Bayesian Melody Harmonization Based on a Tree-Structured Generative Model of Chord Sequences and Melodies

This article describes a melody harmonization method that generates a sequence of chords (symbols and onset positions) for a given melody (a sequence of musical notes). A typical approach to melody harmonization is to use a hidden Markov model (HMM) that represents chords and notes as latent and observed variables, respectively. This approach, however, does not consider the syntactic functions (e.g., tonic, dominant, and subdominant) and hierarchical structure of chords that play vital roles in traditional harmony theories. In this paper, we propose a unified hierarchical generative model consisting of a probabilistic context-free grammar (PCFG) model generating chord symbols associated with syntactic functions, a metrical Markov model generating chord onset positions, and a Markov model generating a melody conditioned by a chord sequence. To estimate a musically natural tree structure, the PCFG is trained in a semi-supervised manner by using chord sequences with tree structure annotations. Given a melody, a sequence of a variable number of chords can be estimated by using a Markov chain Monte Carlo method that partially and iteratively updates the symbols, onset positions, and tree structure of chords according to the posterior distribution of chord sequences. Experimental results show that the proposed method outperformed the HMM-based method and a conventional rule-based method in terms of predictive abilities.