Research for Quantitative Character Analysis in Opera through Markov Chains

Abstract

As a mathematical tool for describing random processes, Markov Chain has shown great potential in the field of music analysis. Existing musicology research often focuses on the qualitative analysis of works, while there is still much room for exploration in the quantitative description of musical elements. This study explores the application of Markov chains in music analysis, focusing on contrasting the musical representations of two characters—Carmen and Micaëla—in Georges Bizet's opera Carmen. Using Musical Instrument Digital Interface (MIDI) files, musical elements were encoded into transition matrices for pitch and harmony, capturing the probabilistic relationships between notes and chords. Stationary distributions and entropy were computed to reveal long-term probabilistic behavior and musical complexity, respectively. Results demonstrate that Carmen's music exhibits higher entropy and unpredictable stationary distributions, aligning with her volatile and defiant nature. In contrast, Micaëla's more structured musical transitions reflect her conventional and stable character. By bridging mathematical modeling with artistic interpretation, this study offers a novel quantitative approach to character analysis in opera and lays the groundwork for future interdisciplinary explorations in musicology.