Zipf-Mandelbrot Scaling in Korean Court Music: Universal Patterns in Music

TL;DR

This study demonstrates that Zipf-Mandelbrot scaling laws, previously observed mainly in Western music, also apply to Korean court music, suggesting a universal pattern in musical structures across cultures.

Contribution

It provides the first evidence that Zipf-Mandelbrot law applies to non-Western music, specifically Korean court music, and explores how musical evolution influences these scaling laws.

Findings

Korean court music fits Zipf-Mandelbrot law well, especially for pitch-duration pairs.

Musical evolution over centuries smooths variations, reinforcing Zipfian patterns.

Joint distributions of independent Zipfian data sets also follow Zipf-Mandelbrot law.

Abstract

Zipf's law, originally discovered in natural language and later generalized to the Zipf-Mandelbrot law, describes a power-law relationship between the frequency of a Zipfian element and its rank. Due to the semantic characteristics of this law, it has also been observed in musical data. However, most such studies have focused on Western music, and its applicability to non-Western music remains not well investigated. We analyzed 43 Korean court music pieces called Jeong-ak, spanning several centuries and written in the traditional Korean musical notation Jeongganbo. These pieces were transcribed into Western staff notation, and musical data such as pitch and duration were extracted. Using pitch, duration, and their paired combinations as Zipfian units, we found that Korean music also fits the Zipf-Mandelbrot law to a high degree, particularly for the paired pitch-duration unit. Korean music has evolved collectively over long periods, smoothing idiosyncratic variations and producing forms that are widely understandable among people. This collective evolution appears to have played a significant role in shaping the characteristics that lead to the satisfaction of Zipf-Mandelbrot law. Our findings provide additional evidence that Zipf-Mandelbrot scaling in musical data is universal across cultures. We further show that the joint distribution of two independent Zipfian data sets follows the Zipf-Mandelbrot law; in this sense, our result does not merely extend Zipf's law but deepens our understanding of how scaling laws behave under composition and interaction, offering a more unified perspective on rank-based statistical regularities.