Fourier Transforms

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Well, I know what the Fourier transform is and what it does, but my question is this: What about for locally periodic waveforms? For example, imagine the waveform of a trumpet playing a single note. You might think at first that it'd be periodic, and for any small enough slice of time, it basically would be, but over time the spectrum would change (near the beginning and end especially). I know you could disect the note into small parts and use the Fourier transform on each one, but wouldn't that leave discontinuities where they were joined? Could a time-varying spectrum (or something similar) be used instead?