Automatic Functional Harmonic Analysis

Music scholars have been studying tonal harmony intensively for centuries, yielding numerous theories and models. Unfortunately, a large number of these theories are formulated in a rather informal fashion and lack mathematical precision. In this article we present HarmTrace, a functional model of Western tonal harmony that builds on well-known theories of tonal harmony. In contrast to other approaches that remain purely theoretical, we present an implemented system that is evaluated empirically. Given a sequence of symbolic chord labels, HarmTrace automatically derives the harmonic relations between chords. For this, we use advanced functional programming techniques that are uniquely available in the Haskell programming language. We show that our system is fast, easy to modify and maintain, robust against noisy data, and that its harmonic analyses comply with Western tonal harmony theory. For ages, musicians, composers, and musicologists have proposed theories regarding the structure of music to better understand how it is perceived, performed, and appreciated. In particular, tonal harmony exhibits a considerable amount of structure and regularity. The first theories describing tonal harmony date back at least to the 18th century (Rameau 1722). Since then, a rich body of literature that aims at explaining the harmonic regularities in both informal and formal models has emerged (e.g., Lerdahl and Jackendoff 1996). Such models have attracted numerous computer music researchers to investigate the automation of the analysis and generation of harmony. Most of these theories, however, have proven to be very hard to implement (e.g., Clarke 1986). We are not aware of a model that has a working implementation that effectively analyzes tonal harmony and deals robustly with noisy data, while remaining simple and easy to maintain, and scaling well to handle musical corpora of considerable size. In this article we present HarmTrace (Harmony Analysis and Retrieval of Music Computer Music Journal, 37:4, pp. 37–53, Winter 2014 doi:10.1162/COMJ a 00209 c © 2014 Massachusetts Institute of Technology. Published under a Creative Commons AttributionNonCommercial 3.0 Unported (CC BY-NC 3.0) license with Type-level Representations of Abstract Chord Entities), a system that meets these requirements using state-of-the-art functional programming techniques. HarmTrace allows us to easily adapt the harmonic specifications, empirically evaluate the harmonic analyses, and use these analyses for tasks such as similarity estimation and automatic annotation of large corpora. The HarmTrace harmony model draws on the ideas of Rohrmeier (2007, 2011). Rohrmeier modeled the core rules of Western tonal harmony as a (large) context-free grammar (CFG, see Chomsky 1957). Later, De Haas et al. (2009) implemented this grammar and specifically tuned it for jazz harmony, with the aim of modeling harmonic similarity. The HarmTrace system transfers these ideas to a functional setting, solving typical problems that occur in context-free parsing (e.g., the rejection of pieces that cannot be parsed) and controlling the number of ambiguous solutions. Because it relies on advanced functional programming techniques not readily available in most programming languages, HarmTrace is inextricably bound to Haskell (Peyton Jones 2003). Haskell is a purely functional programming language with strong static typing. It is purely functional because its functions, like regular mathematical functions, are guaranteed to