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Upon analysing the Fourier Transform of a musical sound, are there any other applications of the Fourier Transform so obtained? Any ideas would be appreciated.

Edit 1: To clarify the situation, I read somewhere that upon obtaining the FFT of a music from an instrument, since the wave was now essentially just a vector, it could be used as the initial condition in the heat equation. My question in particular is about how exactly this works, and if any other such applications exist.

Also, how could I explain the mathematical difference between two musical instruments by considering the FFT of their sound?

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