A Victorian drawing machine
The harmonograph was a parlour marvel of the mid-1800s. A table holds a pen on one swinging arm and paper on another. Each pendulum moves at its own rate, and the pen traces the sum of their motions while friction slowly bleeds the swing away. An early design is usually credited to the Scottish mathematician Hugh Blackburn, and the French physicist Jules Lissajous studied the same curves with tuning forks and mirrors.
Why music shows up
Musical intervals are frequency ratios. When two pendulums swing in a simple ratio, the pen keeps returning to the same path, and you get a clean, closed figure. The simpler the ratio, the simpler the figure, which is roughly how the ear hears it too.
| Interval | Ratio | What you get |
|---|---|---|
| Unison | 1 : 1 | an ellipse or circle |
| Octave | 2 : 1 | a figure eight |
| Perfect fifth | 3 : 2 | a three-lobed knot |
| Perfect fourth | 4 : 3 | four lobes, busier |
| Major sixth | 5 : 3 | five lobes |
| Major third | 5 : 4 | a dense, woven figure |
The beauty is in the error
Tune the pendulums perfectly and the pen retraces one line as it shrinks inward. Real harmonographs were never perfect, and that's what made them worth watching. A pendulum a fraction of a percent off its ratio drifts slowly out of phase, so the figure turns as it fades, laying down the woven, spiralling bands these drawings are known for. Musicians hear the same drift as a slow beating between two nearly-in-tune notes.
Set Mistuning to zero and release. Then try 1%. The interval is the same; the drawing is completely different.
What the pen obeys
Each pendulum is a sine wave that decays over time. One swings the pen side to side, one swings the paper up and down, and a third, rotary pendulum moves in a small circle that adds to both. With frequencies f, phases p and friction d:
f₁ : f₂ is the interval you choose, with f₂ nudged by the mistuning. r is the rotary pendulum's share. The phases are random on every release, the way your hand never pulls a real pendulum back to quite the same spot twice.