Plato and the Asymptotic Ideal in Poly Music

The Asymptotic Ideal

Oct 9, 2026 · 

Polytempic polymicrotonality is an ideal. I have always known this. Four voices, four tunings, four tempi, none of them bowing to a master clock or a master fundamental. Every voice is its own world, and every world keeps its own time.

I accept that this music may never be fully realized by human hands. I do not consider that a defeat. I consider it the nature of the thing. The ideal is not diminished because it cannot be reached. It is the reason to keep reaching.

The Pattern in the Republic

In Plato's Republic, Socrates builds the perfect state in speech: the just city, ruled by the philosopher king. Then he is pressed on the obvious question. Can such a city ever exist?

His answer is one of the most honest moments in philosophy. He concedes that it may never exist exactly as described. A painter who paints the most beautiful human being imaginable is not a worse painter because no such person walks the earth. The pattern has value as a pattern. What is done in deed always grasps the truth a little less than what is said in word. The best we can ask is that an actual city come as close to the pattern as possible.

Socrates is not apologizing for the ideal. He is defining its job. The ideal is the measure. Everything actual is judged by its distance from it.

The Asymptote

Calculus gave this idea a shape two thousand years after Plato. An asymptote is a line that a curve approaches forever and never touches. The gap shrinks without limit. It never closes.

\lim_{x \to \infty} \frac{1}{x} = 0, 
\quad \text{yet } \frac{1}{x} \neq 0 
\text{ for every } x

That is my relationship to my own music. The score is the line. Every realization is a point on the curve. Some points sit very close. None of them arrive. Realization is not a yes or a no. It is a continuum, and the work lives along it.

String Quartet 18

My eighteenth string quartet pushes the ideal further than anything I have written. Each of the four voices is in its own tuning. Each moves in its own tempo. And each tempo is constantly changing, independently of the other three.

The meters come from Henry Cowell. In New Musical Resources he proposed time signatures built on divisions of the whole note that our notation never adopted: thirds, fifths, sevenths. I use 3rd notes, 5th, 7th, 11th, 14th, and 15th notes. They are not ornaments on a common pulse. They are the source of the tempo changes themselves.

Every modulation is algebra. If a 1/5 note at the old tempo must equal a 1/7 note at the new one, I solve for x:

x = T \times \frac{7}{5}

At quarter note = 84, that gives 117.6. The primes (3, 5, 7, 11) give the purest ratios. The composites (14 and 15) let me move between those prime worlds in a single step. Four voices carry four separate chains of these ratios, compounding as the piece unfolds. No two chains are alike, and none answers to the others.

The Machine Comes Closest

The computer is the nearest point on the curve. Csound will hold four tunings and four tempo chains without fatigue, without drift, without a single player listening to the others and bending toward them.

But even the machine does not arrive. A tempo of 84 × 7/5 × 11/3 must be rounded to a sample position. A pitch in 31 or 53 divisions of the octave must be rounded to some fraction of a cent. The error is microscopic. It is still error. The computer does not touch the line. It only shrinks the gap further than any human can.

The Human Approach

And yet people have come close. NeoQuartet of Gdansk played Hypercube, my second quartet, in four tempi and four tunings: 12, 19, 31, and 53. Two brass players once navigated polytempo and poly-just intonation with nothing but their ears. They sat further from the line than the computer does. They also gave the music something the computer cannot give: a human being straining toward the form.

So the machine and the performer approach the ideal from different directions. One comes closer in precision. The other comes closer in meaning. Neither arrives, and the music needs both.

The Work Is the Approach

Socrates never built his city. He did not need to. The pattern was the gift, and every just act afterward was measured against it.

I write my scores the same way. Each one is a pattern laid down in theory, more exact than any realization will ever be. The quartets, the symphonies, the algebra of ratios are the line. Every performance and every render is a point on the curve. The distance between them is not failure. It is where the music lives.