> It's also common in audio manipulation, e.g. change to the frequency domain in order to modify pitch, then change back to the time domain.
Funnily enough, yes but actually no. For understanding and mathematical proofs the Fourier Transform is obviously essential. But when you first get into audio DSP programming it might seem that the FFT is crucial as well. But virtually all digital audio filters directly operate on the on the time domain.
The problem is that we always work with a sampled signal. And while the Nyquist theorem tells us that as long as our sample frequency is at least twice as high as the highest frequency in our input that our sampling doesn't lose any information, we still have to be aware of it.
When you convert a fixed frequency sampled signal into the frequency domain, you get a sum of a fixed set (co)sine functions, because we're still sampled. Now some operations you can do in this frequency domain exactly and simply. E.g. if 100Hz is part of our fixed set of functions and we wish to subtract a 100Hz signal, we can directly do that on our coefficient. A pitch shift by an exact multiple of the frequency sample delta is possible exactly too, assuming the lowest/highest frequencies shifted off are inaudible.
But you almost never want to do these operations. A classic example of something you might want to do is a low-pass filter. Simple right? FFT to frequency domain, zero out the coefficients above the cutoff frequency, and convert back. No! Zeroing out coefficients is equivalent to subtracting those specific sine waves. But that is not a low-pass. As an example, suppose our sample frequency is such that the FFT's sine wave coefficients are 10Hz apart, and we wish to do a 30Hz low-pass filter. This means that a 25Hz signal should be completely unaffected, but a 25Hz signal can't be represented just using the 10Hz and 20Hz coefficients - you need higher order terms! Thus if we zero those out, we distort our 25Hz signal.
So to solve this in the real-world you get into the difficult topic of filter design, or other audio DSP algorithms that are vastly more involved than a simple FFT.