Subject: Re: First moment of a spectrum From: "James W. Beauchamp" <j-beauch(at)UX1.CSO.UIUC.EDU> Date: Mon, 12 Jun 2000 10:41:32 -0500
Al, Eric, and all, A number of researchers have been using a definition for the "spectral centroid" as a physical measure closely correlating with perceptual "brightness". I think centroid is the same as the first moment. One definition that we use is SUM(k=1,N) A[k]*f[k] fc = -------------------- SUM(k=1,N) A[k] where k is the spectral component, f[k] is its frequency, and A[k] is its amplitude. N is the number of components. If f[k] is a harmonic series, i.e., f[k] = k*f0, then SUM(k=1,N) k*A[k] fc = ----------------- * fo SUM(k=1,N) A[k] fc can also be thought of as the relative-amplitude-weighted average frequency of the spectrum. A normalized version of fc is NC = fc/fo. The minimum value of fc happens when all A[k] = 0 except A, and then fc = fo. On the other hand if all A = 0 except A[j], where 1 <= j <= N, then fc = j*fo. For these case, NC = 1 and j, respectively. In order to cause the centroid to go to zero when only A is nonzero, we sometimes use the definition fc' = fc - fo = fo*(NC - 1) . I have found the centroid to be a very useful parameter for sound synthesis, particularly for brass sounds. It has also been used a lot as a primary physical correlate in MDS (multi-dimensional-scaling) studies. McAdams, Menneguzzi, and I  used it in a study to determine which parameters are most significant as judged by the ability of subjects to detect their absence. What we did was eliminate centroid changes over the durations of several musical instrument sounds. Our conclusion was that this is a highly detectable modification for most instruments. Lately it appears that some researchers are using higher-ordered moments for automatic instrument classification. Fraser and Fujinaga  have used spectral moments 0 through 3 for timbre recognition and Brown  used spectral centroid and time domain moments for automatic identification of instruments. It appears that the centroid, and perhaps higher-ordered moments, are pretty useful for instrument analysis, synthesis, perception, and identification. Jim Beauchamp j-beauch(at)uiuc.edu  McAdams, S., Beauchamp, J, and Menneguzzi, S., 1999. "Discrimination of musical instrument sounds resynthesized with simplified spectrotemporal parameters", J. Acoust. Soc. Am., Vol. 105, No. 2, pp. 882-897.  Fraser, A. and Fujinaga, I., 1999. "Toward real-time recognition of acoustic musical instruments", Proc. 1999 Int. Computer Music Conf., pp. 175-177.  Brown, J. C., 2000. "Automatic identification of musical woodwind instruments using pattern recognition" (A), J. Acoust. Soc. Am., Vol. 107, No. 5, pt. 2, p. 2818.