Re: First moment of a spectrum ("Alain de Cheveigne'" )

Subject: Re: First moment of a spectrum
From:    "Alain de Cheveigne'"  <>
Date:    Tue, 13 Jun 2000 19:39:06 +0200

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 -------------------------------------------------------------- Alain de Cheveigne' CNRS/IRCAM, 1 place Stravinsky, 75004, Paris. phone: +33 1 44784846, fax: 44781540, email: cheveign(at) --------------------------------------------------------------

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