Harmonograph
A Victorian drawing machine, running live: two damped pendulums steer a glowing nib over the paper and a decaying Lissajous figure spirals into the centre in about thirty seconds. Then the sheet slides away and a new harmony begins. Ratios the ear calls consonant close on themselves after a few turns; detune one by a few cents and the figure never closes, turning slowly instead. Drag on the paper to push the pendulums yourself.
In 1844 the Scottish mathematician Hugh Blackburn hung a weight from a Y-shaped cord and noticed that, set swinging, it traced looping figures that depended on how the two arms of the Y were tuned. Within a few decades the idea had become a piece of parlour furniture: a wooden table with two or three heavy pendulums hanging through it, one moving a pen, one moving the paper. Set them swinging, lower the pen, and a figure appears that no hand could draw — dense, symmetrical, slowly shrinking as friction takes the swing away.
The surprise is what decides the shape. When the two pendulums swing in a simple ratio — once each, twice against once, three against two — the pen returns to exactly where it started after a few turns and the figure closes on itself. Those are the same ratios that make musical intervals sound consonant: 2:1 is an octave, 3:2 a perfect fifth, 4:3 a fourth, 5:4 a major third. The Victorians sold harmonographs precisely as a way to see harmony. Push one pendulum a few cents out of tune and the figure never quite closes; every lap lands a little to one side of the last, and the drawing turns slowly as it decays, like a sound beating.
The tiles under the paper measure this from the same equations that draw the figure. The ratio is reduced to the simplest fraction that fits it; if one fits, the page tells you how many turns until the pen comes home, and if none does, it says the figure never closes. Next to it, the miss after one lap is measured directly: the pen is run for one nominal lap with friction switched off, and the distance by which it misses its own starting track is reported as a share of the figure's width. Zero for a fifth. Not zero for a fifth that is two cents sharp.
The same mathematics turns up well beyond the drawing room. Lissajous figures were how engineers compared two frequencies on an oscilloscope before frequency counters were cheap; the slow rotation of a near-unison figure is the visual twin of the beats a piano tuner listens for; and damped oscillation, the exponential fade every curve here follows, describes a car's suspension, a guitar string and the needle of an old meter settling on its reading.
Two dippers in a tank of light. Where their waves meet, the water doubles — or stops dead. Drop a stone, drag a wake, paint a wall.