On the emergence of Zipf ’s law in music

Abstract

Zipf's law is found when the vocabulary of long written texts is ranked according to the frequency of word occurrences, establishing a power-law decay for the frequency vs rank relation. This law is a robust statistical property observed even in ancient untranslated languages. Interestingly, this law seems to be also manifested in music records when several metrics – functioning as words in written texts – are used. Even though music can be regarded as a language, finding an accurate equivalent of the concept of words in music is difficult because it lacks a functional semantic. This raises the question of which is the appropriate choice of Zipfian units in music, which is extensive to other contexts where this law can emerge. In particular, this is still an open question in written texts, where several alternatives have been proposed as Zipfian units besides the canonical use of words. Seeking to validate a natural election of Zipfian units in music, in this work we find that Zipf's law emerges when a combination of chords and notes are chosen as Zipfian units. Our results are grounded on a consistent analysis of the statistical properties of music and texts, complemented with theoretical considerations that combine different reference models, including a simple model inspired in the Lempel–Ziv compression algorithm that we have devised to explain the emergence of Zipf's law as the consequence of languages evolving into more efficient forms of communication.