Sketch · 2026-05-01
Lissajous, with sound.
Plot a sine wave on the x-axis against another on the y-axis and you get a curve. When the two frequencies sit in a small whole-number ratio, the curve closes after a single period and stays put. Push the ratio off, and it slips, slowly sweeping the square.
Ratios
Headphones recommended. Audio plays in stereo with the X tone on the left and the Y tone on the right.
Note
The curve drawn here is the parametric path (sin(2π·fx·t + φ), sin(2π·fy·t)). When fx and fy are commensurable, the figure closes after the least-common-multiple period and the trace lands exactly on top of itself. The picture stops moving. Your ear hears a single, consonant interval.
Push one slider by a single hertz and the ratio becomes irrational. The curve no longer closes. Each new period lands a hair off the last one, and the figure slowly rotates through every position it could occupy. The audio side of this is the beat frequency: a slow wow as the two phases drift past each other.
Lissajous figures are the reason mid-century electronics shops kept an oscilloscope in X-Y mode on the bench. Feed an unknown frequency into one input and a calibrated reference into the other; tune the reference until the figure freezes; read the ratio off the lobes. Frequency counters made it obsolete in the 1970s, but the trick still works.
Source
Lissajous, J. A. (1857). Mémoire sur l'étude optique des mouvements vibratoires. Annales de Chimie et de Physique, 51, 147–231. The figures were first described by Nathaniel Bowditch in 1815; Lissajous gave them their full optical treatment four decades later.