Time-Frequency Representation of Musical Rhythm by Continuous Wavelets
A method is described that exhaustively represents the periodicities created by a musical rhythm. The continuous wavelet transform is used to decompose an interval representation of a musical rhythm into a hierarchy of short-term frequencies. This reveals the temporal relationships between events over multiple time-scales, including metrical structure and expressive timing. The analytical method is demonstrated on a number of typical rhythmic examples. It is shown to make explicit periodicities in musical rhythm that correspond to cognitively salient “rhythmic strata” such as the tactus. Rubato, including accelerations and retards, are represented as temporal modulations of single rhythmic figures, instead of timing noise. These time varying frequency components are termed ridges in the time-frequency plane. The continuous wavelet transform is a general invertible transform and does not exclusively represent rhythmic signals alone. This clarifies the distinction between what perceptual mechanisms a pulse tracker must model, compared to what information any pulse induction process is capable of revealing directly from the signal representation of the rhythm. A pulse tracker is consequently modelled as a selection process, choosing the most salient time-frequency ridges to use as the tactus. This set of selected ridges are then used to compute an accompaniment rhythm by inverting the wavelet transform of a modified magnitude and original phase back to the time domain.
Leigh M. Smith and Henkjan Honing
Journal of Mathematics and Music, 2(2), 2008 pages 81-97