In this study, sequences of musical notes from various pieces of music are converted into one-variable random walks (here termed ‘music walks’). Quantitative measurements of the properties of each musical composition are then performed by applying Hurst exponent and Fourier spectral analyses on these music-walk sequences. Our results show that music shares the similar fractal properties of a fractional Brownian motion (fBm). That is, music displays an anti-persistent trend in its tone changes (melody) over decades of musical notes; and music sequence exhibits generally the 1/fβ-type spectrum (fractal property), with apparently two different β values in two different temporal scales.
Physica A: Statistical Mechanics and its Applications 380(1) ： p.418-428