Take two vibrations — one horizontal, one vertical — and let a point obey both at once. The path it traces is a Lissajous curve. This is not abstraction: it is what an oscilloscope literally draws when you feed it two signals. The same mathematics that determines whether two notes sound consonant or dissonant also determines whether the curve closes into a knot or fills the entire plane. What you see below and what you hear are the same thing, rendered through different senses.
When the ratio a : b is rational — expressible as a fraction of integers — the curve eventually closes. It returns to its starting point. The simpler the fraction, the fewer loops before closure: 1:2 closes in one breath, 7:11 wanders longer but still comes home.
When the ratio is irrational, the curve never closes. It will, given infinite time, pass arbitrarily close to every point in the rectangle. This is ergodicity: the time average equals the space average. The path of a single point, if you wait long enough, reconstructs the entire available space. An infinitesimal change in the frequency ratio — from 1.5 to 1.5000...1 — transforms a three-lobed figure into a space-filling orbit. The boundary between order and chaos is not a wall. It is a number: rational or not.
Helmholtz understood this in the 1860s. Two tones whose frequencies share a simple ratio produce overtones that align rather than interfere. The basilar membrane in your cochlea, a fluid-filled spiral roughly 35mm long, performs a physical Fourier transform — different positions resonate at different frequencies. When overtones align, fewer sensory neurons fire conflicting signals. The result is what we call consonance. When they clash, we hear beating, roughness, tension. The geometry on screen is the same phenomenon: aligned overtones produce simple, closed figures; misaligned ones produce complexity that never resolves.
Of all irrational numbers, the golden ratio φ = (1 + √5)/2 is the most irrational — the hardest to approximate by rationals, the slowest to converge in its continued fraction expansion (all 1s). Set the frequency ratio to φ and watch: the curve fills its rectangle more uniformly than any other irrational ratio. It is the most ergodic orbit, the most dissonant interval, the farthest thing from music. And yet there is something beautiful in watching it weave — a patient, systematic exploration of all available space, never repeating, never resting, never arriving.
The boundary between a closed orbit and an ergodic one is infinitely thin and everywhere dense. Between any two rationals lies an irrational; between any two irrationals lies a rational. Order and chaos are not separated by a wall. They are woven into each other, inseparable at every scale. Perhaps this is why music moves us: it lives precisely at this boundary, offering just enough structure to be intelligible and just enough complexity to be alive.