so generates two sesquialters, and so on.Footnote 86 For the case of sesquitertian proportions start with repeated products of 3, namely, 1, 3, 9, 81, . . . and then add one-third of a number to get the number below it. E.g. 144 has 144 ÷ 3=48 added to it, yielding 192, as in the third row and fifth column of the second table in Figure 1.

Figure 1

Tables of sesquialter and sesquitertian proportions

Although Boethius and Jacobus would surely have known this technique they use a different one. In describing the genesis of sesquialters Boethius starts with three numbers, apparently in the style of Nicomachus but reversed:

$${4\quad 2\quad 1}$$

Then he says to write down three new numbers: ‘the first is equal to the first, that is 4, the second to the first and second, that is 6, and the third equal with the first, twice the second, then the third, that is 9’ (my italics). So we get

$$\eqalignno{ & {4\quad 2\quad 1} \cr & {4\quad 6\quad 9} $$

This was an ancient technique, known in the second century bce, and termed the ‘Adrastus rule’.Footnote 87 No explanation is given for the ‘twice’. However it seems likely that Nicomachus was thinking of square numbers, which are figurate numbers, a class he considers at length in his Book II from chapter VII.Footnote 88 It can then be seen that the square of one plus a number is the number squared plus twice the original number plus one: the original square plus two sides plus an extra 1 in the corner: (n+1)2=n 2+2n+1 (see Figure 2). Therefore in order to get successive sesqui-nth proportions one needs to start with a number that is divisible by n twice. Obviously the least such number is n 2. Multiplying this by (1+1/n) is the same as dividing by n and then multiplying by (n+1), which gives the next number in sesquialter proportion: (n+1)n or n 2+n. The third number is this number divided by n and multiplied by (n+1), and is therefore (n+1)×(n+1) or (n+1)2, which is n 2+2n+1. In calculating all his numbers Jacobus, like his predecessors, needed to compose proportions because all other musical intervals have proportions that can be calculated from these basic ones. When we add two tones together on the musical scale, e.g. C–D and D–E we get the interval C–E: two tones; similarly for any other interval. Because these are proportions that are being composed together, instead of addition we have to ‘multiply’ the elements together. When we consider the proportions for the two tones we are composing, namely 9 : 8 with 9 : 8 then the resulting proportion is 81 : 64. Conversely, when we subtract one interval from another we ‘divide’ proportions.

Figure 2

Forming a continued proportion of size 1+1/n, i.e. a sesqui-nth proportion. The areas are n 2 : n 2+n : n 2+2n+1 or n 2 : n(n+1) : (n+1)2.

As mentioned above, mathematicians, and specifically Boethius and Jacobus, generally work backwards, but there is one place where their ratiocinations are clear: the determination of the numbers follows the intuitive rather than the mathematical sequence. This is given by Boethius in his De musica, II. 28, and Jacobus in II. 55, when calculating the proportion of a semitone.Footnote 89

A semitone is the difference between two tones and a fourth, for example the interval between the pair of tones that go from C to E, and the fourth, C–F. The process is illustrated in Figure 3. Since one tone corresponds to 9 : 8, write down 8 as the starting point. By advancing a tone one moves to 9. However an interval of two tones is required. One cannot take 9/8 of 9 as this would not give a whole number, so go back to the beginning and, instead of using 8, use 8×8=64 as the starting point. The proportion is unchanged, the ratio is now changed to 72 : 64. Advancing one tone yields 9/8×64=72 and, because the extra 8 has been used, it is now possible to advance another tone, which gives 9/8×72=81.Footnote 90 This yields the continued proportion 64 : 72 : 81 (where the order of the numbers has been reversed).

Figure 3

Reconstructing Boethius’s numbers for the minor semitone

The technique of looking at just the smaller number in a proportion and multiplying by that number is something that occurs frequently. Boethius sometimes points this out; Jacobus seems not to.Footnote 91 Once Boethius has the sequenceFootnote 92

$$64\quad 72\quad 81$$

he can now consider the enveloping interval of a fourth, whose proportion is 4 : 3 or one and a third to one (1⅓ : 1). He therefore needs to add one third to the starting point in order to find the number n to fit the sequence

$$ 64\quad 72 \quad 81 \,n$$

where n represents adding one third of 64 to 64. Again fractions are to be avoided. Boethius therefore multiplies the first three numbers by 3 and this does not disturb the proportions. This gives

$$ 192 \quad 216\quad 243$$

and now it is possible to take a third of 192 (namely 64) and add it to 192. In this way the final sequence is obtained:

$$192\quad 216\quad 243\quad 256 $$

Here 192 : 256 is in the same proportion as 3 : 4 (= 64×3 : 64×4). It is clear how the proportions work since 192 : 216 is in the same proportion as 8 : 9, and so is 216 : 243.Footnote 93 So finally the proportion for the semitone is revealed by the difference between two tones and a fourth (diatessaron) and is the proportion of 256 : 243. Moreover this is in its lowest terms: 256 and 243 are relatively prime.Footnote 94 Incidentally, this proportion is a super-13-particular proportion, since 256/243=1+13/ 243.Footnote 95

Jacobus follows Boethius in analysing the numerical proportions for all sorts of combinations of musical intervals and then extends Boethius’s work. The beginning of Book II provides descriptions and mathematical details of the basic intervals and their relation to mathematical proportions. As noted above, despite the difficulties of calculating with Roman numerals, Boethius managed to calculate proportions involving numbers as large as about half a million.Footnote 96 Jacobus rehearsed all of these and took matters a step further, going into the millions. This is the culmination of his Book II: it is principally devoted to establishing the proportions corresponding to an exhaustive list of musical intervals.Footnote 97 These combinations go far beyond those to be found in Boethius’s De institutione musica and involve extensive computations.Footnote 98

Now I turn to investigating Jacobus’s cogitations when he conducted his exhausting analysis of numerous intervals in Book II. One way to approach Jacobus’s calculations can be demonstrated by looking at the end of the second book of Boethius, II. 30, or his IV. 2.Footnote 99 This result is taken from the ancient Sectio canonis but with some changes.Footnote 100 Jacobus is showing that the diapason (octave) does not comprise six tones.Footnote 101 He uses the above ideas and rather than immediately attempt to multiply by the fraction 9/8 (corresponding to a tone) six times he first looks at successively multiplying 8s, starting from 1.Footnote 102 This gives 1, 8, 64, 512, 4,096, 32,768 and finally 262,144, as shown in the first row of the table in Figure 4.Footnote 103 Then, starting with the last number, he will get whole numbers each time he multiplies by 9/8 (up to six times) so that the final proportion (where he has had to divide by 8 six times) will have (whole) numbers.Footnote 104 This yields the numbers in the penultimate column in Figure 4. Eventually he arrives at 531,441 and so the proportion for six tones is 531,441 : 262,144. In modern terms we would simply look at 96/86, which is indeed 531,441/262,144.

Figure 4

A version of the Boethian figure for comparing six tones with an octave

Boethius is always interested in getting such pairs of numbers in their lowest terms (ensuring they are co-prime), that is to say, such that there is no number (apart from 1) that will divide both of them. Now the only number (apart from 1 and 9) dividing 9 is 3 and the only numbers (apart from 1 and 8) dividing 8 are 2 and 4, so 9 and 8 have no common divisors.Footnote 105 A fortiori the powers of 9/8, in particular 96/86, are automatically in their lowest terms: there are no common divisors of top and bottom numbers. It follows that in order to get a ratio corresponding to 96/86 involving only whole numbers then the smallest choice has to be 531,441 : 262,144.

Musically speaking, we can think of 262,144 as corresponding to the highest of the seven notesFootnote 106 and 531,441 as the lowest, or alternatively regard these numbers as being the relevant lengths of a string on a monochord. To get an octave below we need to double the length of the string. Thus 262,144 is doubled to 524,288. Finally, it is clear that the octave (corresponding to 524,288) is smaller than the six tones (corresponding to 531,441). The resulting proportion of 531,441 : 524,288 is called the Pythagorean comma.Footnote 107

Jacobus uses this approach in discussing the comma further (II. 44). He directly quotes Boethius III. 10, saying ‘The [Pythagorean] comma is the smallest interval of sounds perceptible to the senses’,Footnote 108 but then goes on to discuss its proportion. Like Boethius he considers the Pythagorean comma in a number of different ways and enumerates several equivalent definitions: A comma, as is said, is that particle or interval by which a perfect tone exceeds a minor tone or the sesquioctave proportion two minor semitones, or a hexatone [exceeds] a diapason.Footnote 109 As has already been seen, the proportion of a Pythagorean comma is 531,441 : 524,288. Jacobus proceeds to calculate the characterisation of the comma as the difference between six tones and an octave. From Figure 4, six tones correspond to 531,441 : 262,144, and an octave to 2 : 1. The difference is therefore that between 531,441 : 262,144 and 2 : 1, that is to say, 531,441 : 524,288.Footnote 110 Jacobus, however, does not present this calculation. Instead he simply lists the numbers:Footnote 111

$$531441\quad 524288 \quad 497664\quad 472392$$

Where do these numbers come from? Jacobus seems to be using the same technique as above but with larger numbers and to be working from right to left. However, all Jacobus points out is ‘Between the first and the fourth numbers there is a sesquioctave proportion in which lies the perfect tone; between the second and <the fourth> is the proportion of two minor semitones in which lies a minor tone’.Footnote 112

The following explanation may give some idea of the cogitations of Jacobus – and also demonstrate his facility with large numbers. The perfect tone is 9 : 8 and a minor semitone is 256 : 243. So he starts with

$$256\quad 243 $$

In order to get another minor semitone, starting from 256 he needs to find a number in the proportion 256 : 243. As noted above, 256 : 243 is in its lowest terms, so it is necessary to multiply both numbers by 243, obtaining

$$ 256\, \times\, 243 243\, \times\, 243\ {\rm or}\ 62208\quad 59049 $$

All numbers can then be divided by 243 and a further number put on the left in the proportion 256 : 243, thus

$$256\, \times\, 256 256\, \times\, 243 243\, \times\, 243\ {\rm or} \ 65536\ 62208\ 59049 $$

Finally a number in the proportion of a tone (9 : 8) to the last number (59049) is required. Now the last number is only divisible by 3 (in fact ten times) and not by 8, so all numbers need to be multiplied by 8. This yields

$$ 524288\quad 497664\quad 472392$$

and finally Jacobus can take 9/8 of the last number, i.e. 472,392×9/8=531,441, yielding his list:

$$531441\quad 524288\quad 497664\quad 472392 $$

Jacobus has hidden his calculations; he simply observes that the first and the fourth form a tone and that two minor semitones form a minor tone.Footnote 113 He does not mention that the two proportions between the second and third and between third and fourth each correspond to a minor semitone. He does, however, mention that he cannot take the proportions he wants, so he has to multiply all the numbers.Footnote 114

Jacobus gives altogether a list of fifty-one musical intervals he considers and the list may be found near the end of Book II (chapter 126). Here is just one example, but it involves the largest numbers he employs. Halfway through Book II (in chapter 72) he calculates the proportions for a major tone, which comprises two major semitones. A major tone is two major semitones and a major semitone is the difference between a minor semitone and a whole tone. It is therefore in the proportion 9 : 8 ÷ 256 : 243=9×256 : 8×243=2187 : 2048, or 1+139/2048 : 1 since 2048 − 2187=139.Footnote 115 Therefore the composition of two major semitones corresponds to the ratio 2187×2187 : 2048×2048 or 4,782,969 : 4,194,304.

A list of the first ten of his fifty-one interval proportions and modern calculations for them may be found in the Appendix below. In some cases there is more than just the calculation of the proportion of a musical interval. In the case of the major semitone he proves the proportion is greater than 9 : 8 and less than 8 : 7. The modern calculator would simply express all these ratios as decimal fractions and then compare them, but the proof Jacobus gives for the former is remarkably simple. He simply compares the differentiae, i.e. the difference in the number of parts. For 4,782,969 and 4,194,304 it is 588,665;Footnote 116 and for 9 and 8 it is 1. Division is not an option for Jacobus since he is already dealing with fractions. So instead he looks at the size of 588,665 relative to the whole (4,194,304) and compares this with one-ninth. We could say: he scales matters up. So he multiplies 588,665 by 9 to get 5,297,985, but when he multiplies by 8 he gets 4,709,320.Footnote 117 Clearly 4,782,969, the first number of the proportion for a major tone, lies between these two numbers so the proportion is less than sesquiseptima (1 1/7 : 1) and greater than sesquioctava (1 1/8 : 1).Footnote 118

One thing remains of note, the facility that algorism provided to Jacobus. As mentioned above, Jacobus was heavily influenced by Jordanus Nemorarius, but the De elementis arithmetice artis of Jordanus does not treat algorism. Jordanus, however, wrote another work on algorismFootnote 119 and there were other algorisms circulating (in many copies), notably by Johannes Sacrobosco and Alexandre de Villedieu.Footnote 120 The art of algorism comprises how to write and read Arabic numerals and the methods of doing arithmetical computations with them. Jacobus only mentions algorism on two occasions. First, he commends Robert Kilwardby, who includes algorism as a practical art extending the theory of numbers in his De ortu scientiarum.Footnote 121 On the second occasion he revels in the value of arithmetic and shows great enthusiasm for the advantages of algorism: ‘Oh commendable arithmetic! Oh how useful is the science of algorism!’Footnote 122

As I have shown elsewhere, the Arabic numerals (which form the basis of the science of algorism) were only very slowly taken up despite their appearance in western Europe by the eleventh century.Footnote 123 The methods of calculating, in particular multiplying, were much simpler once one was used to operating with, or even just reading, the numerals. Jacobus’s knowledge of algorism surely made an enormous difference to him in doing his calculations and indeed was what made his Books II and III possible, even though it was feasible to perform the same operations without algorism, as Boethius had done in many cases.

Before algorism was available calculations were usually made on an abacus. Arithmetic operations could be performed with Roman numerals but, while addition and even subtraction were not too hard, and while it is possible to multiply Roman numerals, the inherent difficulties are readily apparent.Footnote 124 For example, consider 40 times 90. In Roman numerals this is XL times XC and the, clearly unintuitive, answer is MMMDC. As Murray pointed out long ago, people in the Middle Ages used ready-reckoner tables to multiply Roman numerals.Footnote 125 In this way multiplication was reduced to repeated addition. It is most likely that Boethius used the abacus, but Jacobus used even larger numbers than Boethius.

Hindu-Arabic numerals were certainly regarded in the early days of their introduction into western Europe (and even much later) as quite strange. On occasion they were written in boxes, as is often the case in Paris 7207, though the boxes are sometimes incomplete, missing the bottom side.Footnote 126 Thus Arabic numerals in a medieval manuscript often look more like figures (in the sense of diagrams), rather than abbreviations for (number) words as Roman numerals do.Footnote 127 Charles Burnett and the present author have independently suggested that these new ‘figures’ were indeed more akin to diagrams and therefore not part of the standard alphabet used by scribes.Footnote 128

Another distinguishing feature of algorism is the way that calculations (addition, subtraction, multiplication and division) are made. When one works with an abacus no trace remains of the steps in the calculation once it is complete. By contrast, when using Arabic numerals on paper the techniques of algorism leave traces that can be independently verified: one can check the working. This also reduces the chance of error.

Finally, while the mathematics taught in Paris was very much Boethian,Footnote 129 and although Johannes de Muris (Jean de Murs, c. 1290–5 to after 1344) contended that mathematics was neglected in Paris because the students found it ‘unintelligible or excessively difficult’,Footnote 130 algorism had begun to be taught.Footnote 131 Jacobus used his knowledge of algorism to revive (not merely revise) and extend the work of Boethius.

Exactly how Jacobus acquired his knowledge of algorism we do not know. His work does not contain any references to treatises on algorism. However, there were people in Paris, even before 1300, who knew algorism. That is not to say it was widely known. It is unknown what number or fraction of people taught or knew such matters. It seems very likely that those few knowledgeable in mathematics would have known each other. Moreover, Jacobus was expecting lovers of mathematics – and algorism – to enjoy the results of his cogitations: ‘If I have been lacking, it is because I am not sufficient to fully understand such hard material, which requires many cogitations, and many operations with numbers’, but then he continues: ‘If they [the readers] are lovers of music theory, they will delight in it. They will revel in the proportions of numbers, in the varieties and stupendous comparisons of [huge] numbers.’Footnote 132

All of this seems to confirm his remark that he has had many cogitations.Footnote 133 A table of the first ten of the fifty-one intervals that he lists at the end of II. 126, for which he provided the proportions, may be found in the Appendix. The calculations there have been done in the modern manner, not following the method of Jacobus as presented above. The proportions for the other forty-one intervals are tedious, rather than difficult, for a modern reader equipped with a calculator – or simply pen and paper – to work out. They certainly bear out the comment of Jacobus that the work was long and laborious.Footnote 134

When Jacobus is determining the proportion corresponding to a musical interval he introduces a sequence of numbers but does not explain how they arise; he does, however, make clear the proportions between them. Boethius, on the other hand, has given a clearer insight into whence these numbers come. I am led to believe that Jacobus thought of the length of a monochord stringFootnote 135 for the musical note (or pitch) from which he started his considerations, and then worked along the string assigning the appropriate number to match the musical interval.

In general I conclude that Boethius and later Jacobus wanted to use whole numbersFootnote 136 and the proportions encountered would often have introduced fractions if applied directly. They therefore changed the scale of measurement by multiplying by appropriate numbers – but always ensuring that these multipliers were as small as possible.Footnote 137 When they had climbed up through the various intervals being used they would then have the result that the extreme numbers gave them the proportion sought. The evidence provided by Boethius and Jacobus supports this analysis of how Jacobus obtained the huge numbers in his work in this way. He, however, had the great advantage that he knew algorism, though in his cogitations he still found the work hard. Nevertheless the facility that algorism gave him was what permitted him not only to understand and verify all five books of Boethius’s De musica but also to extend the work of Boethius.Footnote 138

APPENDIX

The proportions of the first ten intervals as determined by Jacobus in his Speculum musicae, Book II, Chapter 126

Footnotes

I am very grateful to Margaret Bent, who introduced me to the ‘elephant in the room’ (Jacobus’s opus) in Oxford in 2015 and subsequently provided trenchant and helpful comments. My colleagues Constant Mews and Carol Williams have offered constant support and encouragement. Constant, Karen Desmond and Philipp Nothaft have provided valuable comments. Finally I am grateful to two anonymous referees who read the article most diligently and provided not only helpful suggestions but also new insights.

1 A comma is the difference between an octave and six tones. See Jacobus, II. 44 and Boethius, III. 4.

2 A diesis is the difference between a fourth and two tones (Jacobus, II. 55).

3 Boethius, II. 27.

4 A major semitone is the difference between a semitone and a whole tone, therefore is in the proportion 9 : 8 ÷ 256 : 243=9×256 : 8×243=2187 : 2048, or 1 139/2048 since 2048 − 2187=139. Jacobus, II. 64.

5 A minor tone comprises two minor semitones for each of which the proportion is 256 : 243; see item III.

6 A major tone is two major semitones.

7 The proportion is given by 2187 : 2048×2187 : 2048=4782969 : 4194304 and 4782969 – 4194304=588665. See also Crossley, ‘A Sense’.

8 A semiditone comprises a tone and a minor semitone.

9 Here again Jacobus, II. 63 uses the fact that 243 ÷ 9/8 (or, equivalently, 243×8/9)=216 and then baldly states that, in its lowest terms 256 : 216 is 32 : 27. (In qua autem determinata et specifica tali proportione fundetur ostendunt primi termini ipsius qui sunt 32 et 27.)

10 A ditone is simply two tones.

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 .

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) 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 .

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), 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 ); 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) and Masi, M., Boethian Number Theory: A Translation of the De Institutione Arithmetica (Studies in Classical Antiquity, 6; Amsterdam, 1983).(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 , 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 ). 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).

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), 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 , 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 , 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 XIIIe 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 .

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 , 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 .

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 . 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. 60r, ‘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. 41v [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. 41v [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. 60r [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).

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 .

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 , 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), 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 .

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 . 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 , 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 .

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. 41r [4.13], and Guy of Saint-Denis, Tractatus de tonis, fol. 61r [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 , 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).

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 .

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 .

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 .

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 .

126 See e.g. Paris 7207, fol. 67r 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 .

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?

I am very grateful to Margaret Bent, who introduced me to the ‘elephant in the room’ (Jacobus’s opus) in Oxford in 2015 and subsequently provided trenchant and helpful comments. My colleagues Constant Mews and Carol Williams have offered constant support and encouragement. Constant, Karen Desmond and Philipp Nothaft have provided valuable comments. Finally I am grateful to two anonymous referees who read the article most diligently and provided not only helpful suggestions but also new insights.

Figure 0

Figure 1Tables of sesquialter and sesquitertian proportions

Figure 1

Figure 2Forming a continued proportion of size 1+1/n, i.e. a sesqui-nth proportion. The areas are n2 : n2+n : n2+2n+1 or n2 : n(n+1) : (n+1)2.

Figure 2

Figure 3Reconstructing Boethius’s numbers for the minor semitone

Figure 3

Figure 4A version of the Boethian figure for comparing six tones with an octave

Figure 4

1