The Final Ritard: On Music, Motion, and Kinematic Models

Motion plays an important role in music, a fact evidenced not only by the wealth of terminology used by musicians and music theorists that refer to music in "motional" terms. Consider, for example, how we speak of music as "slowing down," "speeding up," "moving from F-sharp to G," etc. A considerable amount of theoretical and empirical work tries to illustrate apparent relation between physical motion and music (see Shove and Repp 1995 for an overview). However, it is very difficult to specify-let alone validate-the nature of this long-assumed relationship. Is there a true perceptual experience of movement when listening to music, or is it merely a metaphorical one owing to associations with physical or human motion?

Some scientists have looked at music and motion in a very direct way, for instance, relating walking speed to preferred tempi (e.g., Van Noorden and Moelants 1999) or body size to timing patterns found in music (Todd 1999). However, these direct relationships between the human body and music seem too simplistic to generally hold. Others have approached the relation more as a metaphorical one, arguing that musicians allude to physical motion in their performances, imitating it in a musical way (cf. Shove and Repp 1995). These theories tend to be difficult to express in computational terms.

This article reviews a family of computational models (e.g., Sundberg and Verillo 1980; Feldman, Epstein, and Richards 1992; Todd 1992; Friberg and Sundberg 1999) that do make the relation between motion and music explicit and therefore can be tested and validated on real performance data. These kinematic models attempt to predict the timing patterns found in musical performances (generally referred to as expressive timing). Most of these studies focus on modeling the final ritard: the typical slowing down at the end of a music performance, especially in music from the Western Baroque and Romantic periods. But this characteristic slowing down can also be found in, for instance, Javanese gamelan music or some pop and jazz genres. In this kinematic approach, one looks for an explanation in terms of the rules of mechanics: that is, how expressive timing might relate to, or can be explained by, models of physical motion that deal with force, mass, and movement.

A discussion of these kinematic models is presented below in the form of a story (see Figure 1), with three fictitious characters who represent the different disciplines involved in this research (psychology, mathematics, and musicology). The story is a continuation of Desain and Honing (1993; see also http://www.nici.kun.nl/mmm/tc for additional sound examples), an article that dealt with the state of the art in expressive timing research some ten years ago. In addition, it brought forward a critique on the usefulness of the tempo curve (a continuous function of time or score position) as the underlying representation of several computational models (including most computer music software at that period). The main point of critique was that the predictions made by models using this representation are insensitive to the actual rhythmic structure of the musical material: they make the same predictions for different rhythms. All this suggested the existence of a richer representation of timing in music perception and performance than is captured by an unstructured tempo curve.

The present article attempts to offer an informative but informal discussion of models of the final ritard, including some of the problems that these kinematic models do not address. Experimental support for an alternative view, as briefly presented in the discussion, will be the topic of a forthcoming article.

The Final Ritard: A Tale on Music and Motion

In the following text, P, M, and their musical friend MF continue their enthusiastic search in trying to unravel the mystery of timing in music performance. This time they will find out about the kinematic approach to expressive timing and computational models that are also based on the notion of a tempo curve; as such they are likely to continue their argument.

Figure 1.

The final ritard, a tale on music and motion.

Prologue: What...