Lognormals, Power Laws and Double Power Laws in the Distribution of Frequencies of Harmonic Codewords from Classical Music

Zipf's law is a paradigm describing the importance of different elements in communication systems, especially in linguistics. Despite the complexity of the hierarchical structure of language, music has in some sense an even more complex structure, due to its multidimensional character (melody, harmony, rhythm, timbre...). Thus, the relevance of Zipf's law in music is still an open question. Using discrete codewords representing harmonic content obtained from a large-scale analysis of classical composers, we show that a nearly universal Zipf-like law holds at a qualitative level. However, in an in-depth quantitative analysis, where we introduce the double power-law distribution as a new player in the classical debate between the superiority of Zipf's (power) law and that of the lognormal distribution, we conclude not only that universality does not hold, but that there is not a unique probability distribution that best describes the usage of the different codewords by each composer.