Multiresolution Representations of Musical Rhythm

An alternative to representing beat induction of musical rhythm as a set of rules is to instead represent beat induction as a dynamic process.One example of this approach is to represent musical rhythm as a signal that can be analysed with a standard signal processing technique. In the case shown here, the analysis technique is decomposition by the continuous wavelet transform (CWT) of Morlet and Grossman.
The CWT decomposes a rhythm represented by a train of impulses or a continous trace of event "salience" into a hierarchy of periodicities (a multi-resolution representation). These periodicities have a limited duration in time (hence the term "wavelets"). Where those periodicities continue to be reinforced by the occurrance of each note of the performed rhythm, a limited number of periodicities are continued over time, forming "ridges". The ridges can then be used to identify periods in the rhythm which match listeners sense of beat. These ridges can then be recomposed using the inverse wavelet transform to produce moments in time that match the taps of the beat.

An audio example below demonstrates beat induction of singing. The first MP3 file is the original sung fragment (Thanks are due to the Meertens Institute for supplying the Dutch folk songs from the "Onder de groene linde" collection). The second MP3 file is the same song with the beat induced by the multiresolution representation process mixed in as a hi-hat sound.

Singing Example
Accompanied Singing Example

For further details see: