A machine that draws the shape of a fading sound.
In the drawing rooms of Victorian Britain a harmonograph was a small table of swinging weights that held a pen against a moving sheet. You set it going, stepped back, and it drew — with no hand to guide it — the quiet mathematics of two motions slowly agreeing to stop.
The design is usually traced to 1844 and the Blackburn pendulum — a bob hung so it could swing in two directions at once. Later cabinetmakers and instrument-sellers turned the idea into a parlour amusement: brass, mahogany, and a pot of india ink.
A lateral pendulum rocks the paper east and west. A second, rotary pendulum carries the pen north and south. The nib simply records their sum. Two honest swings, added together, become a figure no one could draw by hand.
Because friction is always at work, each swing loses a little height with every pass. The loops tighten, the rosette draws inward, and after a minute or so the pen comes to rest at the centre — the whole performance spiralling quietly to stillness.
No two plates are ever the same. The figure depends on how the weights were released, the phase between them, and a frequency ratio held near, but never exactly on, a simple fraction. That tiny imperfection is what makes the web slowly turn.
Two sines, gently dying.
Strip away the brass and a harmonograph is four decaying sinusoids — two for each axis. The pen's position at time t is nothing more than their sum, multiplied by an envelope that fades with friction.
The pair of frequencies on each axis sets the Lissajous figure; the term e−δt is the friction that bleeds the swing away; and holding f1:f3 a hair off a whole ratio — say 3 : 2 — makes the rosette slowly precess into its drifting web.
The figure is finished the moment the pendulums forget they were ever moving.