Waveforms, Starrett, Hahn

Starrett:
>> It is not so much that rich tones are composed of a fundamental
>> frequency and its overtones as that they can be decomposed this way.
[snip]
>> Joseph Fourier figured out how to add up cosine and
sine
>> waves, which are mathematical representations of the the simplest
kinds of
>> sounds, to obtain any type of periodic function whatsoever, [snip]
>> Describing a complex waveform in terms of its decomposition into
a
>> sum of sine and cosine waves whose frequencies are simple multiples
of
>> the fundamental (such as A 440) is useful, but not necessary, for
>> describing a tone and understanding why a waveform looks and sounds
as it
>> does.

Hahn:
>There's been much excellent discussion on this topic already; I'd just
>like to interject that the reason Fourier-type decomposition of
periodic
>waveforms is useful is because we believe that the inner ear does
>something very similar, therefore it tells us something significant
>(_pace_ Brian, not _everything_) about how we would hear such a sound.

That's very true -- in fact, if one shifts the phases of the Fourier
components of a waveform, without messing with their amplitudes, one can
end up with a completely different-looking wave, but one that sounds
almost identical (especially at low volumes) to the original. If the ear
were really very sensitive to wave shape, one would hear the sound of a
piano or guitar or plucked bass string as continually changing in
timbre; the inharmonic partials can be interpreted as harmonic ones
whose phases are drifting relative to one another. Perhaps in some
subtle sense the timbre of these instruments does continually change,
and perhaps that's why these are my favorite instruments to play, but
the difference between these sounds and resynthesized versions with
exactly harmonic partials (with _any_ relative phases) is subtle enough
so that one can say the the amplitudes of the Fourier decomposition are
primary, and the shape of the wave secondary, in describing the timbre
of the wave. In fact, it is my understanding that the BBE
sound-enhancing hardware works by applying different phase shifts to all
frequencies, so as to eliminate peaks in the total waveform that at loud
volumes could cause distortion in amplifiers, speakers, and the human
ear, while preserving the amplitudes of all components. If the shape of
the wave were very important to musical effect, this procedure could
hardly be called sound-enhancing!

I just picked up a copy of the 1997 edition of Manfred R. Schroeder's
_Number Theory in Science and Communication_ -- another tour-de-force by
him for those who like mathematics but dislike formal proofs -- and the
introduction to chapter 28 asks the question, "For a given amplitude
spectrum (_magnitudes_ of Fourier transform coefficients), how does one
choose the _phase angles_ of the Fourier coefficients in order to
achieve the smallest range of magnitudes in the corresponding inverse
Fourier transform [i.e. in the resynthesized wave - PE]?" Although he
does not mention BBE, he is addressing essentially the same issue (and
uses Galois sequences to do it).

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End of TUNING Digest 1487
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