Revealing competitive behaviours in music by means of the multifractal detrended fluctuation analysis: application to Bach's Sinfonias

The one-, two- and three-dimensional multifractal detrended fluctuation analysis (MF-DFA) was applied to Bach's Sinfonias, which are characterized by the superposition of three different voices. Each voice, represented as a time series, can be considered as a component of a one-, two- or three-dimensional vector. The one-dimensional MF-DFA was applied to any single voice, while the two- and three-dimensional MF-DFA was applied to the couples of voices and to the triple, respectively. Each voice is characterized by a multifractal degree (MD), indicated by the range of the generalized Hurst exponents; the higher the MD, the larger the amount of heterogeneity and irregularity. Competitive scaling multifractal behaviours in Bach's Sinfonias were revealed; although one (or two) voices showed a relatively high MD, the other two voices, or voice, are characterized by a low MD. Nevertheless, the overall effect of the Sinfonia, measured by the MD of the triple, tends towards homogeneity, or at least to an average between the different competitive scaling behaviour shown by the different voices.

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