Thirty-six million Chinese kids now study classical piano, not counting string and woodwind players. Chinese parents pay for music lessons not because they expect their offspring to earn a living at the keyboard, but because they believe it will make them smarter at their studies. Are they right? And if so, why?
The intertwined histories of music and mathematics offer a clue. The same faculty of the mind we evoke playfully in music, we put to work analytically in higher mathematics. By higher mathematics, I mean calculus and beyond. Only a tenth of American high school students study calculus, and a considerably smaller fraction really learn the subject. There is quite a difference between learning the rules of Euclidean geometry and the solution of algebraic equations: the notion that the terms of a convergent infinite series sum up to a finite number requires a different kind of thinking than elementary mathematics. The same kind of thinking applies to playing classical music. Don’t look for a mathematical formula to make sense of music: what higher mathematics and classical music have in common is not an algorithm, but a similar demand on the mind. Don’t expect the brain scientists to show just how the neurons flicker any time soon. The best music evokes paradoxes still at the frontiers of mathematics.
In an essay for First Things titled “The Divine Music of Mathematics,” just released from behind the pay wall, I show that the first intimation of higher-order numbers in mathematics in Western thought comes from St. Augustine’s 5th-century treatise on music. Our ability to perceive complex and altered rhythms in poetry and music, the Church father argued, requires “numbers of the intellect” which stand above the ordinary numbers of perception. A red thread connects Augustine’s concept with the discovery of irrational numbers in the 15th century and the invention of calculus in the 17th century. The common thread is the mind’s engagement with the paradox of the infinite. The mathematical issues raised by Augustine and debated through the Renaissance and the 17th-century scientific revolution remain unsolved in some key respects.
The material is inherently difficult, although it’s possible to find simple illustrations of what Augustine means by higher-order number. As I wrote in the First Things piece:
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.
It takes years of study, to be sure, to hear the metrical plasticity in Brahms, or to make sense of higher mathematics. But that’s the whole point: The painstaking acquisition of knowledge and technique, and the enhancement of attention span and intuition, are the long-term benefits of classical music study. Humility, patience, and discipline are the virtues that children acquire through long-term commitment. I doubt that blasting your baby with Mozart will do much good. It takes a lot of learning to hear what Mozart is doing, especially because we have lost so much of the musical culture that Mozart took for granted in his audience.
Most important is the spiritual dimension of classical music: it embodies a teleology. Classical music is a journey to a goal, full of suspense and surprises, but always with a purpose. It is no coincidence that the classical style of Western composition was developed for religious music.
Never before in human history has music been so accessible. A touch-sensitive electric piano with sounds sampled from good acoustic instruments, suitable for a beginning pupil, costs about as much as a video game station. If you want to make your kids smarter, throw out the video games and get them music lessons. Get them involved in youth orchestras where available. Make them sweat. One day they will thank you for it.