Predictability and Statistical Memory in Classical Sonatas and Quartets

Abstract

Statistical models and information theory have provided a useful set of tools for studying music from a quantitative perspective. These approaches have been employed to generate compositions, analyze structural patterns, and model cognitive processes that underlie musical perception. A common framework used in such studies is a Markov chain model, which models the probability of a musical event -- such as a note, chord, or rhythm -- based on a sequence of preceding events. While many studies focus on first-order models, relatively few have used more complex models to systematically compare across composers and compositional forms. In this study, we examine statistical dependencies in classical sonatas and quartets using higher-order Markov chains fit to sequences of top notes. Our data set of 605 MIDI files comprises piano sonatas and string quartets by Mozart, Haydn, Beethoven, and Schubert, from which we analyze sequences of top notes. We probe statistical dependencies using three distinct methods: Markov chain fits, time-delayed mutual information, and mixture transition distribution analysis. We find that, in general, the statistical dependencies in Mozart's music notably differ from that of the other three composers. Markov chain models of higher order provide significantly better fits than low-order models for Beethoven, Haydn, and Schubert, but not for Mozart. At the same time, we observe nuances across compositional forms and composers: for example, in the string quartets, certain metrics yield comparable results for Mozart and Beethoven. Broadly, our study extends the analysis of statistical dependencies in music, and highlights systematic distinctions in the predictability of sonatas and quartets from different classical composers. These findings motivate future work comparing across composers for other musical forms, or in other eras, cultures, or musical traditions.