Harmonic clusters and tonal cadences: Bayesian learning without chord identification

Abstract

Authentic and half cadences not only serve as structural cornerstones of tonal music, but are readily perceived by both expert and novice listeners. Yet it is unclear how untrained listeners, who have never studied harmonic theory, come to recognise cadential patterns that are most often codified as short series of chords, such as V⁷–I or ii⁶–V. This study addresses both questions by analysing a corpus of string-quartet excerpts whose authentic and half cadences were identified by two musical experts. A cognitively plausible model of harmonic learning, which is based on clusters of scale-degree distributions abstracted from the data, is proposed. After assessing the correlation of this model’s output with cadential categories, Bayesian or near-Bayesian frameworks are used to learn these categories from the model in both supervised and unsupervised contexts. The model succeeds in not only spotting cadences but also identifying cadential categories from unlabelled data. The model’s relationship to relevant perceptual research, as well as the results’ implications for human learning and detection of cadences, is discussed.

Keywords:

• Cadences
• harmony
• chord analysis
• pattern induction
• perception
• statistical learning

Notes

¹ Parenthesised figures indicate optional additions to the chord.

² A related branch of harmonic theory involves neo-Riemannian and other transformational approaches (Cohn, 1996, 1997, 1998; Hyer, 1989; Kopp, 2002; Lewin, 1982). The most common transformations—parallel, relative and leading-tone exchange—often relate chords that exhibit the same harmonic function (Cohn, 1998). In graphical terms, each harmonic function can be invoked by chords near one another on a Tonnetz (Cohn, 1997).

³ It is true, of course, that cadences are often conceived not as one or two chords but rather as longer chord progressions, such as I⁶-ii-V-I. Nonetheless, as the results below indicate, a two-bar window proves sufficient to learn and detect cadential patterns.

⁴ It should be noted that the Nagelkerke pseudo-R ² value does not, as in linear regression, indicate the proportion of variance explained by the model.