It’s more math theory than music theory but it can be related to music. If you’ve studied music stuff you’ve probably seen the overtone series. Many sounds are produced in a resonance chamber that gives you a spread of frequencies, each with varying loudness. The different loudnesses of these component frequencies are what make different instruments sound different even when they’re playing the same note. This is the same idea of resonance from voice training stuff. What you’re doing when you’re playing with resonance is you’re playing with changing the resonance chamber so that you’re amplifying a new set of frequencies in the jumbled mess of frequencies that is the human voice.

It’s easy to take a bunch of frequencies and then sum them together to make a complex sound, but if I were to put a microphone next to an oboe playing, how could I tell all of the little frequencies that make up that sound? All the microphone sees is vibration. The answer is the fucking BEAST that is the Fourier transform. Absolutely voodoo magic piece of math. You can feed the audio signal straight from the microphone right into the Fourier transform and it’ll spit out all of the component frequencies as well as how loud they are. I could not possibly even begin to exaggerate how powerful this tool is. It. Is. everywhere.

Anyway, mathematicians did what mathematicians do and said “okay Fourier analysis is cool and all, but what is it a special case of? Is it possible to make sense of this stuff in new places outside of where they theory was originally developed?” This is the question abstract harmonic analysis asks.

I’m not a physicist so it’s possible someone here might correct me, but to my limited understanding of physics it’s a lot like how if you start with the theory of general relativity (which allows for any accelerations that you want) and restrict it to just accelerations of 0, you get special relativity for free. Special relativity is just hidden inside general relativity. But historically special relativity came first, so to go from that to the general case, we had to ask the question “what might the theory we already have be hidden inside of?”