1. The Universal Power Law in Music Statistics
L. Liu, H. S. Zhang, J. H. Xin, J. R. Wei, and J. P. Huang
Department of Physics, Fudan University, Shanghai , China
Introduction
CDF of Pitch Fluctuation
Autocorrelation Function
The distributions of a variety of physical and social phenomena follow a power law, including the sizes of earthquakes, the sizes of solar flares, the frequencies of words (Zipf's law), the incomes among the population (Pareto principle), and many other quantities. In our recent work, we analyze classical music from seven famous composers who belong to three different musical styles. Interestingly, we observe a universal power law in the cumulative distribution function (CDF) of pitch fluctuations, although the power-law exponents are quite different among all the composers. Furthermore, a power law behavior is found in the autocorrelation functions of both pitch and duration .
The CDFs can be directly calculated from the pitch fluctuation time series. As shown in Fig.2, both positive and negative tails of CDFs for all selected composers follow a power law function as follows,
where the power-law exponents are different among all the composers.It is noted that the exponents of positive tail and negative tail are similar for a specific composer except Haydn.
Autocorrelation is the cross-correlation of a time series with itself, which can be calculated in a time series X(t) as follows,
As shown in Fig.3, the autocorrelation functions of pitch and duration follow a power law decay with the time lag increase. It is noted that for Beethoven’s compositons, the autocorrelation function of duration differs from power law .
Data Analyzed
We analyze the compositions of seven composers from three commom pratice periods of classical music (see Table 1).
Music data for the selected compositions are sourced from the internet and are provided in MIDI (Musical Instrument Digital Interface) format. The MIDI format file stores music information in digital forms which can give the pitch and duration of every musical note. Here, we denote the pitch time series as f(t) (t=1,2…N), where N is the number of notes in a selected composition. Similarly, the duration time series is denoted by d(t) (t=1,2,…,N). We also introduce a pitch fluctuation to describe the pitch change between two adjacent notes, which is defined as follows,
Table 1 Data analyzed
Fig.3 Autocorrelation of pitch and duration
Fig.2 CDF tails of pitch fluctuation
Conclusion
An universal power law is found in the CDFs of pitch fluctuation for all the selected composers, although the power-law exponents are quite different.
The autocorrelation functions of both pitch and duration show a power law behavior, which means both pitch and duration have long-range correlations.
We conclude that, despite of great difference in commom pratice period, the compositions show the universal properties in statistics.
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Fig.1 Selected composers in pitch-duration space