Transforming Sound from the Time Domain to the Frequency Domain
Probably the most foundational thing any musician or sound engineer could take the time to understand is sound. And probably the most important way one could do that is to understand Fouriers theorem.
If you ever said to your math teacher “how am I ever going to use this in the real world,” you are about to eat those words. I hope they’re delicious.
Fouriers theorem says that any waveform (i.e. timbre) can be made by adding sine waves at various multiples (i.e. harmonics) of the fundamental (i.e. note).
More mathily – Fouriers Theorem transforms sound from the time domain (the way we see and experience it) and rotates it 90 degrees to look at it sideways in the frequency domain (which actually provides insight).
The six arrows represent the first six terms of the Fourier series of a square wave (they are sine waves!). The two circles at the bottom represent the exact square wave (blue) and its Fourier-series approximation (purple).
Put another way, Fouriers Theorem shows us that sine waves are the atoms of sound.
Isn’t that cool? It doesn’t get much more awesome than that people. All of a sudden, sound is much less mysterious.
And the more you think about it, the more situations it helps you make sense of, the more situations you see people who don’t understand this get things wrong, the less mysterious sound becomes.
(So that’s why they test our hearing with “pure” sign waves, because they’re checking our hearing at a given frequency and don’t want our ability to hear overtones to affect the results.)