Musical behaviors—singing, dancing, and playing instruments—date back to Neanderthals, and have been a part of every human culture as far back as we know. People experience great enjoyment and pleasure from music, and music theorists have argued that this enjoyment stems in part from the structural features of music, such as the generation and violation of expectations... Mathematics has often been used to characterize, model, and understand music, from Schenkerian analysis to neural topography; and geometric models of tonality. One particular mathematical relation that has received attention in music is the 1/f distribution, which Mandelbrot termed “fractal.” 1/f distributions have been found to be a key feature of a number of natural and sensory phenomena. Analyzing the frequency of several natural disasters, including earthquakes, landslides, floods, and terrestrial meteor impacts, reveals an inverse log-log linear (fractal) relation between the frequency and the intensity of the events.
Here is the abstract:
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/fβ 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 contribute 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 1/f rhythms to give their compositions unique identities.