So there's this thing called a wave, and waves with wavelengths that form a simple ratio interact to form periodic waveforms that are observably stable.
Pulses too, actually.
That is certainly not the case. The superposition of waves is linear, and waves of separate frequencies do not really "interact" besides being linearly combined. The fact that a waveform is periodic is not perceptible in our ears. We hear the frequencies separately and the ratio between them does not matter.
This is not a theoretical concept, but a physical observation that anybody can check. You can readily try it with a synthesizer. Create two waveforms f(x)=sin(x)+sin(2x) and g(x)=sin(x)+sin((2+e)x) where "e" is a small irrational number. The wave f is periodic and the wave g isn't. They sound just the same: you cannot tell which one is periodic just by hearing them.