Musical rhythm spectra from Bach to Joplin obey a Mf power law

摘要

Much of our enjoyment of music comes from its balance of predictability and surprise. Musical pitch fluctuations follow a 1/f power law that precisely achieves this balance. Musical rhythms,especially those of Western classical music, are considered highly regular and predictable, and this predictability has been hypothesized to underlie rhythm's contribution to our enjoyment of music.Are musical rhythms indeed entirely predictable and how do they vary with genre and composer? To answer this question, we analyzed the rhythm spectra of 1,788 movements from 558 compositions of Western classical music. We found that an overwhelming majority of rhythms obeyed a 1/f~β power law across 16 subgenres and 40 composers, with β ranging from ~0.5-1. Notably, classical composers, whose compositions are known to exhibit nearly identical 1/f pitch spectra, demonstrated distinctive 1/f rhythm spectra:Beethoven's rhythms were among the most predictable, and Mozart's among the least. Our finding of the ubiquity of 1/f rhythm spectra in compositions spanning nearly four centuries demonstrates that, as with musical pitch, musical rhythms also exhibit a balance of predictability and surprise that could contrib ute in a fundamental way to our aesthetic experience of music.Although music compositions are intended to be performed, the fact that the notated rhythms follow a 1/f spectrum indicates that such structure is no mere artifact of performance or perception, but rather, exists within the written composition before the music is performed. Furthermore, composers systematically manipulate (consciously or otherwise) the predictability in Mf rhythms to give their compositions unique identities.