Minimum description length modelling of musical structure

Abstract

This article presents a method of inductive inference whose aim is to build formal quantitative models of musical structure. The models are constructed by statistical extraction of significant patterns from a musical corpus. The minimum description length (MDL) principle is used to select the best model from among the members of a non-parametric model family characterized by an unbounded parameter set. The chosen model achieves optimal compromise between goodness-of-fit and model complexity, thereby avoiding the over-fitting normally associated with such a family of models. The MDL method is illustrated through its application to the Hidden Markov Model (HMM) framework. We derive an original mathematical expression for the MDL complexity of HMMs that employ a finite alphabet of symbols; these models are particularly suited to the symbolic modelling of musical structure. As an illustration, we use the proposed HMM complexity expression to construct a model for a common melodic formula in Greek church chant. Such formulas are characterized by text–tune association that is governed by complex rules. We show that MDL-guided model construction gradually ‘learns’ important aspects of the melodic formula's structure, and that the MDL principle terminates the process when nothing significant is left to learn. We outline how the musical applications of MDL can be extended beyond the HMM framework, and how they address general methodological concerns in empirical music research.