Markov Chain Analysis of Musical Dice Games

We have studied entropy, redundancy, complexity, and first passage times to notes for 804 pieces of 29 composers. The successful understanding of tonal music calls for an experienced listener, as entropy dominates over redundancy in musical messages. First passage times to notes resolve tonality and feature a composer. We also discuss the possible distances in space of musical dice games and introduced the geodesic distance based on the Riemann structure associated to the probability vectors (rows of the transition matrices).

💡 Research Summary

The paper presents a comprehensive statistical‑physics approach to the analysis of tonal music by treating each composition as a first‑order Markov chain derived from a “musical dice game” – a stochastic process that generates note sequences based on transition probabilities. The authors assembled a corpus of 804 pieces written by 29 composers spanning the Baroque, Classical, Romantic, and early‑20th‑century periods. For each work they extracted a pitch‑class sequence (12‑tone chromatic scale, optionally extended to include octave information) and constructed a transition matrix (P) where the entry (p_{ij}) gives the probability that a note of class (i) is followed by a note of class (j). This matrix is the empirical estimate of the one‑step transition kernel of a discrete‑time Markov chain that captures the short‑range statistical dependencies of the melody or harmonic line.

Entropy and Redundancy
Using the transition matrix, the Shannon entropy per step, \