In 1343, Levi ben Gershom (1288-1343) penned the brief treatise De numeris harmonicis at the request of Philippe de Vitry. Levi, known to his Latin speaking readers as Gersonides, among other titles, was a Jewish philosopher, mathematician and scientist and lived his entire life in Provence, mostly in Orange, though he spent time in Avignon, then the seat of the Papacy. The original Hebrew does not survive, and we have come to know the treatise through two Latin translations—Basel, Universitätsbibliothek and Paris, Bibliothèque Nationale de France.[1] The treatise is in the form of a simple proof of a hypothesis presented to Levi by de Vitry, a matter of “scientia musicali.” De Vitry defined “harmonic numbers” for the purposes of the treatise. Levi states the premise thus:

Any of the harmonic numbers may be distinguished by 2 except between 1 and 2, 2, and 3, 3 and 4 and 8 and 9. He [Vitry] defines harmonic numbers as follows: a harmonic number is divisible (except for 1) either by 2 or 3 in succession or alternately one with the other continuously down to one. They are, therefore, in succession: 1, 2, 4, 8 … and 1, 3, 9, 27 … and alternately 6, 12, 18, 24 …[2]

Aside from this neutral and rather detached exposition, Levi does not ascribe any significance to the harmonic numbers as a set of numbers, and, indeed, no trace of them is to be found in the document Ars nova, normally attributed to de Vitry.[3] Accordingly, historians have offered various interpretations of the meaning of this treatise. Hugo Riemann, writing at the end of the nineteenth century, for instance, supposed that the treatise bore a relationship to interval theory.[4] Plausible enough, this thesis was nevertheless supplanted in the 1950’s when Eric Werner advanced his own hypothesis that the significance of the treaty lay in a mathematical justification of de Vitry’s Ars nova metrical system.[5] From the 1970’s on, Wolf Frobenius, drawing on other appearances of the term “harmonic numbers,” particularly as it arose in the work of the Dutch theorist Johannes Boen, has held that the treatise deals with de Vitry’s interest in the Pythagorean tuning system.[6] The latter two discussions will be considered in greater detail in the second portion of the paper. For the first portion, we shall turn to the intellectual context inhabited by Levi.

It has already been mentioned that Levi attained status as a philosopher, mathematician and scientist in fourteenth-century Provence. Just what it meant to be Jewish philosopher at this time, it ends out, is an interesting and complicated issue in and of itself, a factor that casts Levi’s work as a mathematician and scientist in a peculiar light, given the prevailing ideologies of his day.

Levi stood at the end of a long period of transmission of Arabic and Greek learning that began on the Iberian peninsula and, later migrated to the south of France. Greek learning was often referred to as “alien.”[7] In a view not atypical of many Jewish scholars, who, confronted with cutting-edge philosophy for the first time, were disturbed by conflict between philosophy as such and received Jewish teaching, R. Asher ben Yehiel, writing at the beginning of the fourteenth century, expressed the following:

The science of philosophy and the science of the Torah are not the same. The Torah was donated to Moses at Mount Sinai … Philosophy being a natural science, it was inevitable that the philosophers deny the Torah, for the Torah is not a natural science, but rather [a science] received and transmitted … [The Torah and philosophy] are indeed contraries, two rival wives who cannot be at one and the same place.[8]

Why was this so? Greek and Arabic learning first arrived in the Jewish world en masse with the work the twelfth-century scholar Abraham bar Hiyya of Barcelona. His writings were undertaken at the request of high-ranking Provençal Jews who were desirous of an education in these matters. Bar Hiyya’s began the transmission of Arabic sources but specifically for practical ends, such as geometry for the purpose of dividing land and astronomy for the calculation of the calendar. Abraham Ibn Ezra wrote at approximately the same time and expanded the palette of available materials, writing on subjects such as astronomy and astrology. His mystical side was instrumental in fostering the idea that the study of philosophy was not necessarily contradictory to Torah and that the study of the former might even enhance the study of the latter. The late 1140’s witnessed a significant exodus of Jewish intellectuals from Spain to Provence brought on by the Almohad persecutions. The emigrants brought with them a familiarity with Arabic poetry, philosophy and science that was hitherto quite unknown to Jews in living in the south of France. Thus began an enormous translation effort, fusing traditional Talmudic learning with the newly arrived knowledge, previously only available to those skilled in Arabic.[9] By the fourteenth century, a steady demand for this material resulted in the appearance of translations of the logical works of al-Fârâbi and Ibn Rushd, the mathematical works of Euclid and Archimedes, the astronomical works of Ptolemy, Jâbir ibn Aflah and al-Bitrûjî and the natural science, physics and metaphysics of Ibn Rushd’s commentaries of Aristotle.[10]

A watershed moment came in 1204 in the translation from Arabic to Hebrew of Maimonides’ Guide to the Perplexed, originally completed in 1190. Immensely influential, the Guide was written in the form of a cryptic letter to an advanced student, presumed to be perplexed, as it were, by the mutual contradictions of the new knowledge and traditional Jewish teaching, of an up-to-date philosophical interpretation of Torah. It also expedited further translation efforts, undertaken in large part by the Tibbonid family.[11] In accounting for creation, Maimonides considers the Platonic, Aristotelian and Scriptural arguments. He considers the Platonic story to be attractive, as this provides both understanding and explanation, but dismisses it on the grounds that the coeternal existence of formless matter is contrary to the Divine and thereby impinges on the perfection of creation. Moreover, any understanding at all is spurious, since the ways of God are, after all, transcendent and patently unknowable to begin with. Aristotle’s account is perhaps the more dangerous to the teachings of Torah, but although Maimonides accepts uncritically everything that Aristotle has to say about all things sublunar, upholding them as incontrovertibly true, he nevertheless believes that Aristotle runs afoul of Ptolemy’s Astronomy and even violates his own logical precepts. As a result, Maimonides does not feel that Aristotle has much of any use to say regarding the supralunar and cannot be recognized as a proper account of creation and, as is typical of the Guide, philosophical interpretations lose out to spiritual ones.[12]

But Maimonides was not the ridged conservative that R. Asher ben Yehiel was. On the contrary, his role in the propagation of foreign learning among Jewish circles was akin to what Augustine had done for the Christian West some seven centuries earlier. Although Maimonides asserted the primacy of Jewish teaching in spiritual matters, his zeal for Greek and Arabic knowledge also legitimized its place alongside religion, helping to rescue from oblivion what Greek and Arabic writers had accomplished. To be sure, such learning was not universally accepted, as is evidenced by the comments of R. Asher ben Yehiel. On the other hand, however, foreign subjects did attain a certain status within the Maimonidean paradigm. Astronomy, for instance, while still considered “dangerous” by some, was accepted to the extent that it assisted with the calculation of the calendar. Mathematics, understood principally according to Euclid’s Elements, was, by contrast,often decried as a baseless indulgence, robbing the soul of an afterlife. In a remark representative of this position, the twelfth-century Aristotelian Abraham ibn Daud of Toledo complained:

[algebra] consumes his time with number and with strange stories like the following: A man wanted to boil fifteen quarters of new wine so that it be reduced to a third. He boiled it until a quarter thereof departed, whereupon two quarters of the remaining [wine] were spilled, he again boiled it until a quarter vanished in the fire, whereupon two quarters of the rest were spilled. What is the proportion, between the [quantity] obtained and the [quantity] sought?[13]

Maimonides did not hold practical mathematics in such contempt. He did, however, believe that such pursuits were only useful inasmuch as they hone the intellect for purposes of scholarly disputation. In addition, the study of mathematics, after Maimonides, were not undertaken as an end unto itself. Rather, pursuit of mathematics was seen as paving the way to astronomy, physics and, ultimately, metaphysics, the true objective of a proper education.[14] With Maimonides, Greek and Arabic teaching enjoyed a newfound legitimacy, but remained merely the “handmaiden” of religion.

Levi ben Gershom, writing a century after Maimonides, explodes this state of affairs, creating, in effect, a synthesis of the science, philosophy and religion that had so bedazzled generations of Jewish thinkers. Intellectually, he bore the influences of Maimonides as well as Ibn Rushd (Averroes), on whom he wrote numerous commentaries. Levi conceived of man’s active intellect as the entire natural order in toto. By forging this umbilical cord between the celestial and terrestrial worlds, between sub- and supralunar, he thereby of necessity links empirical scientific discovery with absolute knowledge of the divine, that which had previously been considered beyond man’s capacity to know. Thus for Levi,

Science has a religious value, but this value legitimizes it qua an independent inquiry: because all knowledge of God’s works has a religious significance, [Levi] upholds the legitimacy of the autonomous pursuit of science. For [Levi], just as for the Protestant men of science, the religious function of science and its autonomy are dialectically related.[15]

 Troping Maimonides, Levi clearly values the importance of knowing the particulars of the active intellect but parts ways with his predecessor in a significant manner, namely that Levi does not hold the view that the knowledge acquired during life on earth, upon death, dissolves into the active intellect. Instead, he rather asserts, following Ibn Rushd, that an individual’s knowledge of particulars lives on after death, and for this reason the path to ultimate knowledge is paved by science and empirical pursuits.[16]

Empirical inquiry was of paramount importance for Levi. To this end, Levi tirelessly devoted himself to science, even undertaking biological experiments and, on paper at least, developing a parabolic mirror that was meant to be used as part of a rudimentary microscope for the observation of tiny parts of plants and animals.[17] In no other area, though, was Levi’s unified view of philosophy and theology more acute than in his astronomical work. Levi devised an invention called “Jacob’s Staff” which aided him in calculating not only the angular distance between planets but also their sizes. In fact, he was one of the only Medieval scientists to make original contributions in the area of astronomy, an area that before Levi had been valued only as a practical tool or derided as anathema to religion. Moreover, his methodology involved the critique of received wisdom from Ptolemy as well as al-Bitrûjî and is in this regard strikingly modern, given the Medieval deference for “authority.”[18] It is clear that for Levi, precision in astronomical calculations mattered in the utmost: the perfection of the heavens was on display for man to witness, and since eternal felicity rested on knowledge of the active intellect, inaccuracies in astronomical observation thus placed the fate of the soul in jeopardy.

Levi’s philosophical outlook was by no means appreciated in his day and he did not find any sympathetic readers until Leibniz and Spinoza. His magnum opus Milhamot Ha-shem (The Wars of the Lord), in which he adopts a Platonic conception of cosmology defended with Aristotelian science, was viciously maligned by his contemporaries, who referred to his work as “Wars against the Lord.” Yet despite his provincial lifestyle, Levi’s mathematical and astronomical work gained the attention of the Christian world, and he even dedicated the Latin translation of a work on trigonometry to Pope Clement VI. It is also believed that portions of his astronomical treatise were translated under Papal request.[19] As stated at the outset, Levi’s 1343 treatise De numeris harmonicis was written following a solicitation by Philippe de Vitry, a high-ranking Christian in his own right. It is to this treatise and its various interpretations that we now turn.

Philippe de Vitry may have met Levi during a 1342 journey to Avignon while in the service of the French King, although details of their encounter, personal or written are difficult to ascertain.[20] Regardless, the objective is clear enough, as stated above by Levi himself. In the De numeris harmonicis, Levi went on to prove that of the harmonic numbers so defined, only the pairs 1 and 2, 2 and 3, 3 and 4 and 8 and 9 differ by one. The question for musicologists remained as so what the significance of these numbers and of this proof actually was. Hugo Riemann, for his part, felt that this dealt with a theory of intervals in the context of de Vitry and Johannes de Muris’ treatises dealing with the musical instruction. Though he does not offer any discussion of evidence, his assertion ostensibly rests on these numbers’ role in the construction of the traditional diatonic intervals, the diapason (2:1), the diapente (3:2), the diatesseron (4:3) and the tone (9:8).

Eric Werner supplied another interpretation when he asserted that this proof constituted a mathematical foundation of de Vitry’s Ars nova metrical system. After considering the hypothesis and result of the treatise, Werner states plainly that this cannot have anything to do with musical intervals, though there is little that would actually rule this one either.[21] With no criteria for judging whether this has to do with intervals or mensuration, Werner then promptly wonders what the mathematical or musical significance of the harmonic numbers is and what they might mean in terms of metrical divisions. Werner holds that the term goes back to Levi’s mistranslation a similar term used by the Spanish-Hebrew scholar Abraham Ibn Ezra in his mathematical work Sefer ha-Mispar. According to Werner, the term in question is erech haneginot, which Werner takes as equivalent to numerus musicae. Gersonides, who was not fluent in Latin, misapprehended the term and rendered it numerus harmonicus.[22] Frobenius finds fault with this, and mentions how the term came from de Vitry, as is clear from Levi’s definition.[23] As will become clearer shortly, numeris harmonicis were in circulation well before de Vitry.

The thrust of Werner’s argument is that the diverse modes of division (dividing values into 2 or 3, hitherto unthinkable in Franconian notation) found in Ars nova mensuration engenders various numbers of minims at the lowest level of division. The idea is that each possible scheme of division results in a unique number of minims. The proof, according to this theory would allow Vitry

… to show that, within all possible combinations (even beyond 81 [minims]), modifications such as points of addition or alteration would not cancel or confuse basic duple or triple divisions. In practical terms, [Levi’s] theorem meant that, even without signature, duple and triple time could be recognized in each mensural unit and that the sum total of the smallest units (minimae) would determine for each case one, and only one, of the possible basic rhythmic divisions. [Levi] was thus able to give irrefutable mathematical justification to one of the most complex musical systems ever propounded.[24]

There are a number of problems with this assessment. Meyer and Wicker point out that although Werner does not actually specify which mensural system is supposed, he nevertheless implies the five-level system found in de Muris’ Libellus. This, however, may not have been a system that de Vitry had in mind, although it did make its appearance in print in 1340, making it contemporaneous with De numeris harmonicis. [25] Moreover, a three-level system, which de Vitry may have in fact used, does not yield unique numbers of minims, i.e., 3*2 = 6 and 2*3 = 6.[26] In addition, Werner suggests that the harmonic numbers utilized by de Vitry must lie between 16 and 81. There are harmonic numbers in between these values, however, that do not arise in a five-level system, that is, maxima, long, breve, semi-breve, minim. While 16 (2⁴), 24 (2³*3), 36 (2²*3²), 54 (2³*3) and 81 (3⁴) do arise, other harmonic numbers within this domain such as 18 (3²*2), and 27 (3³) require a four-level system and 48 (2⁴*3) and 72 (3²*2³) presupposed a six-level scheme, which would have been unheard of at the time.[27] This makes for quite a cluttered and historically insensitive theory and Werner’s argument and has been correspondingly rejected.

A more accurate theory is that of Wolf Frobenius, namely that the harmonic numbers refer to a tradition of thought dealing with intervals and tuning systems, specifically the Pythagorean sort, something not unlike Riemann’s view after all. Frobenius is quick to highlight that the harmonic numbers are none other than those from Plato’s Timaeus and Aristotle’s De anima, both of which were among the works of these authors that had been transmitted to the Middle Ages, principally, in the case of the former, through the well-known (probably) fourth-century commentary of Calcidius. Frobenius’ evidence rests of the work of two subsequent theorists, the Dutch theorist Johannes Boen and Nicole of Oresme. Meyer and Wicker also point out the work of English theorist Richard Kilwardby, in whose c. 1250 De ortu scientiarum there is an essentially Platonic cosmological conception of music as a speculative endeavour. Music, for Kilwardby, consists of the “harmonic number of the assembled things according to harmonic ratios.”[28] Significantly, although this work espouses a Neo-platonic stance towards music, it still would have been known in the Parisian intellectual circles of the 1320’s, where de Vitry and de Muris were active, and it is only with de Vitry that harmonic numbers take on a purely technical meaning, stripped of any extraneous metaphysical clutter.[29]

For Frobenius, the most conclusive evidence appears in the work of Boen and Oresme. The former, in his c. 1358 treatise on consonances gives an account of consonances that uses what appears to be none other than de Vitry’s own conception:

In addition, that there can be a relationship of tones between noticeable Tonstufen, it is essential, that both abbreviated numerical relationships (numeri primi) be “armonici”, progressively divisible by 3 down to 1 as 1, 3, 9, 27—and such is “masculinus”—or by 2 as 1, 2, 4, 8—and such is “femininus”—, or through both (viceversa) as 6, 12, 18, 24—and such is “promiscuus,” because they derive from the product of a masculinus and a femininus. These definitions of numerus armonicus is extracted generally from a book of Navarre and can be illustrated in a table, as they are here: [See Figure 1].

An the sides of this figure (top and left) you will find the abbreviated numbers (numeri primi) of a given musical proportion. So build the third masculinus (9) and the fourth femininus (8) the whole tone (9:8), likewise build the sixth masculinus and the ninth femininus the small semi-tone (243:256), likewise build the eighth masculinus and the twelfth femininus the apotome (2187:2048) … From here it is to be observed, that 5, 7, 11, etc. are not numeri armonici, thus are also the proportions that they build with whichever other numbers are not noticeable between tone-steps.[30]

It is clear from this discussion that Boen goes about building the common Pythagorean intervals according to the same conception of harmonic numbers put forth by de Vitry and Levi. Note also how the numbers 5, 7 and 11 throw up red flags: the simple fact that they are not harmonic numbers precludes their participation in the formation of Boen’s diatonic system. We will return to the issue of the “book of Navarre” shortly.

Nicole of Oresme’s 1377 Le livre du ciel et du monde, which contains a discussion of the Harmony of the Spheres and is itself a gloss translation of Aristotle’s De caelo et mundo, there is found a similar discussion:

And these [harmonic proportions] in the tone-system are not all consonant, but rather only four are, whose numerical relationships differ only by 1; and so it acts with no other possible proportions in the tone-system. And of these four proportions, one is the octave or dupla in 2:1; the second is the fifth in 3:2; the third is the fourth in 4:3; and the fourth is the whole-tone in 9:8. And so there are only four proportions that are simple consonances, but many more can be made out of these.[31]

 In addition to this corresponding discussion, Frobenius and Meyer and Wicker point the emergence of the relationship (n+1):n. The number 1 had also come to be recognized as a number, which allowed the ratio 2:1 to slip into relationship and shed its unique status as a multiple relationship, as distinct from the super-particular relationship. Therefore, the theory states that harmonic numbers parking of this (n+1):n relationship and only these are the fundamental intervals of the Pythagorean tone-system, the genetic material of pitch space. Levi’s treatise, then, can be read as offering conclusive mathematical proof that indeed only 2:1, 3:2, 4:3 and 9:8 partake of this relationship and, hence, what had been suspected all along was in fact the case. Meyer and Wicker note how de Vitry was likely inspired to seek out such proof following his exposure to Boethius by way of de Muris’ university text Musica speculativa, which includes a watered-down discussion of the Pythagorean myth, bereft of Pythagoras’ actual experiments or any of the more subtle points of Boethius De institutione musica.[32] Perhaps desirous of further substantiation of Boethius’ views, de Vitry took the next step towards a consolidation this picture.

Although the passages from Boen and Oresme are outwardly very convincing, the evidential link between them and de Vitry is rather more problematic. Boen explicitly mentions a “book of Navarre,” which apparently assisted him in his formulation. The book in question is a mysterious book that contained not an explicit account of Levi’s treatise, which Frobenius feels was circulating by the 1350’s, but a definition of harmonic numbers virtually identical to that found in De numeris harmonicis. Frobenius supposes that both Boen and Oresme were active at the University in the 1350’s, and therefore would have come in contact with this book, as well as de Vitry himself. Evidence linking de Vitry to the College, however, is difficult to come by. We know that he became affiliated with the College in an administrative capacity with his ascension to the Bishop of Meaux in 1351.[33] In response to the suggestion that de Vitry was affiliated with the University between the years 1316 and 1322, Sarah Fuller mentions how, although de Vitry’s name was listed at the royal court, any affiliation during this time lacks, “a particle of corroborative evidence.”[34] For these reasons, despite the compelling coincidences found in the work of Boen and Oresme, it is very difficult to tie them together with de Vitry at the College of Navarre.

Levi ben Gershom’s De numeris harmonicis represents an exciting confrontation of Jewish and Christian worlds in the fourteenth century and is the work of a consummate intellectual whose position in his own intellectual context was as precarious as it was brilliant. The actual significance of his treatise, however, has proven over the years to be a matter of some opacity. In an early major study of Levi, Joseph Carlebach resists any interpretive urge whatsoever. In his footnote following Levi’s mention of “unum suppositum in predicta scientia,” Carlebach demurs, stating, “I will not endeavour to interject what the following sentence might mean for the music theory of the Middle Ages.”[35] Others have taken up the task, and our understanding of the treatise has evolved from an intervallic understanding, to a mensural one and finally to an interpretation dealing with the foundation of the diatonic Pythagorean tone-system. Meyer and Wicker note how it is interesting that such an effort on the part of de Vitry came about at a time when major thirds (5:4) and minor thirds (6:5) were beginning to win prominence in English music as well as instrumental music.[36] Though it is perhaps a mere trifle of speculation, it is also interesting to wonder to what extent Levi himself might have been familiar with harmonic numbers, as he was clearly familiar with Plato’s Timaeus. De numeris harmonicis remains an engaging cipher of fourteenth-century musical thought and one whose solution merits a close investigation of the history of ideas.