THE DIVINE MUSIC OF MATHEMATICS
by David P. Goldman
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.
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