Artificial Wasteland · Instruments · hearable mathematics

Period-Doubling, Played

One knob. Turn it up and a steady pulse splits — one beat, two, four, eight — and then breaks into chaos. The famous route to chaos, made of sound. The rhythm you hear repeats with exactly the period of the equation.

This is the bifurcation diagram of x → r·x·(1−x) — for each setting of the knob r (left→right), the dots show every value the orbit endlessly returns to. One dot is a steady beat. Two dots, a gallop. It doubles — 2, 4, 8 — faster and faster, then shatters into the chaos on the right. Drag the dial (or press play) and hear the slice you are standing on.

r = 3.200 · period 2
drag the dial ↔
chaos meter · Lyapunov λλ = −0.92 · locked

Each beat is the next number the equation spits out, turned into a note. When the orbit is periodic, the row of notes repeats — and the louder downbeat marks the bar, so you hear the meter: two beats, four, or the three of the waltz hiding inside the chaos. When it's chaotic, the notes never come back and there is no bar at all.

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What's a fact, what's a choice. The sequence of beats, their period, and where it doubles are the equation — fixed, not chosen. The pitch each number maps to (a pentatonic scale), the tempo, and the root are the free choices, the same for every setting; change them and the melody changes but the period never does.

Here is the deep fact, computed in your browser as you load the page. The places where the period-2ⁿ beat is most perfectly locked sit at parameters Rₙ; the gaps between them shrink by a constant factor every time the beat doubles — and that factor is the same for almost any equation you could have used. It is Feigenbaum's constant δ.

computing…

What you are playing

Take a number between 0 and 1, and a knob r. Multiply: x → r·x·(1−x). Feed the answer back in and repeat, forever. This is the logistic map — the simplest equation that does something genuinely complicated, written down in this form by the biologist Robert May in 1976 as a toy model of a population that grows and starves.

For a gentle knob the population settles to one steady value — one note, struck over and over. Turn the knob up and the steady value goes unstable and splits in two: the population now alternates high-low-high-low, and you hear a two-beat gallop. Turn it further and each of those splits again — four beats — then eight, sixteen, the doublings crowding closer and closer together until, at r∞ = 3.5699456…, the period becomes infinite and the sound falls into chaos: a stream of notes that never repeats, deterministic but unpredictable.

The period of the rhythm is the period of the orbit. To double the beat is to turn the one knob past the next fork in the tree.

And chaos is not the end of structure. Sail out past r∞ into the noise and, at exactly r = 1 + 2√2 = 3.8284271…, the chaos snaps shut into a clean three-beat waltz — a window of order in the middle of the storm, born in a single instant (a tangent bifurcation). Then it period-doubles again — 3, 6, 12 — back into chaos. The waltz preset lands you just inside it.

The number hidden in the doubling

Measure how fast the forks crowd together — the ratio of one gap between doublings to the next — and you get a number: δ = 4.66920160910299…, Feigenbaum's constant. The astonishing part, which the panel above re-derives live, is that this number barely depends on the equation. Swap the parabola for a sine wave, or almost any other smoothly-humped map, and the route to chaos crowds together at the same rate δ. It is a universal constant of the way order falls apart — discovered by Mitchell Feigenbaum in 1975 on a pocket calculator, and proved universal by Oscar Lanford in 1982 with a computer-assisted proof.

The chaos meter under the dial is the other rigorous number here: the Lyapunov exponent λ, the average stretching rate ⟨ln|r(1−2x)|⟩ along the orbit. When λ < 0 nearby beats pull together and the rhythm locks; when λ > 0 they fly apart and no rhythm can hold — that is the definition of chaos. At full tilt, r = 4, it reaches exactly ln 2: each beat throws away one bit of where you started.

What is real, and what is a choice

The orbit, its period, the forks, δ and λ are the equation — not an imitation of it. Every beat is the next iterate of r·x·(1−x), run live; the period you hear is the period the math has; the bifurcation diagram is plotted by iterating the map in your browser. δ is computed here from the superstable cascade and cross-checked against an independent 50-digit computation in the repo; the landmarks r = 3, 1+√6, 1+2√2, r∞ are exact or published values.

The free choices are the sound, not the structure. Mapping each number to a pitch on a pentatonic scale, the tempo, the root note, and the marimba-ish timbre are all aesthetic — chosen once, applied identically at every setting of the knob. They change which tune you hear; they cannot change when it repeats. Near a fork the orbit takes a long time to settle (critical slowing-down), so the live period readout there reads honestly as "settling / chaos" rather than faking a clean number.

Sources

An exhibit in the Wasteland's instrument room — hearable mathematics, where every sound is the object. It is the playable companion to the stratum The Number Hidden in Every Map, which shows the same cascade as a diagram you can zoom and proves δ stops being universal when you change the shape of the hump.