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Re: human versus spectral resolution
At 7:49 AM +0200 4/3/08, Beerends, J.G. (John) wrote:
A good theory is never wrong, it only has a limited context in which it
But a bad theory can be wrong.
The point is not that so much that some theories
are wrong, but that all theories are wrong in
George Box is quoted as saying: "All models are
wrong, some are useful". Donald Hebb made a
similar point (see my chapter on pitch models in
the Springer handbook). In other words: asking if
a theory is wrong is the wrong question to ask.
We should ask whether it's useful.
With respect to the original question, Nordmark
(1968, 1970) made a very useful point. The word
'frequency' is commonly used with two meanings.
One is the reciprocal of a time interval, for
example the interval between two events in the
waveform, or the position along the lag axis of
the peak of an autocorrelation function. This,
he called 'phase frequency' (Kneser 1948). The
other is the locus of a spectral feature
determined by Fourier analysis, called 'group
frequency'. To quote Nordmark: "For a time
function of limited duration, [Fourier] analysis
will yield a series of sine and cosine waves
grouped around the phase frequency. No exact
value can be given [to] the group frequency,
which is thus subject to the uncertainty
In practice, the period of a waveform can be
determined with arbitrary accuracy from a limited
segment of waveform (roughly: 2 periods of the
lowest expected fundamental). This can be done
by a machine, and possibly also by the auditory
system. I don't see a reason to believe that the
auditory system would do better than a machine on
such simple tasks.
This does not mean that it's not worth looking at
how the auditory system does these things. For
one thing, knowing how the auditory system does
a simple task such as frequency analysis can
provide inspiration to help develop algorithms
that can then be run on a computer. Those
algorithms do not need to be faithful models of
the auditory mechanism to be useful. For
another, there are more complex tasks such as
speech recognition for which we don't know the
nature of the problem well enough to get a
machine to do it. So I agree wholeheartedly with
Dick Lyon's position that there's lots to be
learned (from a practical point of view) from
studying the auditory system.
Nordmark's point not well known or understood,
even by savvy signal processing specialists. It's
very common to read that the Heisenberg/Gabor
principle (earlier suggested by Helmholtz) limits
the accuracy of frequency measurement, and that
the fact that one can do better is a paradox.
Nordmark JO (1968) Mechanisms of frequency
discrimination. J Acoust Soc Am 44:1533-1540.
Nordmark JO (1970) Time and frequency analysis.
In Tobias JV (ed) Foundations of modern auditory
theory. New York: Academic Press, 55-83.
Kneser (1948). "Bemerkungen über Definition und
Messung der Frequenz." Archiv der Elektrishen
Übertragung 2: 167-169. [I haven't read this: can
anyone confirm if it's relevant?]