April 2012

The Divine Music of Mathematics
How music theory proves what ancient mathematics thought impossible.

It is consoling to think that the emotions that music arouses in us have something to do with the makeup of the universe. The eternal relation of math and music has been a perennial question since Plato, from Boethius and Cassiodorus in late antiquity, through Dante’s celestial harmony in Paradiso and Shakespeare’s discussion in The Merchant of Venice. The deeper affinity between mathematics and music, though, is less consoling and more challenging: The modern concept of a higher-order number begins with St. Augustine’s fifth-century treatise on music, and a red thread links it to Leibniz’ invention of the calculus in the seventeenth.

Music employs number both in its harmonic foundation and its metrical presentation in time. But what sort of number is it? In the sixth book of his De Musica, Augustine asserted the existence of a higher order of number that in some way stands above the senses, the numeri iudiciales or “numbers of judgment” which “come from God” and enable the mind to judge what it perceives and remembers, as well as what it expects. Augustine’s assertion is arresting in all three of its parts: first, that neither our sense perception nor even our memory explains how we hear music; second, that the faculty by which we judge the numbers (rhythms or harmonies) of music is also a kind of number; and third, that this higher-order number comes from God.

Championed by St. Bonaventure in the thirteenth century and embraced by Nicholas of Cusa in the fifteenth, Augustine’s “numbers of judgment” point to the mathematical revolution of Newton and Leibniz in the seventeenth century. The concept of higher-order number separates the mathematics of classical antiquity from modern mathematics beginning with the calculus. Archimedes encountered solutions to individual problems in the calculus, but the idea that the integral and the differential were a new order of number that could be manipulated like any other number lay outside the boundaries of the Hellenic imagination.

Unlike the fifth-century Roman theorist Boethius, the great classical source for medieval theory, Augustine never directly discussed harmonics. His concern in De Musica was the mathematics of poetic rhythm rather than the divisions of vibrating strings. Yet the problem of higher-order numbers forced itself upon the fifteenth century through musical practice, when musicians began to alter the natural harmonic intervals to suit the requirements of the emerging tonal system.

We can dismiss these facts as happenstance. Or we can inquire as to whether the mind’s perception of music does indeed tell us something fundamental about higher orders of number.

In De Musica, Augustine presents a hierarchy of rhythm that begins with “sounding numbers”—the rhythm we actually hear—followed by “memorized rhythms,” that is, the mind’s recognition and remembrance of a pattern. Rising above all such numbers is what Augustine calls “consideration,” the numeri iudiciales. These “numbers of judgment” bridge eternity and mortal time; they are eternal in character and lie outside of rhythm itself but act as an ordering principle for all other rhythms. Only they are immortal, for the others pass away instantly as they sound, or fade gradually from our memory. They are, moreover, a gift from God, for “from where should we believe that the soul is given what is eternal and unchangeable, if not from the one, eternal, and unchangeable God?”

Book 6 of De Musica resists the usual scholarly approaches in part because it is so hard to identify precedents. Paul Ricoeur observes astutely that Augustine draws more on the Bible than on the Greeks, referring to Genesis. We might also seek Augustine’s source in Ecclesiastes. For the Greeks, time is the demarcation of events. Plato understands time as an effect of celestial mechanics in Timaeus, while Aristotle in the Physics thinks of time as an attribute of movement. To Kohelet, though, time itself is an enigma; as with Augustine, it is the moment itself that remains imperceptible. As Ecclesiastes 3:15 reads in the translation of the nineteenth-century rabbi and polymath Michael Friedländer: “That which is, already has been; and that which is to be has already been; and only God can find the fleeting moment.”

Augustine asserts that some faculty in our minds makes it possible to hear rhythms on a higher order than sense perception or simple memory, through “judgment.” What he meant quite specifically, I think, is the faculty that allows us to hear two fourteeners in the opening of Coleridge’s epic:

It is an ancient Mariner,
And he stoppeth one of three.
“By thy long grey beard and glittering eye,
Now wherefore stopp’st thou me?”

Read by a computer’s text-to-voice program, this will not sound like what Coleridge had in mind. A reader conversant with English poetry intuitively recognizes the two syllables “And he” as a replacement for the expected first syllable in the first iamb of the second line. The reader will pronounce the first three syllables, “And he stoppeth” with equal stress, rather like a three-syllable spondee, or a hemiola (three in place of two) in music. Our “numbers of memory” tell us to expect ballad meter and to reinterpret extra syllables as an expansion of the one expected. The spondees in the second fourteener, moreover, grind against the expected forward motion, emulating the Mariner’s detention of the wedding guest.

Something more than sense perception and logic is required to scan the verse correctly, and that is what Augustine calls “

consideration.” As I observed in “Sacred Music, Sacred Time” (November 2009), De Musica employs poetic meter as a laboratory for Augustine’s analysis of time as memory and expectation, and his approach remains robust in the context of modern analysis of metrical complexity in classical music. To perceive the plasticity of musical time in the works of the great Western composers, to be sure, requires a trained ear guided by an educated mind, but the metrical complexity of a Brahms symphony depends on the same faculty of mind we need to hear Coleridge correctly.

The time of human experience is plastic, as opposed to the putatively objective time that Plato and Aristotle derived from celestial motion. In Time and Narrative, Ricoeur argues that Augustine’s theory of time substitutes a psychology for a cosmology. His three-volume discussion of time, though, ignores De Musica, where Augustine implies an ontology as well as a psychology of time. That is the implication of his assertion that the faculty of consideration itself rises to numbers and assumes objective existence. And that is how Augustine was understood by his foremost medieval popularizer, St. Bonaventure, in Itinerarium Mentis ad Deum (The mind’s journey to God). Bonaventure understands Augustine’s numbers in terms of harmonics as well as poetic meter. Augustine, he writes, “says that numbers are in bodies and especially in sounds and words” which he calls “sonorous.”

Some are abstracted from these and received into our senses, and these he calls “heard.” Some proceed from the soul into the body, as appears in gestures and bodily movements, and these he calls “uttered.” Some are in the pleasures of the senses which arise from attending to the species which have been received, and these he calls “sensual.” Some are retained in the memory, and these he calls “remembered.” Some are the bases of our judgments about all these, and these he calls “judicial,” which, as has been said above, necessarily transcend our minds because they are infallible and incontrovertible.

These imprint on our minds the “numbers of artifice,” which Augustine had not included in this classification because “they are connected with the judicial number from which flow the uttered numbers out of which are created the numerical forms of those things made by art. Hence, from the highest through the middles to the lowest, there is an ordered descent. Thence do we ascend step by step from the sonorous numbers by means of the uttered, the sensual, and the remembered.”

Bonaventure clearly believes that the higher-order numbers apply to harmonics as well as meter. By “numbers of artifice,” as Theo van Velthoven observes, Bonaventure does not imply that the numbers are a mere human construct, but rather that by means of these numbers man creates artefacta. As he reads Augustine, the higher-order numbers express a God-given creative faculty in human beings that makes it possible for us to understand and act upon God’s creation. His meaning is obscure, and usually dismissed as a throwback to Neoplatonic mysticism. Yet two centuries after he wrote, music gave us the first practical instantiation of higher-order number, through the tuning associated with the new polyphony of the fifteenth century.

A brief review of harmonics is in order here. Music derives from nature and is as old as human sentience. In Werner Herzog’s documentary Cave of Forgotten Dreams, a paleontologist toots “The Star-Spangled Banner” on a reproduction of a 35,000-year-old flute carved from vulture bone, unearthed in 2009 in a South German cave. Vibrating strings or air columns naturally divide into halves, thirds, fourths, and fifths; these tones produce the consonant intervals of the octave, fifth, fourth, and major third, and so on. From the tones, a Neolithic musician was able to produce something like a modern major scale. The “natural” fifth in the ratio of 3:2 requites the modest needs of the pentatonic scale that characterizes folk music from Ireland to Japan, but not the more complex demands of Western polyphony. Within a single key, the natural intervals sound tolerable, but if we change keys—as composers began to do in the fifteenth century—the natural intervals sound out of tune. We may infer from surviving works on music theory by Aristotle’s student Aristoxenus that the Greeks employed more sophisticated tuning, but we do not have sufficient documentation of its use to be sure of its aesthetic purpose.

We recognize the earliest use of tonality in the music of the fifteenth century, that is, the employment of different keys. Renaissance composers learned to establish a tonic, or “home key,” and then modulate to one or more contrasting tonics (usually separated by a fifth), and then return to the home key. Tonality thus makes possible what modern music theory calls goal-oriented motion, that is, a journey to a contrasting tonal area and a return. The aesthetic implications of tonality changed music as profoundly as, for example, perspective in painting; the new technique enabled composers to depict Christian teleology in musical time.

The contrast of tonic and dominant as individual melodic tones underlies modal and pentatonic folk music as much as it does Western tonal music. But the emerging musical art of the West found a way to prolong this procession of tones by making the dominant (and other tones) into temporary tonics. Once composers learned to create distance from the tonic, they could also prolong intermediate tonics, and add new layers of complexity to musical teleology. Delayed resolution made it possible to evoke suspense, longing, reverence, and even humor. Once composers learned to create long-range expectations of tonal resolution, they gained the freedom to manipulate these expectations to produce a palette of affects. The plasticity of musical time made possible by tonality, and the perception of the passage of time at multiple levels, gave Western music the capacity to evoke a sense of the sacred.

To stabilize the arrival upon the dominant, the composers of the fifteenth century began to “tonicize” the fifth and other tones of the scale, that is, to make them into temporary home keys. If the leading tone (the half step that rises to the tonic) is B-natural when C is the tonic, Renaissance composers raised the fourth step of the scale to F# to serve as a local leading tone to the dominant G. This move is now so basic to Western music that we expect to hear it in the bridge of every major-key pop standard, but for the fifteenth century it was a startling innovation.

Natural intonation sounds out of tune even at this level of complexity, which requires the introduction of “accidental” tones outside the seven-tone diatonic scale. Musical practice thus provoked a crisis in theory. At the University of Padua, Prosdocimus de Beldemandis showed in 1425 that Pythagorean calculations could divide the octave into twelve half-tones, identifying all the accidentals required to build a scale starting on any tone. Nicholas of Cusa studied at Padua at the time and surely knew Prosdocimus. The trouble was that Pythagorean arithmetic produced intervals that sound audibly out of tune.

We do not know exactly when musicians began altering, or “tempering,” musical intervals to subordinate nature to the requirements of artifice. But it could have been no later than 1444, when Cusa wrote his dialogue De Coniecturis. In this work Cusa first mentions the irrational character of the half-tone: “The fact that the precise [size] of the half-tone is hidden to reason [ ratio] is due to the fact that it cannot be attained except by the coincidence of even and uneven [i.e., irrationality].”

What Cusa mentioned there in passing is spelled out in the 1450 dialogue Idiota de Mente (The layman on the mind). In this work he paraphrases Augustine, and makes his understanding of the half-tone explicit. There exist musical numbers, he asserted, “too simple for the understanding [ ratio] of our mind to attain.” He was speaking of the use of irrational numbers in musical harmonics, leaning perhaps on Bonaventure. The solution to the problem of the unequal half-tones derived from Pythagorean ratios was the division of the whole tone by its geometric mean.

Cusa compares the semitone to the diagonal of a square, that is, an irrational number: “When I consider nothing more than unity in number, then I see uncomposed compositeness, and the coincidence of unity and multiplicity,” he writes.

But if I perceive more acutely, then I see the composite unity of octave, fifth, and fourth. That harmonic proportion is the unity without which number cannot be understood. Furthermore, I perceive from the relationship of the half-tone and the half of the double, which is like the proportion of the side of the square to its diagonal, a number which is too simple for the understanding (ratio) of our mind to attain. (Emphasis added)

What Cusa evidently means here is that the pitch of the half-tone F# that lies between F and G, for example, is the square root of the ratio of the pitch G to the pitch F. It is like the proportion of the side of the square to its diagonal, that is, it is a square root. That is a bit convoluted, but correct. (In modern equal temperament, first described by mathematicians influenced by Cusa in the second decade of the sixteenth century, the twelve half-tones of the octave are specified by powers of the twelfth root of two. If A is 440 cycles per second, the pitch of B is 440 cycles per second multiplied by 22/12 and the B b in between A and B is 440 multiplied by 21/12. The factor 21/12 is the square root of the factor 22/12.)

The anguish caused by the appearance of irrational numbers in musical tuning cannot be overstated. Mathematicians knew very well how irrational numbers looked, for example, in the form of square roots in geometric construction. Greek mathematicians had proved by the fifth century b.c.e. that no ratio of two integers could express the relationship of the hypotenuse to the side of an isosceles right triangle. But the notion of “irrational number” seemed an oxymoron to the great fifth-century mathematician Eudoxus, who argued that the irrationals were not even numbers to begin with, but only “magnitudes.”

The mathematical problem had deep philosophical implications. Aristotle understood that irrational numbers implied the “actual infinity” that he abhorred. A number that cannot be expressed by a ratio of integers requires an endless series of approximations that never quite arrive at the true value. The square root of two, for example, appears in modern decimal notation as

1.4142135623731 . . .
or 1 + 2/5 + 1/10 +2/100 + 1/1000 + 3/10,000 + 5/100,000 + 6/1,000,000 ad infinitum.

The series goes on forever, so that its members cannot be included in an instantiated universal populated by objects of sense perception. “The fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately,” wrote Aristotle in the Metaphysics. Greek mathematicians knew that an infinite series of rational numbers converged on an irrational value, for example, Archimedes’ famous approximation of the circumference of a circle through a many-sided inscribed polygon. But they thought of such procedures as a set of tricks to arrive at approximations. It was not until the seventeenth century that Leibniz explicitly embraced “actual infinity” in the calculus, which gives a precise value for the sum of many converging infinite series.

A generation before Cusa, Indian mathematicians had discovered the infinite series that converges on the transcendental number , but it is improbable that Europeans knew of this result. To medieval mathematicians, the irrationals were surds, or “deaf” numbers, that is, numbers that could not be heard in audible ratio. The association of rational numbers with musical tones was embedded so firmly in medieval thinking that the existence of an irrational harmonic number was unthinkable—until Nicholas of Cusa.

The fifteenth-century embrace of irrational harmonic numbers, moreover, threatened the classical and medieval concept of universal order. Since before Plato’s Timaeus, Hellenic thought had identified the naturally occurring harmonic ratios of music with the cosmological principles of creation. Plato took from the Pythagoreans the belief that nature unfolds herself in harmonic ratios such that the musical consonances parallel the harmonies of the spheres. On the authority of Boethius and Cassiodorus, medieval theorists taught that musical harmony binds together the World Soul and identified what we now call cosmology with musica mundana, the music of the universe. They identified the tones of the musical scale with the planets, which emit an unheard music of the spheres.

The universal aesthetic experience of mankind, as ancient as the earliest known cave paintings, seemed to cohere with the composition of the heavens, through the application of perceptible harmonic ratios. What the senses perceived intuitively made the cosmos intelligible. Augustine’s “numbers of judgment” preoccupied mystics like Bonaventure, but not the medieval mainstream. Yet the Pythagorean idea became a curiosity, while Cusa’s defense of irrational numbers led in a straight line to Leibniz and the calculus.

To the extent that medieval musicians drew on Augustine, they reduced his idea to a mechanical formula, often with infelicitous results. The so-called isorhythmic style of the fourteenth-century Notre Dame school arranged musical meter according to the harmonic ratios. Unlike the still-popular free polyphony of the fifteenth and sixteenth centuries, the learned style of Guillaume de Machaut and his contemporaries has no purchase among the listening public. Like the twentieth-century serialism of Schoenberg and Webern, it is Augenmusik, music for the eyes but not the ears. The Church abandoned it entirely in favor of the modern style of Palestrina. By the same token, modern attempts to apply the “harmony of the Spheres” to science, for example Kepler’s Universal Harmony, are curiosities rather than contributions. The various attempts to impose mathematics on music (or music on mathematics) produced, respectively, bad music and bad mathematics.

The practices of musicians, Cusa in effect argued, trumped the doctrines of the theorists: If musicians already were employing surds in the new music of the Renaissance, such numbers could not in fact be “deaf.” The musical provenance of irrational numbers thus forced the issue upon the mathematicians. The human mind, Cusa argued, could not perceive such numbers through reason ( ratio), the measuring and categorizing faculty of the mind, but only through the intellect ( intellectus), which depended on participation ( participatio) in the mind of God.

Cusa’s use of Augustinian terminology to describe the irrationals—numbers “too simple for our minds to understand”—heralded a problem that took four centuries to solve (and, according to a handful of holdouts, remains unsolved to this day). Not until the nineteenth century did mathematicians arrive at a rigorous definition of irrational numbers, as the limit of an infinite converging sequence of rational numbers. Sense perception fails us; instead, we require an intellectual leap to the seemingly paradoxical concept of a convergent infinite series of rational numbers whose limit is an irrational number. Irrational numbers thus lead us out of the mathematics of sense perception, the world of Euclid and Aristotle, into the higher mathematics foreshadowed by Augustine.

Leibniz averred, “I am so in favor of the actual infinite that instead of admitting that nature abhors it, as is commonly said, I hold that nature makes frequent use of it everywhere, in order to show more effectively the perfections of its author.” Gilles DeLeuze, in Leibniz and the Baroque, reports that Leibniz “took up in detail” Cusa’s idea of “the most simple” number: “The question of harmonic unity becomes that of the ‘most simple’ number, as Nicolas of Cusa states, for whom the number is irrational. But, although Leibniz also happens to relate the irrational to the existent, or to consider the irrational as a number of the existent, he feels he can discover an infinite series of rationals enveloped or hidden in the incommensurable.”

Cusa’s influence is evident on the music theorists of the next two generations. The first geometrical construction of the division of the half-tone appears in a 1494 treatise by the editor of Cusa’s collected works, the French humanist Lefèvre d’Étaples; it seems clear that Cusa was his source. And from Faber the idea makes its way into several widely circulated treatises of the next generation.

We see Cusa’s influence prominently in the music theory of the late fifteenth century. In the 1496 edition of his Practica Musicae, the most widely read musical text of the century, the Milan maestro di capella Franchino Gaffurio reported that organists reduced the fifth (implicitly all fifths) by a small and somewhat indefinite amount, called the participata. Regarding the thirds and sixths, Gaffurio calls them “irrational intervals,” because they are defined by numerical ratios so complex that they represent “a kind of irrationality.” For that reason, Gafurio concludes, musical proportion “participates” in arithmetic, harmonic, and geometric division; that is, the musical numbers determined by simple harmonic ratio “participate” (are tempered) to correspond to geometric division. The use of the term “participation” to describe temperament stands out, for Cusa had used the same word to explain the mind’s capacity to comprehend irrational numbers, through participation in the mind of God.

The medieval synthesis of Aristotelian ontology and Platonic cosmology offered embittered resistance to the Augustinian revolution of the fifteenth century. The theorists of the Counter-Reformation denounced the new tempered tuning as a violation of the natural order. In a survey of sixteenth-century responses to the problem, Peter Pesic of St. John’s College quotes the monk Michael Stifel, whose 1544 treatise is one of the first to unambiguously identify such numbers as irrationals. Stifel examined the geometric division of the whole-tone he had learned from Faber, but concluded that

we are forced to deny that irrational numbers are numbers. Namely, where we might try to subject them to numeration and to make them proportional to rational numbers, we find that they flee perpetually, so that none of them in itself can be precisely grasped. . . . Moreover, it is not possible to call that a true number which is such as to lack precision and which has no known proportion to true numbers. Just as an infinite number is not a number, so an irrational number is not a true number and is hidden under a sort of cloud of infinity.

Sixteenth-century theorists, Pesic reports, clung to Aristotle’s abhorrence of the “actual infinite.” The leading music theorist of the sixteenth century, Giuseppe Zarlino, reviewed the mathematics of equal temperament but rejected them—in a famous denunciation of Galileo Galilei’s musician father—in favor of a “natural” tuning that musical practice had long since rendered impractical. In fact, Zarlino’s compositional examples in his own textbook violate the rules he propounds in defense of “natural” tuning. Cusa’s influence had dominated the music theory of the late-fifteenth century but virtually disappeared during the sixteenth century after the Aristotelian counter-attack. The actual infinite had few defenders in the sixteenth century. Not until the seventeenth century do we find an explicit defense of musical temperament as a better expression of nature, in a musical treatise by Agostino Steffani, court composer at Celle and a friend of the court’s most famous member, G. W. Leibniz.

What are we to make of the long journey of Augustine’s “numbers of judgment” from poetic scansion to the calculus? Nicholas of Cusa believed that the same aesthetic judgment that stands above perception and memory of poetry also enables us to transcend sense perception in our understanding of the mathematics of the infinite. In the light of later philosophy, this view does not seem untoward. The founder of the neo-Kantian school, Hermann Cohen, made his reputation on the claim that the infinitesimals of the calculus prove the existence of Kant’s synthetic reason, for it proves the existence of things that are in the mind but not in the senses.

The musical roots of the calculus give some support to the view that Kant’s synthetic reason excises the divine from Augustine’s theory of divine illumination. In a sense, Augustine’s theory triumphed over Aristotle’s during the nineteenth century, but at the cost of substituting the hardwiring of the brain (synthetic reason) for participation in the mind of God.

But that is not the end of the story. Leibniz’ infinitesimals, as I reported in “The God of the Mathematicians” (August/September 2010), lead us eventually to George Cantor’s discovery of different orders of infinity and the transfinite numbers that designate them; Cantor cited Cusa as well as Leibniz as his antecedents, explaining that “transfinite integers themselves are, in a certain sense, new irrationalities. Indeed, in my opinion, the method for the definition of finite irrational numbers is quite analogous, I can say, is the same one as my method for introducing transfinite integers. It can be certainly said: transfinite integers stand and fall together with finite irrational numbers.” Cantor’s transfinites—the “actual infinity” denied by Aristotle—contain an infinity of lower-order numbers, and therefore might be thought of as an instantiation of Augustine’s “numbers of judgment.”

We can therefore draw a red line from the musical innovations of the 1430s to the present-day frontier of mathematics and the philosophical quarrels that continue to erupt over it. Augustine’s “divine numbers” still are unsettling. Even if we can specify the irrationals as definite, the transfinite numbers remain in some way beyond our ken, hidden away, as it were, in the mind of God.

Kurt Gödel and Paul Cohen proved that although we know that there are an infinite number of transfinite numbers, we cannot find out exactly what they are or in which order they appear, not, at least, within any known system of mathematical logic. Mathematicians disagree about what this result (the independence of the Continuum Hypothesis) means. Did Gödel and Cohen demonstrate that the ultimate number in the mind of God—infinity—must remain in God’s mind, inaccessible to human thought? Or did they merely play a game with an arbitrarily devised set of rules? Gödel clearly believed that his results had ontological significance.

We are left with problems in epistemology and ontology we might begin to address if we knew whether they were problems in epistemology or in ontology, or both. I believe that we should understand Augustine’s concept of divine number as both an epistemology and an ontology: His numeri iudiciales link the higher faculties of the human mind to the workings of nature that lie beyond ordinary sense perception. This “both,” as opposed to the “either,” rests on a theological surmise, that the numbers of judgment come from God.

In light of the extraordinary influence of Augustine’s idea, we might think about the problem this way: Even if the ultimate foundations of reality remain hidden from us, we nonetheless possess a creative faculty that gives us insight into the infinite. We employ the same faculty at play in music as we do in probing reality through mathematics. And this faculty whose workings we observe in the laboratory of music offers an intimation of our role as junior partners in creation.

We might find a foundation for this ontology in Ecclesiastes, which asserts that time is neither the cosmological ideal of Plato nor the perceived motion of Aristotle, but rather God’s time: “That which is already has been; and that which is to be has already been; and only God can find the fleeting moment.” The connection between the actual infinity and the biblical concept of a higher order of time becomes clear when we consider that adding to a series of numbers always implies an action in time. This was the case in the first philosophical formulation of the problem of actual infinity in Zeno’s paradox, in which Achilles cannot overtake the tortoise. It is still the case in modern mathematical logic, namely in Alan Turing’s reformulation of Gödel’s First Incompleteness Theorem as a problem of infinite computation time. The biblical assertion that time is a divine mystery may have been the most fecund statement about nature in scientific history.

David P. Goldman is Tablet magazine’s classical music critic and the author of How Civilizations Die (and Why Islam is Dying Too).