While much has been written about the absorption of Aristotelian texts into the curriculum of the University of Paris, much less attention has been given to the role of Jean des Murs (c. 1290–5 to after 1344), whose interest in the teaching of Boethius on music went much further than was normally the case in the University of Paris in the early fourteenth century.Footnote 1 In recent years, there have been several editions of his Musica speculativa secundum Boetium.Footnote 2 There have been relatively few efforts, however, to appreciate the originality of a treatise which, we argue, is much more than a derivative summary of all five books of the Boethian De institutione musica.Footnote 3 We hold that Jean draws on a mathematical frame of mind to present the key arguments of Boethius as eighteen conclusions, but framed within epistemological principles provided by Aristotle.
THE STUDY OF THE DE INSTITUTIONE MUSICA OF BOETHIUS IN THE THIRTEENTH CENTURY
The interest of des Murs in the Boethian De institutione musica Footnote 4 went much deeper than the limited focus on its first two books mandated by the University of Paris.Footnote 5 Far fewer copies of the Boethian treatise survive from the thirteenth century than from the twelfth.Footnote 6 A vivid example of how Aristotle’s criticism of Pythagorean tradition in the De caelo affected the standing of Boethius in the 1270s is provided by Johannes de Grocheio’s criticism of the notion of an audible music of the spheres, upheld by followers of John of Garland.Footnote 7 Grocheio himself only refers to the first two books of the De musica.Footnote 8
That only the first two books of Boethius were studied in the late thirteenth century is also evident from comments made by Jacobus (c. 1260–after c. 1330) in his vast Speculum musicae, written sometime after 1325, that in his youth he had heard lectures on the first two books of the De institutione musica.Footnote 9 Afterwards, having acquired a complete copy of this work of Boethius ‘from a certain worthy man (quodam valente)’, he started to read much more widely to deepen his understanding.Footnote 10 Jacobus reports that while there are some people who extract a few things from that treatise, ‘they say things that hardly accord at all with the intention of Boethius’.Footnote 11 The possibility must be considered that Jacobus was directing his polemic against those, like Jean des Murs (whose Musica speculativa Jacobus quotes almost verbatim),Footnote 12 who were interpreting Boethius in a very different way from traditional abridgements of the De institutione musica.
If Jacobus was born around 1260, his interest in exploring all five books of Boethius may have been influenced by the Dominican author Jerome de Moravia (of Moray), who seems to be one of the earliest theorists to initiate renewed interest in all five books of the Boethian treatises.Footnote 13 Jerome incorporates almost all of Books I–IV into his Tractatus de musica, with a few paragraphs from Book V.Footnote 14 He was particularly interested in combining what Boethius had to say with more recent treatises on mensural music by John of Garland, Franco, and Petrus Picardus, all masters of what came to be known as the ars antiqua. Exactly when Jerome completed his Tractatus is not certain, but it must have been after 1272 (as he incorporated part of the commentary on the De caelo of Thomas Aquinas, composed between late 1271 and his death on 7 March 1274) and before 1306, when the manuscript was bequeathed by Peter of Limoges (c. 1240–1306) to the library of the Sorbonne. Jerome’s importance in broadening attention to beyond the first two books of the De institutione musica deserves attention in appreciating the achievement of Jean des Murs in creating an epitome of the entire treatise, not just of its first two books.
THE NOTITIA ARTIS MUSICAE AND THE TWO VERSIONS OF THE MUSICA SPECULATIVA
Jean des Murs does not explicitly acknowledge Boethius by name in his first exposition of the theory and practice of music, the Notitia artis musicae, a work dated to 1319 according to a false explicit, but in fact 1321.Footnote 15 In this treatise, he sets out his understanding of the theory and practice of music, introduced by a prologue in which he draws on the teaching of Aristotle in the Metaphysica to emphasise that while theoretical understanding is superior to experience, all art must be based on experience.Footnote 16 The first of its two parts begins with discussion of the generation of sound and its relationship to number (chapters 1–2), followed by discussion, drawn from Boethius, of what Pythagoras had discovered about musica, namely the relationship between consonances and numerical proportions (chapter 3). Chapters 4–5 are similarly based on core teachings of Boethius (taken mostly from Books I–II) about the impossibility of dividing the tone into two equal parts and the overlapping of the hexachordal system on an octave-based system. The second part of the Notitia is not related to Boethius, being concerned with the measure of time, and is organised into nine ‘conclusions’. Jean’s way of identifying time according to both imperfect and perfect measures would be fundamental to the stylistic revolution of the ars nova. He was taking care to explain how he could respect the teaching of the ancients while insisting that no one should ever say that they had ever fully defined the discipline: ‘For opinions and revolutions of knowledge run as if coming back in a circle.’Footnote 17 Jacobus referred to this very passage of the Notitia in his Speculum musicae, but added a reference to Jean’s source, ‘from Aristotle in the book of Meteorology’, in order to mock his argument, since he thought it absurd to compare new revolutions in musical knowledge to the passage of the seasons.Footnote 18 Jacobus was suspicious of how Jean des Murs seemed to disparage tradition in speaking of revolutions in this way.
The decision of Jean des Murs to write the Musica speculativa as a synthetic overview of the core teachings of Boethius can be seen as a response to criticism that he had not drawn sufficiently from Boethius in his Notitia. According to a rubric recorded in two manuscripts of the initial version of the Musica speculativa, des Murs composed it in June 1323, while teaching at the Sorbonne.Footnote 19 This version, edited by Witkowska-Zaremba from mostly fifteenth-century manuscripts conserved in Central Europe, but taken there by masters and students from Paris during the Great Schism (1378–1417), begins with a prologue (Etsi bestialium voluptatum), which argues from the authority of Aristotle in the Ethics that the senses, when used in moderation, are not bad in themselves, but can lead, in the case of music, to the greatest delight. The primary example is that of Ulysses, who, as Aristotle reports in the eighth book of the Politics, delighted in song.Footnote 20 This leads Jean to rectify what he sees as the prevailing neglect of Boethius:
Because the books of the ancient philosophers, not just about music but also [those] of the other mathematicians, are not read and because of this it happens that they are abhorred as unintelligible or much too difficult, it seemed to me a good thing that, from the Musica of Boethius which, according to the power given to me by God, I have studied closely and understood to some extent with God’s favour, I have been devising a short treatise in which I shall try to make manifest the more beautiful and essential conclusions pertaining to the very art of music with clarity of discussion and vividness of meaning.Footnote 21
In this initial version (A) Jean then begins the first part of the treatise with four distinctly non-Boethian principles which will guide his argument:
-
All teaching and every discipline arise from pre-existing knowledge.
-
No other [knowledge] is found prior to that from sensory knowledge.
-
States [of any knowledge] rest on manifold experience as its endpoint.
-
The experience of sensible things creates art.Footnote 22
Jean’s development in the Musica speculativa of these principles or axioms marks a significant evolution from the Notitia, in which he had only spoken in general terms about how musical art must rest on experience. They draw on Aristotle’s Posterior Analytics and Metaphysics but are formulated in condensed form. The sense of the third axiom (Experientiae multiplici ut in termino status acquiescere) is clarified by the closing assertion of the Notitia, that ‘no one should say we have attained the state of music and its immutable end’.Footnote 23 These principles, with their implicit argument that music must rest on experience, run directly against the opening claims of Boethius about musica mundana and humana as harmonies not based on audible sound, unlike musica instrumentalis. Significantly, Jean avoids all reference to the opening discussion of Boethius about different kinds of musica, the first two of which cannot be known by experience.
The Musica speculativa differs from the Notitia in not having any discussion of time and mensuration, but rather demonstrates profound familiarity with the argument of all five books of the De Musica. Instead, Jean expands on the core elements he had provided more briefly in the Notitia, structured around four propositions:
-
Pythagoras passed on to us the art of sounds
-
He attached the force of numbers for the sake of symphony
-
The three harmonies are perfect consonances
-
These three consonances [melodiae] offer clarifying numbers.Footnote 24
What Jean here calls three harmonies or melodies are the perfect consonances of the fourth, fifth and octave (diatessaron, diapente and diapason). His theme is that they first exist as sounds rather than as numbers, although Pythagoras was primarily concerned with defining the consonances in terms of numeric proportions.
Jean’s decision to modify the Musica speculativa by going back to the introduction of the Notitia has never been fully explained. According to a coded rubric in one important manuscript of the second (B) version of the treatise (Paris, BnF lat. 7378A; P3 in Falkenroth’s edition),Footnote 25 Jean des Murs revised this version in 1325.Footnote 26 This means that he was writing soon after Pope John XXII had issued his decree (of 1324/25) condemning the new style of music, ars nova.Footnote 27 In this revised preface, Jean goes back to the core opening arguments of the Notitia, omitted in Version A, to produce a more rounded justification for his manner of proceeding. In the later version, he did not make any allusion to contemporary neglect of Boethius, perhaps because the claim was no longer relevant. Instead he went back to his earlier argument that while theory is superior to practice, all knowledge rests on experience. He now included, however, the four central principles that he had introduced without explanation in Version A. Des Murs may have produced a transitional version prior to producing Version B, as found in Milan, Biblioteca Ambrosiana H. 165 inf., fols. 1r−16r.Footnote 28 It begins in the same way as the Notitia about Aristotle’s teaching on the priority of theory over practice (there also included in the prologue) and then follows with chapters about the relationship of music to sound introduced with a slightly revised statement: ‘Since musica (music theory) is adaptation (adequatio) of sound related to number, it is necessary to consider both, namely number and sound.’Footnote 29
THE CONCLUSIONS AND DIAGRAMS OF THE MUSICA SPECULATIVA
The bulk of the first part of the Musica speculativa is devoted to four propositions and eighteen conclusions concerned with the proportions of various consonances as presented by Boethius, demonstrating the competence of Jean des Murs as an accomplished mathematician. The second part is, according to Jean, more about practice, although it might be more accurate to say ‘the theory of practice’.
In addition to Paris, BnF lat. 7378A, a fourteenth-century manuscript (c. 1362), identified by Lawrence Gushee as particularly important in the manuscript tradition of Version B of Musica speculativa, the manuscript BnF lat. 7207 (c. 1430–60, of Italian origin) is remarkable for its clarity and accuracy, in particular of its diagrams. Footnote 30 Some manuscripts do not include the diagrams at all, e.g. Oxford, Bodleian Library, Bodley 77, fols. 95ra–99va; others contain mistakes in the figures in both senses of the word: numbers in the text and the diagrams themselves.Footnote 31 Des Murs frequently advises the reader to consult the diagrams, as will be seen below.
The approach of Jean des Murs in formulating propositions and conclusions was surely influenced by his study of Euclid, probably at the University of Paris. He had knowledge of not only the arithmetic but also the algebra of al-Khwārizmī,Footnote 32 and he would demonstrate his mathematical abilities more fully in subsequent writings on astronomy.Footnote 33 He constantly refers his readers to the diagrams.Footnote 34 The Conclusio sexta remark is particularly striking and he rightly says ‘everything is clear in the figures’, and the diagram explaining the sesquioctava proportion is perhaps the most complicated in the work (see Figure 1).Footnote 35 Since readers, medieval and contemporary, have often found these figures hard to understand, a few words of explanation may be helpful. Most of the diagrams comprise semicircles, often with their diameters. The diameters may be regarded as the string on a monochord and the semicircle is simply to give an idea of the extent. In his edition of Boethius Friedlein used brackets instead of semicircles (presumably because of typographical constraints), which may be clearer.Footnote 36 However the lengths of the semicircle diameters in Jean des Murs manuscripts bear no direct relation to the length of the string.Footnote 37 The redrawn diagrams in the present article do usually bear such a relation, as will be explained below.
Figure 1 Figura D from Paris BnF lat. 7207, fol. 295v, courtesy of the Bibliothèque nationale de France
The propositions and conclusions in Part I of the Musica speculativa can easily be related to corresponding elements in the De institutione musica of Boethius, but the same cannot be said of Part II; see Table 1 and below. In Proposition 4 Jean starts with the diagram with which Boethius ends De institutione arithmetica,Footnote 38 which is mentioned again in De institutione musica, I.10: ‘Haec ultima figura est in suprema calce Arithmeticae.’Footnote 39 Jean then proceeds to extract all he can from this figure; at the same time he makes his audience (or students) so familiar with this figure that the succeeding ones should be less forbidding and more obvious.
Table 1 Comparison of Jean des Murs, Musica speculativa and Boethius, De institutione musica
1 See also the last chapter of Jean’s Arithmetica speculativa in H. L. L. Busard, ‘Die “Arithmetica speculativa” des Johannes de Muris’, Scientiarium Historia, 13 (1971), 103–32, at p. 132.
2 Jean supports the Pythagoreans and rejects Ptolemy, while Boethius simply states both views (see the main text).
The first propositio (see above) is not purely mathematical in nature; along with the third propositio it relates the Pythagorean tradition of associating numeric proportions with musical consonances. In the second propositio he lists the basic consonances and their proportions: octave (diapason): double; fifth (diapente): sesquialtera; and fourth (diatessaron): sesquitertia, commenting that the remaining basic interval, the tone, is not itself a consonance but is in sesquioctava proportion. The third propositio expounds the connections between pitches (voces) and consonances. In the fourth propositio he explains the relation between the basic consonances (plus the tone) with a Boethian-style diagram, Figura A (see Figure 2).Footnote 40 He uses the description from propositio tertia in terms of pitches with associated intervals in this diagram but inserts the numbers 6, 8, 9, 12, which he had treated in propositio secunda. In this way he guides the reader from the familiar area of the musical scale to the numeric proportions. The first five Conclusiones all use these numbers, and the accompanying figure, for their elucidation.
Figure 2 Figura A: (a) from Paris, BnF lat. 7207, fol. 294v, courtesy of the Bibliothèque nationale de France; (b) redrawn with modified lengths. The orientation varied in medieval texts; see n. 42 below
It should first be pointed out that there is a significant difference between proportions and ratios.Footnote 41 Medieval music theorists rarely use the word ratio, except in the sense of reason, favouring proportio for the numerical aspects. A ratio is a relation between two numbers, for example 3 : 2. On the other hand, a ratio represents a proportion; thus the sesquialtera proportion is represented by 3 : 2 but also by 6 : 4, 18 : 12, etc. Further, medieval writers did not necessarily specify the order of the terms in a proportion, so, unlike modern mathematical language, Jean does not need to specify the relative positions of the items he is comparing.Footnote 42 Perhaps the easiest way to express this is that a sesquialtera proportion holds between two items (numbers, strings, pitches, for example) when the larger is 1½ times the smaller. Here it seems appropriate to add a further note on the language. First, Jean names all the proportions that he uses, and this means he does not need to write down specific numeric ratios. We shall try to adhere to his style; hence terms involving ‘sesqui’ will be common, where ‘sesqui-nth’ (where nth means nth part) means the proportion of (1 plus 1/n) : 1 (or viceversa).Footnote 43 Thus he will say, for example, that two lengths ‘are in sesquialtera proportion’ and this simply means that the ratio of the larger to the smaller is 3 : 2.
Just before the first Conclusio Jean says: ‘now is the time to extract the mysteries of this figure and the marvels contained, one by one’;Footnote 44 thus it is appropriate to consider how the figures are constructed. As noted above, they comprise semicircular arcs and straight lines, usually accompanied by words describing the proportion involved. Jean admits the figures look complicated: ‘But this figure [Figure 2] can quite reasonably be said to be like a bow’, which it is indeed: it has an overarching curved bow. He then goes on to say that many forms lie hidden in it.Footnote 45
Here is one way to understand the construction in which we use the numbers that come from the last figure in the De institutione arithmetica of Boethius for ease of navigation.Footnote 46 The proportions between 6, 8, 9 and 12 are, successively, sesquitertia, sesquioctava and sesquitertia, corresponding to a fourth, a tone and another fourth.Footnote 47 So, to construct the figure, start with the interval of a fourth on the left. Next to it draw a semicircular arc for the tone and then another arc for the second fourth (from numbers 9 to 12). Returning to the left draw the next arc higher in the figure, that for the fifth (going from the number 6 to the number 9). Now go back to the left the distance of a tone and draw an arc from there (at the number 8) to the far right (number 12). This is another fifth. Finally draw the largest arc, corresponding to the octave (diapason).
In manuscripts containing the Boethian-style figures the arcs for the fourths and the tone are usually all of the same size. However, in our redrawn version of Figura A in Figure 2 (and subsequent figures except where noted otherwise) the lengths have been modified so that putting the length corresponding to, say, the tone, next to the length corresponding to the fourth gives exactly the same length as the fifth and likewise a fourth plus a fifth gives an octave.
Now we proceed to consider some of the conclusions in order. In his first Conclusio, that the proportion for the octave is greater than that for the fifth, des Murs deftly introduces superparticular proportions in the following way, exemplifying as well as defining: a superparticular proportion is less than a multiple and contains one whole (but never two wholes) plus one part,Footnote 48 for example, one and a half (sesquialtera), one and a third (sesquitertia), etc.
The first five conclusions are all evident from Figura A (see Figure 2 above) and the numbers previously singled out by Boethius: 6, 8, 9 and 12.Footnote 49 For this figure he chooses to start with 9, which is in sesquialtera proportion to 6, then add a third part of 9 (namely, 3), giving 12 (forming a sesquitertia proportion), which is the double of 6, ‘as shown in the figure’.
The sleight of hand is in the choice of starting with 6. Numbers in Boethian texts were usually only whole numbers. To get a sesquitertian proportion the starting number needs to be divisible by 3. However, if we start with 3 and then go, via a sesquitertian proportion, to 4, we cannot go to a whole number by a sesquioctava proportion; instead we would get 4 × 9/8 = 4½. So go back to the beginning and start with an even number divisible by 3. The smallest is 6. One can then proceed along the sequence 6, 8, 9, 12. Boethius developed this technique, though he did not explicitly explain it, and Jacobus revived it;Footnote 50 Jean des Murs explained the technique, as will be seen shortly.
The technique appears to be related to, indeed may have led to, Euclid VIII.4, which is a proposition on how to combine sequences of ratios into a continuing sequence.Footnote 51 In this case the sequence of ratios Muris considers is 1 : 2 (octave), fifth (2 : 3), fourth (3 : 4) and tone (8 : 9).Footnote 52 So he starts with 6, which is the least number divisible by 2 and 3 (the right-hand numbers of the first two ratios), giving 6 : 9 :12, and the first and last numbers give the ratio for the octave, the first and second for the fifth (diapente). Here the fourth (diatessaron) is already represented by 9 : 12. Finally, the tone is incorporated by including 8 to give 6 : 8 : 9 : 12. (In fact, the fifth is represented twice, by both 6 : 9 and 8 : 12 and likewise the fourth by 6 : 8 and 9 : 12.) One virtue of these four numbers is that all possible ratios between them correspond to the octave, fifth, fourth or tone.
Conclusio sexta is of a different nature; it comprises a long attempt (following Boethius, De musica, III.1) to prove that the tone cannot be divided into two equal parts; we shall deal with this after Conclusio quinta decima.
When he comes to Conclusio septima, which says that a fourth is two tones plus a semitone, des Murs reveals his hand and gives the reason for starting with the particular numbers he chooses and the key to the general method. He is not only more explicit than Boethius, but he invites the reader, who is addressed in the second person, to join in. First, however, he simply presents the numbers – those in the top line of the table in Figura E (see Figure 3).Footnote 53 Then he says that these are the smallest whole numbers (primi numeri) by which the ratios can be represented.Footnote 54 He continues: ‘And such is the art of finding such consonances.’Footnote 55 Then he explains, and we paraphrase:
When you want a number of tones or sesquioctava proportions, start from the first octuple, which is 8 and advance one to 9. If you want two, start from the second octuple (i.e. octuple of octuple = 8 × 8 =) 64; if three, start from the third (8 × 8 × 8 =) 192. So, from 64, one tone takes you to 64 × 9/8 = 72 and another to 72 × 9/8 = 81.
Figure 3 Figura E redrawn
The general principle is that if you want to take a sesqui-‘something’ proportion then you multiply by the ‘something’ so that the division gives a whole number: thus for sesquioctava: multiply by 8; for sesquitertia, multiply by 3. He then interpolates a remark that if you go up by a sesquitertia proportion from 64 you will get a fourth (diatessaron), but 64 does not have a (whole number) third part. So multiply everything by 3 and consider 64 × 3 =192, which does have a third part, and this is shown in the first four numbers in the top line of Figure 3 (Figura E).Footnote 56 He does not explain the remaining two lines of numbers, but the second line is simply obtained by multiplying the top line by 8/3.Footnote 57 The third line is obtained by multiplying the third line by 3 so that all the numbers are whole. Conclusio octava, which says that a fifth is three tones plus a semitone, is then obvious from the preceding discussion. Putting together parts of the last two arguments then yields Conclusio nona that two semitones and five tones comprise an octave.Footnote 58
Conclusio decima simply points out that since a semitone is not half a tone the difference between a tone and a semitone is a different kind of semitone. The former is called a minor semitone or diesis (256 : 243) and the latter a major semitone or apotome (2187 : 2048),Footnote 59 but the calculations only come in Conclusiones un- and duo-decima respectively. The first calculation is clear from the table in Figura F (see Figure 4). For the major semitone one wants to add sufficient to form a tone from a minor semitone. So he starts with 243, but this has no (whole number) eighth part. Applying the method of Conclusio septima, he multiplies by 8, getting 1944. Then, multiplying 256 by 8 yields 2048, giving 2048 : 1944 for the minor semitone. For the tone this means taking a sesquioctava proportion from 1944, which is 1944 × 9/8 = 2187, giving 2187 : 2048 for the major semitone, and he then adds these intervals, as is shown in Figure 4.Footnote 60
Figure 4 Figura F redrawn
The thirteenth conclusion shows that six tones are greater than an octave: the difference is a comma, whose proportion is 531441 : 524288. The technique is again to multiply by 8 the appropriate number of times so that one can take sesquioctava proportions. Jean’s treatment is quite succinct.Footnote 61
In establishing Conclusio quarta decima, that an interval of five tones is greater than an interval of two fourths, although the respective numbers may be found in Figura H they are not mentioned – nor needed – in the text.Footnote 62 The explanation is a simple use of the previous result, but following the main conclusion he revisits the result, this time starting from five tones, again using a figure and saying: ‘For this see the figure below.’Footnote 63 There he shows a fourth ascending from the lowest number (diatessaron intensa) and another fourth descending from the highest (diatessaron remissa) and these clearly do not fill up the five tones. He only mentions at the end that the numeric differences may be found in Figura I (see Figure 5): ‘as can be seen in this figure’.Footnote 64
Figure 5 Figura I redrawn
This is the first time he uses the terms intensa and remissa, but they will be more frequent in Part II.Footnote 65 The inclusion of the differentia is somewhat problematic. It would appear that des Murs simply means the difference between the two components: the five tones and two fourths. However, he associates a number 5961/12 with this difference. One possible interpretation is as follows. The differentia is simply the difference between the two ends of the diatessara. It has no other significance. If it were moved to left or right the difference in pitches that it indicates would change. Footnote 66 If five tones were equal to two diatessera then the two arcs would meet and the differentia would be zero.Footnote 67
Conclusio quinta decima shows the comma is ubiquitous: it is the difference between a tone and two minor semitones and the difference between an octave, which comprises five tones and two (minor) semitones, and six tones.
Des Murs then has a series of thirteen corollaries, which he claims are evident, which is true of (1) to (5). (6) A true semitone, viz. a genuine half of a tone, does not have a proportion, that is to say, it cannot be expressed by a ratio of whole numbers. From his Pythagorean point of view he therefore infers (7) that a true semitone does not exist in the nature of things.Footnote 68 Corollary (8) seems to have been corrupted in many manuscripts, and should probably read ‘The comma is not in a multiple or superparticular proportion’, which is clear since it is in the proportion 1 and 7153/524288 : 1, roughly 1 7/500: 1, which is a superpartient proportion.Footnote 69 Version A indicates that ‘the comma does not exist in number but is greater than 75 and less than 74’. This seems to be a corruption of Boethius, III.12, whose chapter heading says: ‘In what proportion of numbers the comma may be, and that it is greater than 75 to 74 and less than 74 to 73.’Footnote 70 (9) The minor semitone is greater than 20 : 19 and less than 19 : 18. This is curious since Boethius III.13, says it is greater than 20 : 19 and less than 19½ : 18½, which is indeed less than 19 : 18. However, this comes from the sixth conclusion (see Table 3 in Appendix 2). Corollary (10) incorrectly states that the major semitone is greater than 15 : 14 and less than 14 : 13; in fact both proportions are greater than a major semitone.Footnote 71 Corollaries (11) to (13) concern the comma and all require complicated calculations and are perhaps later additions, though we expect Jean des Murs would have been capable of them given his mathematical expertise.
Now we return to Jean’s discussion of the division of the tone into two equal parts and the (misleading) role of the differentia. In his Conclusio sexta he starts from Boethius I.17, at the end of which he will again say that ‘everything is clear in the figures’.Footnote 72 The tone is in the proportion of 9 : 8 but there is no (whole) number between 8 and 9, so he considers another representation of the same proportion, namely 18 : 16. Like Boethius he points out, as is clear, that 17 is 1/16th more than 16 but 18 is 1/17th more than 17, so the proportions of 18 : 17 and 17 : 16 are different. For this he presents one of his most complicated figures, which is really two figures: one starting with 16 and looking at a sesqui-16th and the other from 17 with a sesqui-17th proportion. For the latter, where he takes successive sesqui-17th proportions he does not resile from working with fractions, getting successively 17, 18, 191/17 and then 191/8, the last figure being obtained by taking a tone from 17, giving 91/8 × 17. He thus constructs Table 2.
Table 2 The lower right of Figura D reconstructed from Paris BnF lat. 7207, fol. 295v
1 Missing in Witkowska’s edition (p. 181).
2 This row is missing in Paris BnF lat. 7207 (Figure 1).
The lower rows of this table are obtained as follows. First, multiply the initial row by 8 to eliminate the fraction in 191/8; perhaps the third entry is omitted because it involves the awkward fraction 81/7. The row below is obtained from the first row by multiplying by 17, which then gives a whole number, 324, in the third place but leaves a fraction of an eighth in the final one. So finally multiply by 8 to resolve this. Since these numbers are not included in the main text one wonders if they were interpolated later.
The problem with this approach to trying to show that the tone cannot be equally divided is that one has to show that all representations of the sesquioctava proportion, 9 : 8, 18 : 16, 27 : 24, etc., fail to provide an exact halving. Boethius was misled into looking at the additive difference between the numbers in a ratio while compounding of ratios actually requires multiplication. This is similar to the awkwardness noted above at the end of the fourteenth conclusion with the differentia (marked as 596½).
The sixteenth and seventeenth conclusions give remarkably simple mathematical results: a double octave corresponds to a quadruple proportion and an octave plus a fifth is in the proportion 3 : 1.
This concludes his abridgement of the mathematical calculations of Boethius, which Bower described as ‘relentlessly technical’.Footnote 73 The final, eighteenth, conclusion deals with a contentious item in Boethius V. 7–10, namely whether an octave plus a fourth is a consonance. Jean des Murs espouses the Pythagorean view that a fourth plus an octave cannot be a consonance since 2 : 1 and 4 : 3 give the proportion of 8 : 3, a superpartient proportion, and therefore is neither a multiple nor a superparticular proportion since it comprises 2 and two parts (specifically two-thirds) rather than just one extra part. He then puts forward Ptolemy’s view that it is a consonance and claims to refute it.Footnote 74 He claims that this is similar to whether the fourth below a fifth is a consonance.Footnote 75 This, he says, clearly is a consonance (e.g. C–F–C), although a fourth above a fifth (e.g. C–G–C) is better and is sweet-sounding.Footnote 76 He then says this revolves around whether the fourth precedes the fifth in the way that a sesquialtera proportion ‘precedes’ (priorem esse) a sesquitertian one, finally appealing to his Figura K, which reprises Figura A (see Figure 2 above),Footnote 77 but now with the number 3 also included to give a double octave, perhaps imitating the way Boethius ends his De institutione arithmetica (not the De institutione musica). The numbers are 3, 6, 8, 9 and 12, which give rise to all the other intervals (consonances) he has just explained. His argument revolves around the point that in the hierarchy of consonances the fourth only arises out of the octave and the fifth.Footnote 78
Part II shifts from consideration of music as based on harmonic proportions to the actual divisions of the monochord, ‘in which all consonances and their parts and parts of their parts are laid out’. Jean argues that this will provide ‘a foundation for constructing a variety of instruments, knowledge of ones yet unknown, and discovery of ones that are new’.Footnote 79 This was radically different from Boethius, whose focus was firmly on the past and was little concerned about the sounding music of instruments.
It is often difficult to see such direct connections with the De institutione musica of Boethius as one can in Part I. Nevertheless, many of the constructions can be found in IV.5 of Boethius, but there they are involved in the partition of the monochord according to the diatonic genus. What Jean does is to show the general method, whereas Boethius constructs notes on the musical scale successively. As Matthieu Husson puts it, Jean’s ‘contraction produced a systematisation of the content of the original [Boethius]’.Footnote 80
Propositio prima sets up the basic consonances. Jean starts with a string (chorda) ab, which represents an octave (see Figure 6).Footnote 81 Note that the letters are as used in geometry, as for example by Jordanus Nemorarius in his Arithmetica, and do not denote pitches.Footnote 82 Taking another string that is half the length, up to c, but still labelling the other end a, means that ab is twice the length of ac and so ab sounds an octave lower than ac when plucked. He then takes a third string which he divides into three equal divisions by inserting points d and e, giving a sequence a, d, e, b.Footnote 83
Figure 6 Redrawn from the lower part of Figura L
Since the lengths of ab and ae are in the ratio of 3 : 2, the plucked lengths resound a fifth. Finally dividing ab into four equal parts, where the points of division are f, c and g, one can get lengths in the ratio of 4 : 3, yielding a fourth. Finally, since a tone is the difference between a fourth and a fifth, plucking the string lengths ae and ag resounds a tone. Des Murs also includes a bow diagram showing all the fourths and fifths and differences of a tone between them, so the figure becomes very complicated.Footnote 84 With the four strings corresponding to the lengths, he links them with curves to indicate the various intervals. Then, striking the strings will generate the basic consonances ‘so that you [singular] can judge the consonances, which you did not know of before, by your ears’.Footnote 85
Jean recalls that the four strings, taken together, reconstruct the Tetrachord of Mercury, as recorded in Boethius I.20, who, referring to Nicomachus, says that in the beginning there was ‘simple music’ performed on four strings. These provided only consonances: the outer two resounding an octave and the middle two a fourth or fifth up or down from the ends.Footnote 86 This of course corresponds to Figura A (Figure 2 above). Des Murs recommends listening to the strings being struck many times so that ‘from the information of the intellect and the senses, you will marvel at the appearance of the consonances and judge the natural consonances as marvellous’.Footnote 87
Propositions 2–4 show how to use fourths and fifths, and later tones, to create ascending or descending intervals. Note that the diagrams have the higher notes to the left, the opposite way to present practice. The second proposition concerns creating a tone at the top or bottom of an interval on the monochord (see Figure 7). The process obviously follows Boethius III.9.Footnote 88 To obtain a high (acutus) tone one starts with a note b and goes up a fifth to c (i.e. to the left), and then goes down a fourth to d. The interval between b and d then yields the desired high tone.Footnote 89
Figure 7 Redrawn from the left part of Figura M
The third proposition is to construct a high minor semitone (Figura N). The principle is the same: one adds or subtracts the appropriate intervals. In this construction he follows Boethius III.10 to the letter.Footnote 90 After this the construction of a low (gravis) minor semitone follows, but now in a much-abbreviated fashion. This is again following Boethius III.9 exactly; even the letters are the same.Footnote 91 It seems odd that des Murs should follow Boethius so closely rather than simply use the second technique for both high and low minor semitones.
Next comes the construction of a low minor semitone (which is much simpler) followed by the construction of major semitones, both high and low, and then the same for the comma, this time using the result of the fifteenth conclusion in the first part of the work. This latest (propositio quarta), because of the thoroughness of Part I, is easily accomplished by subtracting a minor semitone from a major one.Footnote 92
After this he omits the discussion of the impossibility of the equal division of the tone (or indeed of any superparticular proportion) into two that Boethius includes in his III.11: Jean has already dealt with this in Part I. Nor does he revisit questions such as the relative sizes of the semitone versus three commas (cf. Boethius III.14 and 15). Instead he proceeds at once to the discussion of the monochord (cf. Boethius, Book IV) and how it embraces all harmonic genera, in order to satisfy the wishes of his readers, whom he now addresses as ‘you’ [plural] with his fifth proposition: ‘Having done these preliminaries you may wish to know about the monochord.’Footnote 93 In Part I he had demonstrated the theory of consonances (speculatione consonantiarum) and their reduction to recognisable figures (sensibiles figuras) ‘that greatly please mathematicians, since by them truth, which is in the intellect, is appropriately reduced to sight and sound’.Footnote 94 Having broken down consonances into tones and major and minor semitones, they can now be represented by intervals on the monochord,Footnote 95 but, before he gives the details, he asks the reader to note three preliminaries.
First, every division of the monochord, which implicitly and virtually contains every kind of instrument, comes from the tetrachord.Footnote 96 The tetrachord is a group of four pitches (strings) bounded by the fourth (diatessaron); in the diatonic genus this is filled in with two tones and a minor semitone.Footnote 97 In proceeding to break down consonances into smaller components he considers a double octave, which reduces to two octaves; a single octave, which reduces to a fifth and a fourth; but then says that the fifth presupposes the fourth while the fourth presupposes no other consonances, just the tone and semitone.Footnote 98 This seems to suggest that the fourth has primacy, which would contradict his argument in Conclusio octava decima that ‘a fourth does not exist as a proportion in itself, but only arises as a consonance out of an octave and a fifth’.Footnote 99
The second preliminary concerns putting together fourths: if they are joined together (i.e. conjunct, as in G–C then C–F) they do not make an octave but if juxtaposed (i.e. are disjunct, as in G–C then D–g) then they do make an octave.Footnote 100
This is a prelude to the third point, which is that, although a fourth contains two tones and a semitone, the ancients arranged the order of these three components in different ways and he presents a diagram (Figura R) showing the three ways. These different ways arose from the three genera: diatonic, chromatic and enharmonic. Des Murs, however, is surprised that, of the three ancient modes, only the first was widely used ‘throughout the world by the faithful’.Footnote 101 Despite him not knowing of these modes being used anywhere in the world, they can still be represented on the monochord, but he does not consider these further, thereby allowing himself to omit any account of the ancient Greek genera as treated in Boethius IV.3–17.
Propositio sexta signals his agreement with Boethius on how the divisions of the monochord should be made, and he restricts himself, at this point, to the double octave. He then includes a figure for the double octave that Boethius considers, complete with the relevant numbers and Greek note names (see Figure 8).Footnote 102 This is the only place where he uses the Greek names that feature so strongly in De musica, I.24 and IV.5–11.
Figure 8 Redrawn from the unlabelled figure <Bis diapason. Monochordum Boetii>. Notice that the ends of the arcs coincide with the appropriate notes
He concludes Musica speculativa by talking about his own version of what he calls the monochord.Footnote 103 After explaining how his instrument is similar to that of Boethius, he describes how he has changed the placing and number of the strings in his monochord, ‘because the way of singing in vogue in the time of Boethius has now changed’ and ‘a more refined form of music is now delighting not just the learned, but the common crowd, particularly the young, and women also’.Footnote 104 But he then goes on to say that he does not know how this has happened unless it is by ‘natural industry regulated by some higher circle’ but things are continually changing and perhaps at some time they will come full circle and things will be as they were before.Footnote 105
He then turns to the monochord and shows how to determine the lengths of the strings relative to each other for all the consonances and for the tone and semitone, with these last two extending the treatment of Boethius IV.18. This section seems largely to reprise propositio prima.
Jean explains that his own instrument, with nineteen strings, covering two octaves and a fifth (although he says it could be more), is in the shape of a triangle, in which the third side is not strictly a line, but more like the circumference of a circle, which in Euclid IV.5 is defined by three points (though here the curve is determined by nineteen points).Footnote 106 In manuscripts with diagrams there then follows an image (see Figure 9) of the strings on the monochordal instrument (presented in the edition of Witkowska-Zaremba, p. 204, from a manuscript of Version A, MS Prague, National Library V.F.6, copied in 1431; Pr3 in her edition).Footnote 107 In Paris, BnF lat. 7378A, the text (Version B) concludes with a diagram of this roughly triangular monochordal instrument on fol. 45v.Footnote 108 Whereas Boethius had confined himself to abstract proportions and a single string, Jean explains how a monochordal instrument can take many shapes, and does not even have to be flat, yet virtually contains within itself all other instruments (a variation of what Grocheio had said about the primacy of stringed instruments, namely the psaltery, the cithara, the lyre, the Saracen guitar and the vielle, which he claimed contains other instruments virtually in itself).Footnote 109 He concludes by declaring ‘I want to describe to you the different shapes in which this monochord can be drawn by means of an example. Their shapes follow in this order.’Footnote 110 The final sentence of the text (about different shapes of instruments) seems to refer to other images found on folio 45v of Paris, BnF lat. 7378A, but missing from all other manuscripts, namely images of a vielle, a lira, psalterium, canon, guiterna and chitara – very close to the instruments mentioned by Grocheio.Footnote 111 It may be that they were too difficult for most subsequent scribes to reproduce. They demonstrate how Jean des Murs wanted to show that an actual instrument of the monochord could be reconfigured in many different ways, and how the core mathematical insights of Boethius could apply to a range of new instruments that he never imagined.
Figure 9 The strings of the monochord-style instrument of Jean des Murs drawn to scale (it resembles a psaltery rather than a conventional monochord)
Throughout his text Jean des Murs relies heavily on diagrams but also reveals the methods that Boethius (and his predecessor Nicomachus) used. His diagrams put before the eyes of his readers a clear vision of the way musical intervals work together. With the exception of the discussion of the equal division of the tone, des Murs progresses systematically through the standard consonances, putting in just enough explanation to be clear while including nothing superfluous; he has made an epitome in both senses of the word. He makes the essence of Boethius as clear as it can be, without ever committing himself to the assumptions about musica as cosmic, human and instrumental that open that treatise. At the same time, he shows much more awareness of the way musical harmonies were always the product of experience, and how new musical instruments reconfigure the core principles of the monochord.
CONCLUSION
The Musica speculativa of Jean des Murs, initially completed in Paris in June 1323, marked a major step forward from the Notitia artis musicae, the treatise by which Jean first established his reputation as a music theorist, but which drew on Boethian ideas in only a limited way. In the thirteenth century, Boethius had tended to be referred to only briefly, mainly for what he said about the three different types of musica (cosmic, human and instrumental), conceptions that were challenged by the increasing awareness that Aristotle placed on the experience of music as sound. While Jean des Murs had started to present his ideas about the theory of music in the Notitia as resting on experience, he had drawn mainly on the first two books of the De institutione musica of Boethius. By contrast, in the Musica speculativa he condenses the arguments of the entire work, utilising a series of diagrams. He also replaces the brief discussion of types of musica by Boethius with his own reflections, shaped by Aristotle, on how all knowledge rests on experience. In the revised version of the Prologue (1325), he refines core arguments about the relationship between sound and number that he had raised in the Notitia artis musicae. He shows his readers, increasingly steeped in an Aristotelian perspective, that there is still much of value in the detail of what Boethius had to say about proportions.
In his Speculum musicae, a treatise that asserts a much more strongly Platonic version of Boethius than does Jean, Jacobus made specific hostile allusions to both the Notitia artis musicae and the Musica speculativa as reflecting what he considered perspectives not faithful to what he thought to be the original intention of Boethius. Jacobus refers to a ‘quidam doctor modernus’ who is clearly Jean, since part of his Conclusio octava decima in the Musica speculativa is quoted almost verbatim.Footnote 112 Around 1300 it appears it was common not to name contemporaries who were being criticised, which suggests that Jean and Jacobus were contemporaries. It also seems plausible that it was the criticisms of Jacobus himself (before his writing of at least some of Book VII of the Speculum musicae) that forced Jean des Murs to take full account of what Boethius had to say. The Speculum musicae offered the response of Jacobus, not just to new tendencies in musical notation, but to ways of reading Boethius. Both men were fascinated by the need to come to grips with what Boethius was arguing in the De musica. Yet it was des Murs who better succeeded in showing to future generations how core ideas transmitted by Boethius still had value, even within an Aristotelian perspective of how all knowledge ultimately rested on experience. For Jean des Murs, this was the true significance of Pythagoras in his discoveries about music.
APPENDIX 1 Translation of the Prologue of Version A (1323)
Based on the Latin text as edited by Witkowska-Zaremba, pp. 171–2 and Meyer, pp. 134–6.
Although the excesses of bestial pleasures, through which taste and touch overthrow the intellect with their unrestrained assaults, are not undeservedly censured (according to the first book of Aristotle’s Ethics:Footnote 113 ‘indeed many bestial people therefore choosing the life of sheep…’), the ordered and moderated delights of sight and hearing, which serve the intellect in due measure in clearer and fuller ways, are not condemned because of this (as Aristotle says in the fourth book of the Ethics concerning the temperate man:Footnote 114 ‘but he moderately pursues whatever there is for health and good habits, as he ought’); further, sight is much praised over hearing, in that it makes us come to know things as much as possible and shows us the many differences between things.Footnote 115 Likewise, as witnessed in experience, voices and sounds completely created by the subtlety of human artifices are a means to lead to the sweetest delights of the intellect. And, after the labours of serious work, which cannot be endured by human nature unceasingly, it grants the benefit of well-deserved rest to the listener. And perhaps also Ulysses wished to indicate that poetically (according to Aristotle’s account in the eighth book of the Politics Footnote 116 ), when he said that it is the noblest demonstration when those gathered together with men rejoicing hear a nightingale on the roof.
In addition, Boethius elegantly teaches in the prologue of his De institutione musica that ‘harmonic sounds may hold much efficacy over human appetites in directing them to different kinds of habits’.Footnote 117 Inspecting them will be of interest for those who turn themselves [to it]. In the same place one will even be able to see how much men of the polis ought to care about the preservation of well-moderated music.Footnote 118 Indeed, since in these days the books of the ancient philosophers, not just about music, but also [those] of the rest of the mathematicians, are not read, and because of this it happens those are abhorred as unintelligible or much too difficult, it seemed to me a good thing that, from the De institutione musica of Boethius, which, according to the power given to me by God, I have studied closely and understood to some extent with God’s favour, I have been devising a short treatise in which I shall try to make manifest the more beautiful and essential conclusions pertaining to the very art of music with clarity of discussion and vividness of meaning.
APPENDIX 2 Translation of the Prologue of Version B (1325)
Based on the Latin text of Version B, ed. Falkenroth, available at http://www.chmtl.indiana.edu/tml/14th/MURMSPE.
Since music is about sound related to numbers or viceversa it is necessary that music consider both, namely number and sound. For before anything is numbered it is necessary for it to exist.
Therefore, it is necessary for sound to be generated before it is numbered.
Three things are necessarily required for the generation of sound: a beater, the thing beaten and the medium of beating. The first is rapidly breaking up the air, the second is a naturally resonant body, the third is air violently broken up. There cannot be a beat without motion, therefore neither can there be a sound without motion. Therefore, a sound exists by the breaking up of the air from the impact of the one beating on the thing beaten. For it is impossible for there to be sound when there is only one of these.
Of sounds, one is low, another high. The low is what is generated from slower and more sparse motions, the high indeed is what is generated from faster, more concentrated motions. Although there may be several separate movements in both, making several sounds in number (as is evident in the beating of gut strings). Yet they are heard without audible interruption as if it were like one continuous sound because of the speed of contiguous movements, just as colour existing on the surface of the tip of the moving crocus seems to be a circular line although it is only a point.
A high sound indeed comes from a low sound through the addition of motions; likewise, a low sound comes from a high sound by subtraction. Therefore, in high sounds there are more motions, in low ones fewer.
But all paucity is related to plurality by a certain number. Therefore, it is necessary for a low sound to be related by comparison to a high one, just as the number of motions of the one is compared proportionately to the number of motions of the other.
Since it has been shown that music exists from sounds that are proportional to each other in a certain way according to the number of motions found in them, it is not pointless to enquire generally into the proportions of numbers.
But every number compared to another is either equal or unequal to it for they correspond to a certain quantity.Footnote 119 If equal, they are one and the same in quantity. Therefore, they are not varied. A consonance is not made from equal sounds; they are either greater or lesser. If, however, unequal, they are either greater or lesser. But one being greater than the other is described in five ways, as multiple, superparticular, superpartient, multiple superparticular and multiple superpartient. It is said to be less in just as many ways by the assigned phrase.Footnote 120
A multiple is when the greater number contains the whole of the lesser in itself, twice, thrice or fourfold and whose kinds are the proportions double, triple and so on to infinity.
A superparticular [proportion] is when the greater contains the lesser in itself and some part of it; its kinds are sesquialtera, sesquitertia, and so on.
A superpartient is when the greater contains the lesser and some parts of it; its kinds are superbipartient, supertripartient and so always.
A multiple-superparticular is when the greater contains the lesser twice or three times [and so one] and some part of it; its kinds are double sesquialtera, double sesquitertia, triple sesquialtera and thus of whichever part to infinity.
A multiple-superpartient is when the greater number contains the lesser twice or three times [and so on] and whatever of its parts; its kinds are double superbipartient, double supertripartient and in adding whatever kind of multiple to whatever of the superpartient.
Examples of the various kinds [of proportion] are shown in numbers. [Jean places his version of Table 3 here.]
Table 3 Des Murs on proportions
Note: We have moved the last row to the top, since des Murs starts with the numbers in this row. Then he gives two examples of each kind of proportion. We have also addedan explicatory column showing the factor by which the number at the very top should be multiplied in order to get a number in the appropriate proportion. For example, to get a number in double sesquialtera proportion to 12 one multiplies by 2½, yielding 24 + 6 = 30; similarly for the other numbers in this row: 24, 36, 48, yielding 20, 60, 90 respectively. The reason des Murs starts with 12 is that he wishes to get whole numbers in the appropriate proportion and, since he uses ⅔ and ¾ he needs to have numbers that are divisible by 3 and 4, and clearly 12 is the smallest such number. The other numbers are then simple multiples of 12 going up to 12 × 4.
That all teaching and every discipline arise from pre-existing knowledge.
That no other [knowledge] is found prior to sensory knowledge.
That it [sensory knowledge] rests on manifold experience as its endpoint.
That experience of sensible things creates art.
Monash University
Open access
