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.

翻译：齐夫定律最初在自然语言中发现，后推广为齐夫-曼德尔布罗特定律，描述了齐夫型元素的频率与其排序之间的幂律关系。由于该定律的语义特征，其在音乐数据中也得到了观测。然而，此类研究大多集中于西方音乐，其对于非西方音乐的适用性尚未得到充分探究。我们分析了43首跨越数个世纪、以传统韩国记谱法“井间谱”记载的韩国宫廷音乐“正乐”。这些曲目被转译为西方五线谱，并提取了音高与时值等音乐数据。以音高、时值及其配对组合作为齐夫型单元，我们发现韩国音乐同样高度符合齐夫-曼德尔布罗特定律，尤以音高-时值配对单元为甚。韩国音乐在长期集体演化过程中，消弭了特异性变异，形成了具有广泛理解度的形式。这种集体演化似乎在塑造满足齐夫-曼德尔布罗特定律的特征方面发挥了重要作用。我们的研究结果为音乐数据中齐夫-曼德尔布罗特标度律的文化普适性提供了新的证据。我们进一步证明，两个独立齐夫型数据集的联合分布遵循齐夫-曼德尔布罗特定律；在此意义上，我们的结果不仅拓展了齐夫定律的适用范围，更深化了对标度律在复合与交互作用下行为机制的理解，为基于排序的统计规律性提供了更为统一的视角。