A nonlinear look at music

Nonlinear systems commonly come close to repeating themselves without being truly periodic. (See the article by Adilson Motter and David Campbell, Physics Today, May 2013, page 27.) One tool for analyzing such behavior is the recurrence plot (RP), a two-dimensional graph that uses color or other means to quantify how similar a system is at any two times t₁ and t₂. But for long time series, patterns embodied in an RP invariably get lost or distorted. Music tracks, for example, are digitized typically at 44.1 kHz, so the waveform of even a five-minute piece comprises some 13 million points. In such a data set, features over short intervals reveal the tonality, whereas features over longer periods manifest structural elements. Miwa Fukino, Yoshito Hirata, and Kazuyuki Aihara of the University of Tokyo propose a novel way to capture both aspects: an RP of RPs. The researchers...