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

Byeongchan Choi, Junwon You, Myung Ock Kim, Jae-Hun Jung

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.

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

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.
Paper Structure (23 sections, 4 theorems, 46 equations, 5 figures, 4 tables)

This paper contains 23 sections, 4 theorems, 46 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dataset and Zipfian Units
  3. Data Preprocessing and Definition of Zipfian Units
  4. Zipf-Mandelbrot law
  5. Zipf-Mandelbrot law and its slope in the log-log plane
  6. Paramter estimates
  7. Data Analysis
  8. Summary of statistics
  9. Pitch.
  10. Duration.
  11. (Pitch, duration).
  12. Experimental settings and overview
  13. How to read Tables \ref{['tab:zm_fit_summary']}--\ref{['tab:song-all']}.
  14. Global union-level fits
  15. Instrument-level fits
  16. ...and 8 more sections

Key Result

Proposition 5.2

Fix $r>0$ and $\varepsilon_1,\varepsilon_2>0$. Then there exist $M_1,M_2\in\mathbb{N}$ such that for every $N_1\ge M_1$ and $N_2\ge M_2$ there exists an integer $c\in\mathbb{N}$ for which In particular, the curve $\{(r,A^{-1}(r))\;|\;r\ge 0\}$ is the continuous analogue of the rank--frequency plot obtained by sorting lattice samples by decreasing $f$.

Figures (5)

  • Figure 1: Interpretation of Jeongganbo notation. Korean traditional pitch names (yulmyeong) are mapped to corresponding Western pitches, with Hwang (黃) corresponding to E$\,\flat$ 4. Duration is represented by the spatial structure of the Jeonggan: one square cell indicates one beat, which may be subdivided into two or three smaller units and further subdivided recursively. The placement and grouping of pitch symbols within these divisions specify both pitch and duration, allowing Jeongganbo to be interpreted and transcribed into Western staff notation.
  • Figure 2: Overview of the process for converting Jeongganbo into Zipfian units. Jeongganbo is first transcribed into Western staff notation, enabling computational analysis of Korean court music using the music21 library. Subsequently, (pitch, duration) pairs are extracted as Zipfian units, and their frequencies are computed. Ornamentation symbols are ignored.
  • Figure 3: Union-level Zipf--Mandelbrot fits for (pitch, duration) units and piecewise-linear fits for marginal pitch-only and duration-only rank--frequency curves (log--log scale).
  • Figure 4: Instrument-level Zipf--Mandelbrot fits for (pitch, duration) units (log--log scale). For each instrument, the normalized fit (left) and raw fit (right) are shown side by side.
  • Figure 5: Application of pitch names according to octave

Theorems & Definitions (7)

  • Remark 5.1
  • Proposition 5.2: Sorting converges to the inverse-area curve
  • Proposition 5.3: Closed form of the sorting function
  • Proposition B.1: Sorting converges to the inverse-area curve
  • proof
  • Proposition B.2: Closed form of the sorting function
  • proof