Scope

The Victorian Pendulum's Drawing

Live audio capture from Spiralyst Lab.

A harmonograph is a Victorian drawing machine: swinging pendulums guide a pen, and as they slowly lose energy the traced figure spirals gently inward. Spiralyst recreates it in pure math — two decaying sine waves, one per axis — producing endless looping rosettes.

The Parlor Wonder

In the late 1800s the harmonograph was a popular parlor wonder: a table-sized contraption of swinging pendulums linked to a pen, left to draw on its own. Set the pendulums going and the pen wove intricate, symmetrical figures; as friction bled the swings of their energy, each loop drew slightly smaller than the last, so the whole pattern spiraled inward to a still point. No two runs were ever quite alike.

The figures are close cousins of the Lissajous curve — in fact a harmonograph trace is essentially a Lissajous figure that fades. When the two pendulums swing at a simple frequency ratio the figure nearly closes on itself; nudge the ratio off a whole number and it never repeats, slowly rotating into a rosette of overlapping loops. The decay is what turns a flat closed curve into a three-dimensional-looking spiral on the page.

It is the one mode in this category with no microphone in it: the figure is generated entirely from its own parameters. But every one of those parameters can be wired to the music, so the frequencies, phases, and damping can breathe with a track — a mathematical pendulum that dances to whatever is playing.

The Math

$$x(\theta) = \sin(a\theta + p_1) \cdot e^{-d\theta}$$

The horizontal pen position: a sine wave of frequency \(a\) and phase \(p_1\), multiplied by an exponential decay that shrinks the swing as the angle \(\theta\) advances.

$$y(\theta) = \sin(b\theta + p_2) \cdot e^{-d\theta}$$

The vertical position, with its own frequency \(b\) and phase \(p_2\) sharing the same decay. Integer ratios of \(a/b\) make nearly-closed figures; off-integer ratios spiral into rosettes.

$$\theta \in [0,\; \text{cycles} \cdot 2\pi]$$

The pen sweeps through this many full turns. More cycles means a longer, denser trace before the decay winds it down to the center.

How Spiralyst Lab draws it

Spiralyst Lab traces a single damped sine wave on each axis — one pendulum per direction, not the multi-pendulum sum of a physical harmonograph — and draws the path as a polyline. Freq A and Freq B set the ratio, Phase A and Phase B rotate and open the figure, Decay sets how fast it spirals inward, and Steps sets how far it runs. Every slider is audio-bindable; the Live monitor toggle wires Freq A and Freq B to bass and treble amplitude so the figure morphs between integer ratios as the music plays.

Did you know?

Real harmonographs were both science demonstrations and parlor entertainment, and the spirograph toys many of us grew up with are their geared, pocket-sized descendants — the same mathematics of coupled rotation, captured in plastic gears.

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