Leigh M. Smith

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 accelerandos and ritardandos, 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 is then used to compute an accompaniment rhythm by inverting the wavelet transform of a modified magnitude and original phase back to the time domain.

Model cortical responses for the detection of perceptual onsets and beat tracking in singing

We describe a biophysically motivated model of auditory salience based on a model of cortical responses and present results that show that the derived measure of salience can be used to identify the position of perceptual onsets in a musical stimulus successfully. The salience measure is also shown to be useful to track beats and predict rhythmic structure in the stimulus on the basis of its periodicity patterns. We evaluate the method using a corpus of unaccompanied freely sung stimuli and show that the method performs well, in some cases better than state-of-the-art algorithms. These results deserve attention because they are derived from a general model of auditory processing and not an arbitrary model achieving best performance in onset detection or beat-tracking tasks.