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 start by making RPs of overlapping short sections of a music track, then evaluate the similarity of the RPs. The figure on the left shows the RP of RPs for the first movement of Beethoven’s Piano Sonata no. 23 (Appassionata); blue denotes the greatest similarity. Each short RP spanned a 4 s window offset from the previous one by 1 s. The banding indicates the recurrence of sections with different themes. Applying a similarity threshold for the large-scale RP accentuates that recurrence in the figure on the right. The diagonal traces circled in yellow, for example, show where a 40-bar stretch is repeated, albeit shifted by a major third, while the chorus circled in red repeats its motifs with increasing tempo. With the flexibility to define “closeness” in multiple ways, the technique is versatile, able to emphasize other features and analyze other types of data. (M. Fukino, Y. Hirata, K. Aihara, Chaos 26, 023116, 2016, doi:10.1063/1.4941371 .)