Neil P. McAngus Todd
Dept. of Music, City Univ., Northampton Square, London EC1V 0HB, England
The rhythm of a piece of music or a poem can be described by hierarchical tree structure. A new application of wavelet analyses to rhythm in music and speech is described whose result is a hierarchical structure [N. P. McAngus Todd, J. Acoust. Soc. Am. 92, 2380 (A) (1992)]. This structure has two complementary components that are obtained from the projection of loci, in the frequency-time plane, of zero crossings of the slope of the energy flux surface in the energy-time plane: (a) a segmentation structure, corresponding to positive second derivatives at the zero crossings; (b) a stress structure, corresponding to negative second derivatives at the zero crossings. The value of the stress is given by the second derivative, which has a ``Mexican hat'' impulse response. These structures resemble the grouping/metrical structures of the theory of F. Lerdahl and R. Jackendoff [A Generative Theory of Tonal Music (MIT, Cambridge, MA, 1983)]. Example analyses of musical and speech performance are demonstrated for this method and evidence is provided that the resultant structures have a reasonable perceptual basis. The possibility of using this technique for the synthesis of rhythm using a cascade structure is discussed.