# Autocorrelation and the Study of Musical Expression

Skip other details (including permanent urls, DOI, citation information)This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. Please contact mpub-help@umich.edu to use this work in a way not covered by the license. :

For more information, read Michigan Publishing's access and usage policy.

Page 357 ï~~Autocorrelation and the Study 0 Peter Desain and Siebe d+ Music Department City University Northampton Square GB-London EC 1V OHB Centre for Know UtrechtI IL NL ABSTRACT: In performances musical structure is and other parameters. method was designed p h autocorrelation. Peaks the autocorrelation functi repeated standard components u autocorrelation the musical structure. Cai function this domain. Pa remove the multiples of a fundamental period. Introduction In musical performances the performer uses variations of often posed hypothesis is that these expressive variations w M. w........ - -- -- L.w.. w. Â~ _ |....w___ t- _ -- -

Page 358 ï~~Regularity in musical structure will be reflected as pei use autocorrelation as a statistical method to find peri the lengths of structural components. We will assum homogeneous, at least for some time span and at some Autocorrelation If a signal is periodic with a period P, it will resemt statistical measure of resemblance is correlation. By c the same signal delayed by different lags we obtain a When the signal contains a periodic component with;I occurs at this value. Considering our domain we ha autocorrelation [Bowermann 1979, Priestley 1981). T to show changes in periodic structure. We realise this the autocorrelation at t is the autocorrelation in the v where W is the window size and X the signal. Th( otherwise a change in the level of a component with are still many 'old' periods contained in the window. "V in the examples we used a factor p-4. A second reaso signal cannot be assumed to be stationary, which meat C)q - qq ''l.

Page 359 ï~~Partial atjtocorrelation A problem occurs in interpreting the autocorrelation ci it will also be the same after period 2P,, 31', etc. r these levels over and above the regularity originating autocorrelation. Partial correlation determines the corn( the influence of other variables on both of them. In thc of smaller periodicities on the autocorrelation for a ceprr)isdfnda(Bwma19]: p(1,1) Sp (1) k- i p (k) Xp(k- 1 g)p(k-j) P ( k) I k-1 Xp(k- 1 1flpQ) j=1 p(k1) p (k-lgJ) e p (k,k) p(kel,k-j)." This formula depends on a statistically sound autocor uoreltndiet.b tit sp sile ortan

Page 360 ï~~However, the method has severe intnnsic limitations. expressive components at each structural level are more c independent combination of the components, an assumpti this method. Furthermore, no phase information is retair for small windows and we can use the method only for ge the piece, not for testing them statistically. -Vp eq Z3 C'4 f._. O 26 2 4 -Z - =t \Â~.--. - -4 Is oor 3a22222