McLaren on Fourier analysis

> The problem with this way of thinking about music
> and sound is that the method you're using (Fourier
> analysis) is incapable of ever giving you *anything*
> BUT a set of perfectly harmonic frequencies. If you
> Fourier-analyze a set of inharmonic frequencies, your
> output will be a set of perfectly harmonic frequencies--
> albeit an infinite number of 'em.

I am not familiar with the works that Mr. McLaren is cricizing, but I
feel that there are is some mathematical imprecision here.

There are various types of Fourier analysis. It seems that the type of
analysis referred to here is Fourier series analysis. McLaren claims
that "If you Fourier-analyze a set of inharmonic frequencies, your
output will be ..." This is a somewhat nonsensical statement in that
you cannot perform a Fourier series analysis on a non-periodic
waveform. The technique is only *defined* for periodic waveforms. So
McLaren's general point, that Fourier series analysis "limits the types
of sounds which can be usefully analyzed" has the right spirit, but is
in detail wrong. In fact Fourier series analysis limits the types of
sounds which can be analyzed at all, usefully or otherwise.

My point is that there are no spurious results to the analysis, since it
is only defined for periodic signals. Periodic signals are those that
can be expressed as a superposition of harmonically related signals.

Other types of Fourier analysis can deal with aperiodic signals. For
instance, if you take a long-term FFT of a gong sound, you may be able
to see its inharmonic structure quite nicely. The tradeoff that you
have made here is that you have very little time resolution with a
long-term FFT, so you do not know much about how the spectrum
dynamically evolves.

Ben Denckla

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