I messed around with combining tones of different frequencies and realized that mixing say, 100Hz with 101 Hz, which are different by one cycle per second, have an interference pattern that results in a beat of exactly 1 second.

beatoff1.mxb

And 100Hz with 102Hz results in a beat of 1/2 second.

I tried a few more, and yes, the difference in cycles per second seems to always result in an interference period of 1 second divided by the difference in seconds, or a difference of x means the beat is x Hertz.

So how about 100, 101, and 102?
That's boring, because the respective differences of 1 and 2 just add up and it's not worth even hearing.

And 100 + 102 + 104 is pretty much the same

100 + 101 + 103, however, gets interesting.
The interference of 1 and 3 means three interference peaks per second, with the third being pronounced.

100 + 102 + 103 ends up being pretty similar.

I figured 100 + 101 + 104 would be similar, but actually the fourth beat gets cancelled out and muted, so it sounds like rest - beat - beat - beat - rest - beat - beat - beat.

100 + 102 + 105, a difference of 2 combined with a difference of 5 (and the 'harmonics' differing by 3) creates a more interesting rhythm.

As you would guess by now, 100 + 103 + 105 is the same.

This pattern clearly suggests prime numbers, since multiple will always just cancel each other out or reinforce each other where there are already peaks.
So, as a last sound sketch, let's throw in the first, oh, 8 prime numbers as harmonic intervals above the fundamental.

Still a solid one-second beat with lots of 'jigginess' in the middle.

As a last test, let's combine 100 with 103 and 107, so the primary intervals are 3Hz and 7Hz, with a secondary interval of 4Hz. I can't explain why, but the resulting interference has a beat of 5 per second. It may be some kind of illusion, where the beats aren't of uniform length.

An interesting phenomena in all of the sounds is the resulting pitches which sometimes seem an entire whole step above the fundamental. There is, of course, mathematical explanations for all of this. I wish I knew them, but I don't.