Music of the Spheres
When a computer generates a random number, what comes out is not, technically, random. Mathematicians, who worry about such things, call such numbers “pseudo-random.” A pseudo-random phenomenon is produced by an arbitrary and unpredictable process, a function so arbitrary and complicated that it is essentially impossible to predict the output without reproducing the function itself. For most purposes, the output may as well be random, but it is not truly random, because the same input will invariably produce the same output.
To illustrate what I’m talking about, suppose you needed a random number from your computer—to shuffle a deck for your next Windows solitaire game, for example. Your computer might do something like this: read the internal clock, express that time as a single number, divide that number by the number of milliseconds your computer has been on since its last boot, divide that quotient by pi, and take the hundred thousandth digit of the result. If it needed more “random” numbers, your computer could continue with the hundred thousand and first digit, and so on. Arbitrary input values, like the clock times in this example, are called “seeds.” Technically, the results are not random, since anyone starting with the same seeds, that is, asking for a random deck shuffle at exactly that time, after running their computer exactly that long, would get the same deck. But for all practical purposes, the chances of any two players getting those same numbers are so low, and the results are so unpredictable, that the deck generated by such a process may as well be random.
Pseudo-random behavior is not limited to computers. You can see it as well in any chaotic system, like the swinging of a pendulum dangling from the end of another pendulum, or the curls of cream poured into a swirling cup of tea. Theoretically, the behavior is reproducible, but in practice, microscopic changes in input generate huge changes in output, and actually reproducing the result is impossible. The apparent randomness stems from the extraordinarily complicated nature of the function. The Mandelbrot set consists of complex numbers which result in convergent behavior when employed as a starting point in the iterated function z2+c. There is nothing special about the function z2+c; other functions, like 2z2+c, or z3+c, possess equally complicated and elegant sets of convergent starting points.
Simple systems can produce surprisingly complicated—and often beautiful—results. Small changes to such systems can produce equally complicated and beautiful results, or they may cause some kind of catastrophic failure. The entire result is typically more esthetically appealing in explicit form, but the information is quite literally contained within the function and the starting point(s) upon which it acts.
I bring up beauty in order to apply the notion of pseudo-random behavior to describe my preferred metaphor for the music of J. S. Bach, my favorite composer. Bach’s work often strikes me as the musical equivalent of the Mandelbrot set, an abstract beauty generated from simple seeds. This is not entirely accidental; baroque-era composers often used a very similar seed technique to produce their music. Take a handful of promising starting notes; these define the basic theme of the music. Extend the phrase by adding the most appealing note you can find to the end, and another, and another. After a measure or two, introduce harmonic lines to your growing melody, and away you go. Rigid rules define which notes may be used next: no parallel fifths, no octaves, no repeated notes. As long as you stick to the permitted notes, you’re guaranteed of producing something acceptable.
What distinguishes Bach from his colleagues is that he seems always to choose the “right” note to use next, instead of merely choosing a permissible one. Lacking Bach’s ear, other composers picked legal notes, but not always the best ones; as a result, they wrote music that merely sounds all right. Bach wrote music that sounds right. The metaphor is an illusion, of course. Bach himself could and did produce entirely different pieces from the same starting notes, often as a form of exercise. But his sure skill makes his pieces seem like the only possible result. Even when he does not employ this technique, the result seems like he has. It’s as though all the beauty of the music lay packed within the first few notes, and he has merely unfolded it for the rest of us to hear.
It is not for the pious themes of much of his music that Bach’s work is often called “the music of God.” Rather, it is for the way his music seems to replicate the Creation, spinning vast, complicated beauty from the simplest starting point, as theologians often imagine God unfolding the universe with clockwork precision.