Modelling Rhythm Perception by Continuous Time-Frequency Analysis

The use of linear phase Gabor transform wavelets is demonstrated as a robust analysis technique capable of making explicit many elements of human rhythm perception behaviour. Transforms over a continuous time-frequency plane (the scalogram) spanning rhythmic frequencies (0.1 to 100Hz) capture the multiple periodicities implied by beats at diierent temporal relationships. Wavelets represent well the transient nature of these rhythmic frequencies in performed music, in particular those implied by agogic accent, and at longer timescales , by rubato. The use of the scalogram phase information provides a new approach to the analysis of rhythm. Measures of phase congruence over a range of frequencies are shown to be useful in highlighting transient rhythms and temporal accents. The performance of the wavelet transform is demonstrated on examples of performed monophonic percussive rhythms possessing intensity accents and rubato. The transform results indicate the location of such accents and from these, the inducement of phrase structures. 1 Rhythm as a Signal Despite a history of computational approaches to rhythm perception and production, the treatment of a rhythm as a signal applicable to digital signal processing methods has not been widespread. Notable recent exceptions have been Todd's approach of applying banks of Mexican hat lters MH80] to an acoustic signal towards an auditory primal sketch of rhythm MT94], and Desain's use of au-tocorrelation to determine periodicities in rhythm DdV90]. Engineering approaches of representing an acoustical phenomenon as a signal have brought a conceptual rigour and considerable application to the eld of acoustics and music. Careful concep-tualisation of musical rhythm in signal processing terms furthers our understanding and utilisation of rhythm in musical contexts. Equally careful consideration of psychological and musicological issues is necessary to build accurate, useful signal representations. The use of signal processing techniques uncovers the information inherent in the rhythm prior to our perceptual processing. The time perception eeects such as masking Cow84] and top-down expectancy from subjective rhythmisation Fra82], are considered in this paper as processes that occur in parallel and with respect to a rhythmic frequency map. 1.1 Rhythm Frequencies Theorists have argued for representations of musical rhythm with respect to the tactus LJ83], and that a rhythm can be decomposed into a hierarchy of rhythmic strata of increasing time spans, subdividing the tactus. With a constant tempo, and no dynamic or timing accents, a repetitive beat (an isorhythm) can be considered as a single period-icity with an underlying frequency f …