An Information Theoretic Approach to Chord Categorization and Functional Harmony

Abstract

We present new tools for categorizing chords based on corpus data, applicable to a variety of representations from Roman numerals to MIDI notes. Using methods from information theory, we propose that harmonic theories should be evaluated by at least two criteria, accuracy (how well the theory describes the musical surface) and complexity (the efficiency of the theory according to Occam’s razor). We use our methods to consider a range of approaches in music theory, including function theory, root functionality, and the figured-bass tradition. Using new corpus data as well as eleven datasets from five published works, we argue that our framework produces results consistent both with musical intuition and previous work, primarily by recovering the tonic/subdominant/dominant categorization central to traditional music theory. By showing that functional harmony can be analysed as a clustering problem, we link machine learning, information theory, corpus analysis and music theory.

Keywords:

• functional harmony
• corpus analysis
• cluster analysis
• information theory
• information bottleneck

Acknowledgements

The authors would like to thank Martin Rohrmeier, Eytan Agmon, Tom Gurion, Tom Yuval, and Dror Menashe for their assistance with the project. Finally, the first author would specifically like to thank Carmel Raz for her extraordinary help in the research phase of the project.

Notes

No potential conflict of interest was reported by the authors.

1 Note that the important idea of graded or ‘fuzzy’ membership of chords in functional categories (Agmon, 1995) is formalized here by using the language of probabilities and random variables. While this is not the only approach, there is a long tradition of using this particular formalism for this purpose in the machine-learning community (Hastie, Tibshirani, Friedman, & Franklin, 2005) and music (Temperley, 2007). The problem of finding such categorizations is often described in the machine-learning literature as ‘distributional clustering’ (Pereira, Tishby, & Lee, 1993). As we will see, this definition will be instrumental for the rest of the theory developed here.

2 As we develop our formalization, we will notice further advantages of using mutual information. For example, this choice significantly simplifies some of the algorithmic steps.

3 The first 70 chorales were compiled with the help of undergraduates in Tymoczko’s MUS306 course at Princeton University, as well as several graduate students (including Hamish Robb and Luis Valencia). For the remaining 301 chorales, Tymoczko corrected analyses produced by Heinrich Taube’s ‘Chorale Composer’ software, as improved by Simon Krauss, an undergraduate thesis student of Tymoczko’s. All data were then thoroughly cross-validated using Michael Cuthbert’s music21 toolkit, with all discrepancies further analysed to locate possible errors. The 49 major-mode chord forms include three triadic and four seventh-chord inversions for each of the 7 scale degrees. The 91 minor mode chords include three inversions of the 13 triadic forms residing in the natural, harmonic, and melodic minor scales (two triadic forms for every chord except the tonic), and all the corresponding seventh chords except for i, the minor triad with a major seventh, which does not often appear in classical music.