Harmonograph

Two pendulums hold the pen, two more swing the paper. Set them to a musical ratio and mathematics draws you a knot.

ratio 1 : 1

drawn 0%

Tap the canvas for a brand-new random machine

Musical ratios

Simple ratios close into tidy loops. √2 : 1 is irrational, so the curve never quite repeats — it just keeps filling in forever.

Switch on 🔊 Hear it and each ratio plays as the actual interval: pitch = 110 Hz × frequency, so 2 and 3 really do sound a perfect fifth apart.

The four pendulums

Pen — X frequency 2.00

Pen — Y frequency 3.00

Paper — X frequency 2.01

Paper — Y frequency 2.99

Phase offset 0.00 rad

Friction (decay) 0.0035

Drawing length 2400

What you're drawing

x(t) = A·e^(−dt)·sin(f₁t + p) + A·e^(−dt)·sin(f₃t)
y(t) = A·e^(−dt)·sin(f₂t)     + A·e^(−dt)·sin(f₄t)

Four decaying sine waves. That's the entire machine — and every picture on this page.

A drawing machine from the 1840s

A real harmonograph is a table with a pen on one swinging arm and the paper on another. Start them both moving and the pen traces the combination of two motions. Friction slowly shrinks each swing, so the figure spirals gently inward and the machine signs off on its own.

Victorian scientists built these to make a picture of something you can only hear: the relationship between two musical notes. That's the surprising part — consonance is a small whole-number ratio, and small whole-number ratios are exactly what draws a closed, symmetric figure.

Try this in order — with the sound on

Hit Octave 2:1. Clean, simple, two-lobed — the most consonant interval after unison, and it sounds like one note rather than two.
Now Fifth 3:2. Still closes up, but with more structure. You are looking at a perfect fifth and listening to the same numbers.
Now Whole tone 9:8. Busy and tangled — and it sounds tense too. Dissonance is complexity you can see and hear, from one pair of numbers.
Finally √2 : 1, the tritone-ish irrational. It never closes. Turn the drawing length up to 12,000 and watch it fill the plane forever without repeating.

Why the near-misses are the prettiest

Notice the defaults: 2.00 against 2.01. Setting the paper pendulum almost equal to the pen pendulum makes the whole figure slowly rotate as it draws — the same beat-frequency effect that makes two slightly out-of-tune guitar strings wobble. Nudge any slider by 0.01 and the drawing blooms into a rosette.

Turn the sound on and you can hear that directly: 2.00 and 2.01 are 220 Hz and 221.1 Hz, and the difference between them — 1.1 Hz — is exactly how many times per second the tone pulses. Slide the paper pendulum further from the pen pendulum and the pulsing speeds up while the rosette gains petals. The wobble in your ear and the rotation on the canvas are the same number.

Made something good? Save PNG keeps it. These make excellent posters.