Function- and Rhythm-Aware Melody Harmonization Based on Tree-Structured Parsing and Split-Merge Sampling of Chord Sequences

This paper presents an automatic harmonization method that, for a given melody (sequence of musical notes), generates a sequence of chord symbols in the style of existing data. A typical way is to use hidden Markov models (HMMs) that represent chord transitions on a regular grid (e.g., bar or beat grid). This approach, however, cannot explicitly describe the rhythms, harmonic functions (e.g., tonic, dominant, and subdominant), and the hierarchical structure of chords, which are supposedly important in traditional harmony theories. To solve this, we formulate a hierarchical generative model consisting of (1) a probabilistic context-free grammar (PCFG) for chords incorporating their syntactic functions, (2) a metrical Markov model describing chord rhythms, and (3) a Markov model generating melodies conditionally on a chord sequence. To estimate a variable-length chord sequence for a given melody, we iteratively refine the latent tree structure and the chord symbols and rhythms using a Metropolis-Hastings sampler with split-merge operations. Experimental results show that the proposed method outperformed the HMM-based method in terms of predictive abilities.