Temporal Patterns in Polyphony

This paper formally characterizes the expressiveness of three approaches for polyphonic pattern representation and matching: R (relational patterns); H (Humdrum); and SPP (Structured Polyphonic Patterns). Relational networks have the highest expressiveness but H and SPP admit faster matching algorithms. It is shown how H and SPP can be cast as different restrictions of R, both providing an expressive subset of full relational networks. In addition, the intersection of H and SPP yields yet another language: SPPseq, a restriction of SPP based on sequences of layered components. This new language is expressive enough to capture basic polyphonic patterns such as suspensions and parallel fifths and may be a new, more efficient approach to pattern extraction. The formal arguments contained in this paper are illustrated with musical examples extracted from J.S. Bach chorale harmonizations. 1 Motivation Polyphony forms a large part of the western musical heritage and its essence — having multiple concurrent streams of musical events (with the temporal relations this implies) — is encountered in most kinds of modern music. However, there are few computational approaches for the expression and efficient matching of polyphonic patterns. This paper formally compares the expressiveness of three such languages and proposes a new one, establishing the hierarchy of Figure 1. To facilitate this presentation, arguments are restricted to patterns containing only two voices; results may however be generalized to denser polyphonic textures. As a motivating example, consider the two-voice suspension of Figure 2. This typical polyphonic pattern is expressed in Figure 3 in the languagesR (relational patterns); Humdrum; and SPP (Structured Polyphonic Patterns). As illustrated by the R expression (Figure 3i), even this simple pattern requires sophistication: variables to be instantiated by three events; inequality statements ensuring that the mapping from variables to events is injective; temporal relations between events (discussed below); and pitch relations such as consonance and dissonance. This paper restricts its attention to the following binary temporal relations: i) m(a, b) (a meets b: a finishes when b begins), ii) the symmetric st(a, b) (a and b start together), iii) sw(a, b) (a starts while b is sounding) and iv) the symmetric ov(a, b) (a and b overlap: they sound together as some point in time). Figure 4 restates the relations in the notation of Allen [1] and Figure 2ii illustrates their musical relevance. E. Chew, A. Childs, and C.-H. Chuan (Eds.): MCM 2009, CCIS 38, pp. 32–42, 2009. c © Springer-Verlag Berlin Heidelberg 2009 Temporal Patterns in Polyphony 33 Suspension • Embellished tritone resolution • Layered passing tones • Dislocated chord • SPPseq SPP H