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signal processing Is a "rolling" FFT possible and could it be of use?

Here is one article that does pitch shifting of audio signals. The way how to get higher frequency resolution is well explained:

It works great in practice and you can even get better resolution out of it compared to what you would expect from a rolling window by using the phase difference between the fft's.

That approach goes by the name Short-time Fourier transform. You get all the answers to your question on wikipedia: