Re: First moment of a spectrum

["Alain de Cheveigne'" <Alain.de.Cheveigne@xxxxxxxx>]

Dear Al & all,

Your interpretation is correct as others pointed out.  If the spectrum were
cut out of a piece of tin, the first moment would correspond to the
frequency at which it balanced.  However there are various ways of mapping
frequency and amplitude to the shape (or gauge) of the piece of tin, and
that makes things a bit complicated.

The value of the first moment is not the same if you calculate it from a
power spectrum, a magnitude spectrum, a cubic-root power spectrum, etc..
That's easy to understand: a power spectrum emphasizes high-amplitude parts
of the spectrum, and so "pulls" the moment towards them. It is also not the
same if you count frequency on a linear scale or a log scale, or along some
psycho-physically motivated scale like the Munich or Cambridge scales (to
use Hartmann's terminology).  That too is easy to understand.  Imagine two
equal amplitude components, one at 1000 Hz and the other at 4000.  The
moment is 2500 Hz calculated with a linear axis, and 2000 with a log axis.

It is not obvious which choice should be made.  In signal processing one
usually uses a linear axis, but if the aim is to predict a psychophysical
dimension it might make sense to scale the frequency axis logarithmically,
as this better matches the distribution of channels in the auditory
periphery.  However a log frequency scale gives excessive weight to very
low frequency components (and infinite weight to the DC component).  Bark
and Cam scales behave more reasonably in that respect, but they are more
complex.

For the amplitude axis, power is nice from a signal-processing point of
view, but it puts very strong emphasis on high-amplitude parts of the
spectrum.  Magnitude is more reasonable in that respect, and cubic-root
power possibly even better (it is a better ingredient to calculate a
loudness predictor).  von Bismark (1974, Acustica 30, 159-172) proposed a
predictor of sharpness that in effect scales frequency to a sharpness scale
g, and then weights it by loudness density.

The message is that there's more than one way of doing it.  The simpler
methods are nice because they are simple, but it is good to keep in mind
that they are not unique, nor necessarily the best predictors of a
perceptual quantity.

Alain

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Alain de Cheveigne'
CNRS/IRCAM, 1 place Stravinsky, 75004, Paris.
phone: +33 1 44784846, fax: 44781540, email: cheveign@ircam.fr
http://www.ircam.fr/equipes/pcm/cheveign
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