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

2026-02-15

AI Summary

This study presents the first systematic investigation into the applicability of the Zipf–Mandelbrot law in non-Western music, focusing on 43 pieces of Korean court music (Jeong-ak) spanning several centuries. By digitally transcribing traditional notation into Western staff notation, the authors extracted pitch and duration data to construct joint pitch–duration units for statistical fitting analysis. The results demonstrate that these joint units conform significantly to a Zipf–Mandelbrot power-law distribution, outperforming models based on either dimension alone. This finding not only affirms the cross-cultural universality of this scaling law but also suggests collective evolutionary processes as a potential mechanism underlying its emergence, thereby advancing our understanding of the joint distributional behavior in composite ranked data.

Technology Category

• Natural Language Processing: Lexical Semantics and Morphology
• Data Mining & Knowledge Management: Data Compression
• Application Domains: Humanities & Computational Social Science

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Ethical and legal aspects of web-scale data analysis, uses and collection practices
• Security and Privacy: Large-scale security measurements

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.

Problem

• Zipf-Mandelbrot law
• Korean court music
• cross-cultural universality
• musical scaling
• non-Western music

Innovation

• Zipf-Mandelbrot law
• Korean court music
• pitch-duration pairing
• cross-cultural universality
• joint Zipfian distribution