Scope
The Dance of Two Frequencies
Live audio capture from Spiralyst Lab.
A Lissajous figure plots one signal against another instead of against time. Here the left audio channel drives the horizontal and the right drives the vertical, so two related tones trace a stable looping curve whose shape reveals their frequency ratio and the phase between them.
A Figure From Two Signals
Take away the time axis, feed one signal into the horizontal and another into the vertical, and a moving dot stops drawing a graph and starts drawing a figure. When the two signals are sinusoids whose frequencies form a simple ratio, the dot retraces the same closed curve over and over — an ellipse for a one-to-one ratio, a figure-eight for one-to-two, and ever more elaborate knots and ribbons as the ratio grows. The phase offset between them tilts and opens the figure.
Jules-Antoine Lissajous studied these curves in 1857 by bouncing light off mirrors mounted on tuning forks, using them to compare frequencies by eye in an era before electronics. The same patterns appear in any system where two oscillations meet at right angles, which is why they turn up everywhere from pendulums to laser shows.
In audio they have a practical life as the vectorscope. Plotting left against right turns the display into a phase-and-balance meter: a mono signal collapses to a single diagonal line, a wide stereo mix blooms into a cloud, and a channel flipped out of phase swings to the opposite diagonal — an instant read on whether a mix will survive being summed to mono.
The Math
$$x = A\sin(at + \delta),\quad y = B\sin(bt)$$
The classic Lissajous curve: two sinusoids on perpendicular axes. The frequency ratio \(a/b\) sets which figure you get, and the phase \(\delta\) rotates and opens it.
$$(x, y) = (L[n],\, R[n])$$
In the app the two axes are simply the left and right audio channels — so the figure is whatever relationship the music's stereo image happens to have, moment to moment.
How Spiralyst Lab draws it
Spiralyst Lab reads stereo samples from the system tap — auto-upgrading the capture to stereo if it was mono — and plots each sample pair as a point: left channel horizontal, right channel vertical. Amplitude scales the figure and Auto-gain normalizes it to a rolling peak so it stays sized to the canvas at any volume. With true stereo music you get a living vectorscope; with two band-bound tones you get the textbook closed curves.
Did you know?
This X-Y view is the canvas for oscilloscope music, where artists compose stereo audio whose left-versus-right plot draws recognizable shapes and animations — the waveform you hear is the picture you see. The one-to-two ratio that makes a figure-eight is the same math behind it.
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