Curves

Harmonograph

Damped pendulums steer a pen across paper: the musical interval between them draws a figure that slowly turns as it fades.

  • vector geometry
  • looping animation
  • works with sound
  • SVG export
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The harmonograph is a Victorian drawing machine: one pendulum steers a pen, another rocks the table with the paper. As both swing and die down, the pen traces a curve that winds inwards like a shell. Its shape depends on the ratio of their frequencies: 2 : 1 gives one figure, 3 : 2 another, so the eye sees what the ear hears, a musical interval.

How it works

Each pendulum is a damped oscillation. Pendulum X moves the pen left and right, pendulum Y moves the table back and forth, and the paper records their sum. A real pendulum hangs in a gimbal and also wanders across its own axis; “Cross swing” sets how much. With “plus a rotary table” the table also circles, driven by a third pendulum whose x and y are a quarter period apart.

x(t) = Σ Aᵢ · sin(fᵢ·t + pᵢ) · e^(−dᵢ·t), y(t) likewise fifth: f₂ : f₁ = 3/2 · (1 + detune) A3 → E4: 2^(7/12) ≈ 1.4983 instead of 1.5

With zero “Damping” the pen draws Lissajous figures, closed at any rational ratio of frequencies. Once “Detune” pulls them slightly apart, the figure no longer closes but slowly turns as it fades, showing whether the pendulums are in tune. The “from notes” interval uses equal temperament, as on a piano: the fifth A3–E4 is 2^(7/12) ≈ 1.4983, almost two cents flat of a pure fifth, so the figure turns even with no detune. “Duration” is how many swings the pen gets to draw.

A bit of history

These curves were studied in 1815 by Nathaniel Bowditch and in 1857 by Jules Lissajous, who watched a spot of light reflected from mirrors on two sounding tuning forks. In 1844 Hugh Blackburn, professor of mathematics at Glasgow, described a pendulum on a V-shaped suspension that traces them; he is usually named as the inventor, though the credit is uncertain. In the 1870s the London maker Samuel Tisley built harmonographs, and by the 1890s they were a parlour craze.

What to tweak

  • “Interval” at “unison 1:1” with “Phase” at 0.25 gives a circle that shrinks into a spiral; “octave 2:1” draws a figure of eight at phase 0 and a parabola at 0.25.
  • Choose “from notes” and type a chord such as C4 E4 G4: the third note drives the table when “Pendulums” is set to “plus a rotary table”.
  • With “Detune” at zero and a pure interval the figure stops turning and only shrinks; raise it and the turns fan out into a rosette.
  • “Taper” thins the line as the swing dies: bold outer turns, a cobweb in the middle.
  • Add the “Glow” effect on top for the trace of an oscilloscope beam; the “XY oscilloscope” layer draws the same figures from stereo sound.

Parameters

Interval
unison 1:1 · octave 2:1 · fifth 3:2 · fourth 4:3 · major third 5:4 · minor third 6:5 · major sixth 5:3 · from notes
Notes
For the “from notes” interval: the ratio of their frequencies, e.g. A3 E4; a third note drives the rotary table
Pendulums
two swinging · plus a rotary table
Cross swing
How much each pendulum also swings sideways
Detune
Moves the frequencies off the exact ratio, so the figure turns
Damping
Duration
How many swings the pen draws
Phase
Offset between the pendulums, as a share of a turn
Line width
Taper
The line thins as the swing dies down
Colouring
one colour · gradient over time
Line colour
Speed
Turns of the phase per animation loop