Whole-number frequency ratios become curves you can see — and, for the Lissajous figures, intervals you can hear.
x(t) = sin(a·t + φ) y(t) = sin(b·t)
Lobe count = the ratio. Count the bumps along the top edge (that is b) and along the side (that is a). A 2:3 figure has 2 vertical × 3 horizontal lobes.
Rational ratio ⇒ closed curve. Because a and b are whole numbers, the pen always returns to its start — the curve closes into a stable loop.
Phase φ rotates and opens the figure: at 0 it can look like a line; near 0.5π it fills out.
Two tones whose frequencies form a simple ratio share many overtones, so we hear them as consonant: octave 1:2, perfect fifth 2:3, perfect fourth 3:4, major third 4:5. The same small integers that draw the tidy figure also make the pleasant chord. Press Play to hear the two tones a×220 and b×220 Hz.
A real harmonograph hangs pens and paper from swinging pendulums. Each axis is a sum of damped sine waves: A·sin(f·t+p)·e-d·t. Friction (damping) shrinks the swing, so the loops spiral inward. A tiny detune from a perfect ratio makes the pattern slowly rotate — that drift is what turns a plain loop into a rosette.