Perceptual Smoothness of Tempo in Expressively Performed Music

Abstract

We report three experiments examining the perception of tempo in expressively performed classical piano music. Each experiment investigates beat and tempo perception in a different way: rating the correspondence of a click track to a musical excerpt with which it was simultaneously presented; graphically marking the positions of the beats using an interactive computer program; and tapping in time with the musical excerpts. We examine the relationship between the timing of individual tones, that is, the directly measurable temporal information, and the timing of beats as perceived by listeners. Many computational models of beat tracking assume that beats correspond with the onset of musical tones. We introduce a model, supported by the experimental results, in which the beat times are given by a curve calculated from the tone onset times that is smoother (less irregular) than the tempo curve of the onsets. Tempo and beat are well-defined concepts in the abstract setting of a musical score, but not in the context of analysis of expressive musical performance. That is, the regular pulse, which is the basis of rhythmic notation in common music notation, is anything but regular when the timing of performed notes is measured. These deviations from mechanical timing are an important part of musical expression, although they remain, for the most part, poorly understood. In this study we report on three experiments using one set of musical excerpts, which investigate the characteristics of the relationship between performed timing and perceived local tempo. The experiments address this relationship via the following tasks: rating the correspondence of a click track to a musical excerpt with which it was PERCEPTUAL SMOOTHNESS OF TEMPO 2 simultaneously presented; graphically marking the positions of the beats using an interactive computer program; and tapping in time with the musical excerpts. Theories of musical rhythm (e.g., Cooper & Meyer, 1960; Yeston, 1976; Lerdahl & Jackendoff, 1983) do not adequately address the issue of expressive performance. They assume two (partially or fully) independent components: a regular periodic structure of beats and the structure of musical events (primarily in terms of phenomenal accents). The periodic temporal grid is fitted onto the musical structure in such a way that the alignment of the two structures is optimal. The relationship between the two is dialectic in the sense that quasi-periodical characteristics of the musical material (patterns of accents, patterns of temporal intervals, pitch patterns, etc.) induce perceived temporal periodicities while, at the same time, established periodic metrical structures influence the way musical structure is perceived and even performed (Clarke, 1985, 1999). Computational models of beat tracking attempt to determine an appropriate sequence of beats for a given musical piece, in other words, the best fit between a regular sequence of beats and a musical structure. Early work took into account only quantised representations of musical scores (Longuet-Higgins & Lee, 1982; Povel & Essens, 1985; Desain & Honing, 1999), whereas modern beat tracking models are usually applied to performed music, which contains a wide range of expressive timing deviations (Large & Kolen, 1994; Goto & Muraoka, 1995; Dixon, 2001a). In this paper this general case of beat tracking is considered. Many beat tracking models attempt to find the beat given only a sequence of onsets (Longuet-Higgins & Lee, 1982; Povel & Essens, 1985; Desain, 1992; Cemgil, Kappen, Desain, & Honing, 2000; Rosenthal, 1992; Large & Kolen, 1994; Large & Jones, 1999; Desain & Honing, 1999), whereas some recent attempts also take into account elementary aspects of musical salience or accent (Toiviainen & Snyder, 2003; Dixon & Cambouropoulos, 2000; Parncutt, 1994; Goto & Muraoka, 1995, 1999). An assumption made in most models is that a preferred beat track should contain as few empty positions as possible, that is, beats on which no note is played, as in cases of syncopation or rests. A related underlying assumption is that musical events may appear only on or off the beat. However, a musical event may both correspond to a beat but at the same time not coincide precisely with the beat. That is, a nominally on-beat note may be said to come early or late in relation to the beat (a just-off-the-beat note). This distinction is modelled by formalisms which describe the local tempo and the timing of musical tones independently (e.g., Desain & Honing, 1992; Bilmes, 1993; Honing, 2001; Gouyon & Dixon, 2005). The notion of just-off-the-beat notes affords beat structure a more independent existence than is usually assumed. A metrical grid is not considered as a flexible abstract structure that can be stretched within large tolerance windows until a best fit to the actual performed music is achieved but as a rather more robust psychological construct that is mapped to musical structure whilst maintaining a certain amount of autonomy. It is herein suggested that the limits of fitting a beat track to a particular performance can be determined in relation to the concept of tempo smoothness. Listeners are very sensitive to deviations that occur in isochronous sequences of sounds. For instance, the relative JND constant for tempo is 2.5% for inter-beat intervals longer than 250 ms (Friberg & Sundberg, 1995). For local deviations and for complex real music, the sensitivity is not as great (Friberg & Sundberg, 1995; Madison & Merker, 2002), but it is still sufficient for perception of the subtle variations characteristic of expressive performance. It is hypothePERCEPTUAL SMOOTHNESS OF TEMPO 3 (b) (a) Onsets Beat track Onsets Beat track Steady tempo