Cross-Cultural Scale Entropy Quantification Research: An Interdisciplinary Exploration of Mathematics and World Music

This study used information entropy as the core mathematical tool and focused on six typical global cultural scales: the Chinese pentatonic scale, the Indian 22-shruti scale, the Arabic 17 tone and other rhythmic scales, the European natural major scale, the Japanese mode scale, and the Indian pentatonic scale. By quantifying the uncertainty of interval distribution, the mathematical laws behind different cultural music aesthetics have been revealed. This study first constructed a frequency system for each scale and conducted interval statistics. Then, the Shannon entropy formula was used to calculate the entropy value, and the entire process was automatically calculated through Python programming. Finally, the deep logic of entropy value differences was explained in combination with cultural background. The results indicate that entropy is highly correlated with the number of core notes and the diversity of intervals in the scale: low entropy scales (2.0-2.8 bits) correspond to the cultural aesthetics of regular harmony, while high entropy scales (3.1-3.7 bits) relate to improvisation and diverse music styles, providing a quantitative analysis framework for cross-cultural music research.