1. 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?

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3. A periodic wave no matter how complex can be explained by fourier transforms, if you take the note of a trumpet, and repeat it, and it is the same each time then cutting out the silences in between would yield a complex periodic wave which - as you've probably guessed by now, can be explained by fourier analysis.

Tip, you need to consider the waveform from highest frequency to the lowest, the lowest in this case would be the repitition with which the note is blown.

4. Well, I know that you can take the fourier transform of a whole song, but doing so wouldn't be very useful. Also, taking the transform of a repeating note would give results, but those wouldn't be so easily applicable to notes of different lengths, amplitudes, etc. Even if you ignored amplitude (by making that a seperate curve, for example), some instruments have component frequencies with varying decay rates. Trying to model such an instrument would require more than just repeating the note.

5. The fourier transform works for any continuous function, you only need a periodic function if you are using a fourier series representation.

6. Well, I sai fourier transform, but I meant to say fast fourier transform (or at least discrete fourier transform). What I'm basically trying to do is to take a recording of a note and decompose it in such a way that its characteristic changes over time are preserved.

7. How do you think synthesizers work? You can make a good synthesizer sound like almost anything if you know what you're doing.

Acoustic sounds normally have an attack, sustain and decay and there are probably other superficial divides you could emplace. These are kind of general patterns to the amplitude of the wave that you might expect, starts off high and gradually gets smaller. Higher harmonics might attenuate more quickly so there might be a kind of dispersive effect going on. The exact timbre of a trumpet note is very complex, it would be impossible for a trumpet player to get it exactly the same twice when viewed closely in a waveform analysis.

8. Frequency synthesisers merely generate (mostly now ) by digital means a set of time slots, of which each one can be programmed to produce a positive or negative output voltage which is then filtered to produce what appears to be a continuous analog waveform. There are different types of synthesisers, some can be 'programmed' by sampling and then stretching or compressing the time slots to achieve different frequencies (such as in a digital piano), others produce only simple waveforms such as sine, square, sawtooth, pulse and some other basic waveforms. The sound card in your PC does exactly the same thing. With the right programming it too, can be used as a synth.

9. Yes, I actually know quite a bit about the various methods of sound sythesis (additive, subtractive, sampled, etc.), the AHDSR envelope (which is why I said "ignoring amplitude chages"), etc. I've also written a (nearly) fully functional software additive synthesizer (no percussion). What I'm asking is simply whether a fourier series that changes over time makes sense. I want to take something like that and try to make a synthesizer that falls somewhere between additive and sampled sythesis.

10. Well my understanding is that you could describe beethoven's 9th symphony using fourier, though I myself have only used it (a long time ago & before the PC was around) on simpler waveforms to produce test waveforms, it just becomes more and more complex, have I not heard somewhere of fourier analysis software? - even though I have no experience of such software.

11. The trumpet sound is a non-stationary random signal in that, like you said, it's frequency content changes over time. I don't have any experience with instrumental sounds but have worked a little with voice data, which is also non-stationary. To track the spectrum the approach with voice is too segment the signal into sections about 20ms or less in duration as voice is quasi-periodic in this range. Yes you will get discontinuities which will cause leakage in the DFT, I think this can be minimised by applying a window function such as Hamming etc, or perhaps by zero padding ie pre-whitening

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