When I think of ring modulation – or multiplication of two bipolar audio signals – I usually think of a complex, polyphonic signal being ring modulated by an unrelated sine wave, producing an inharmonic effect. Indeed, this is what “ring modulator” means in many synthesizers’ effect racks. I associate it with early electronic music and frankly find it a little cheesy, so I don’t use it often.

But if both signals are periodic and their frequencies are small integer multiples of a common fundamental, the resulting sound is harmonic. Mathematically this is no surprise, but the timbres you can get out of this are pretty compelling.

I tend to get the best results from pulse waves, in which case ring modulation is identical to an XOR gate (plus an additional inversion). Here’s a 100 Hz square wave multiplied by a second square wave that steps from 100 Hz, 200 Hz, etc. to 2000 Hz and back.

As usual, here is SuperCollider code:

(
{
    var freq, snd;
    freq = 100;
    snd = Pulse.ar(freq) * Pulse.ar(freq * LFTri.ar(0.3, 3).linlin(-1, 1, 1, 20).round);
    snd ! 2;
}.play(fadeTime: 0);
)

Try pulse-width modulation, slightly detuning oscillators for a beating effect, multiplying three or more oscillators, and filtering the oscillators prior to multiplication. There are applications here to synthesizing 1-bit music.

Credit goes to Sahy Uhns for showing me this one some years ago.

EDIT 2023-01-12: I have learned that Dave Rossum used this technique in Trident, calling it “zing modulation.” See this YouTube video.