Physical motion as a metaphor for timing in music: the final ritard

Abstract

Both in music theory and in music psychology the metaphor that links expressive timing and movement of a physical mass under forces is abundant, and has inspired new theoretical results. However, the attractiveness of the analogy and the "naturalness" of the explanation may fool the investigator. We believe music plays on, and makes use of the architecture of our perceptual systems, and that perceptual theories, in principle, can go quite far in bringing forward explanations of the many subtleties involved in the performance of music. Nevertheless, it is sensible to use the physical motion approach wherever it can contribute to our understanding of musical expression, and to precisely investigate the consequences of the claims made by the proponents of this approach.

1 Introduction

In music theory, when one talks about rhythm, timing and tempo, often the analogy with physical motion is made. For example, in comparing tempi one uses terms like walking or moving. And to characterize the progression of music (be it harmonic, melodic, rhythmic or timing wise) the notion of motion or `flow' is frequently used. In music psychology the physical metaphor is abundant as well, and has inspired some new theoretical results, and form a healthy alternative to a wholly mentalistic approach to music. For example, in the work of Large (Large & Kolen, 1994) the human ability to track the tempo has been modeled as a resonator, using the theory of non-linear coupled oscillators (from physics), and introduces new ways to describe the behavior of these models mathematically.

2 Physical aspects of music performance

And indeed, the wholly mentalistic approach to music perception and production that AI researchers often tend to take (i.e. modeling the process as an abstract parsing or generation process based on explicit mental syntactic representations) needs to be complemented by theories that are much more based on the physical properties of the human body in interaction with a musical instrument. This direction views human movement and gestures in music performance as embodiment of musical thought (Clarke, 1993; Davidson, 1991) and thus views the similarity between musical expression and human movement as necessity - not a metaphor- and in that way proposes a healthy alternative to the abstract mentalistic approach.

Furthermore, physical constraints can be observed in the performance signal. The restrictions that the system of musician and instrument impose on a performance (e.g., the influence of distances between string positions in guitar playing), and the effort involved to get the desired result are audible. It might well be that the hypothesis that musical expression is communicating abstract musical structure, needs some amending in which the body-instrument expression is recognized as well as a source of music appreciation. Note that the abstract and often inaccessible nature of computer generated music may well be caused, at least partly, by ignoring these constraints - there is no sense of an instrument being played. And it is the same lack of appropriate constraints that will give away a studio musician who tries to imitate a guitar on a keyboard synthesizer. The results of the few composers who have tried to capture these aspects in their composition algorithms, however unsystematic, have proven to be immediately effective (e.g., Garton, 1992).

3 Critique on the use of the physical metaphor

Although the physical metaphor can be very useful, as argued above, we have to be cautious as well. The attractiveness of the analogy (e.g., it directly fits a large amount of the terminology used in music theory and performance practice) and the "naturalness" of the explanation, may fool the investigator. The fact that a model has such a basis does not in itself make it a better model. It may give a good description of the data, but it, in principle, does not teach us anything about the underlying mechanisms: a good approximation is not necessarily a good explanation. And to either validate or falsify these models as models of the underlying perceptual mechanisms, more data and arguments need be brought to bear than has been done by the proponents of the physical motion theory.

Sundberg & Verillo (1980), for instance, make the analogy between the performance of a final ritard (the typical slowing down at the end of a piece) in piano music as alluding to stopping after running. Based on a model of a physical mass under constant deceleration, square-root curves are fitted through measured tempi in piano performance of pieces from the baroque period. Kronman & Sundberg (1987), Feldman, Epstein & Richards (1992) and Sundberg & Verillo (1995) make the same parallel between musical performance and physical motion. Feldman, Epstein & Richards (1992) measured beats in recordings of music from a wide range of periods and styles. They propose a model of musical "motion" in which the progression of music over time is conceived of as being controlled by the mental analog of a mechanical force. Todd (1993) even relates expressive timing directly to the functioning of the vestibular system, the human sense of acceleration that stems from a small organ located in the ear. He argues that "the reason why expression based on the equations of elementary mechanics sounds natural is that the vestibular system evolved to deal with precisely these motions" (Todd, 1992).

Figure 1. Final ritard in performances of the first Variation from Beethoven's WoO 70 at three tempi (averaged over three repeated performances).

4 The Final Ritard

Restricting our analysis to the study of the final ritard, we suspect aspects not addressed by the physical motion models to play a key role; we expect a dependency between the structure (rhythmical and metrical) of the material, the global tempo and the shape of the ritard.

Considering the rhythmic structure, a ritard of many notes of equal (score) duration can have a deep rubato, while a ritard of a few notes, with possibly a more elaborated rhythmical structure, will be less deep. This is a necessary consequence of the need to leave the rhythmic interpretation intact (not to break the rhythmic categories) while decelerating fast. Models of tempo tracking and rhythmic quantization (e.g., Longuet-Higgins, 1976; Desain & Honing, 1989) will necessarily predict at least the borderline case for which the rhythmical structure can still be perceived. Apart from explaining the dependency of a ritard on the material played, this will yield a constraint on the form of ritards. Such restrictions are not made by a physical motion model, since any metaphorical mass, force and amount of deceleration is equally likely. We expect the final ritard to only coarsely resemble a parabola, the detail depending heavily on the rhythmical material in question. This may yield an explanation for the common musician's intuition that music from different composers and style periods require different final ritards to work well musically (Clynes, 1987).

As an example, take the fragment shown in Figure 1. It can be seen that the shape of the final ritard is affected by the rhythmical structure in the last bar. How to characterize this interaction is still an open question.

Considering the metrical structure, these perceptual models may very well predict a more or less stepwise deceleration, directly linked to the meter of the piece - a phenomenon that has been observed informally (Clynes, 1987) but that could have never been predicted by a physical motion model.

Considering the global tempo, the course of evolution of a physical motion model is fully described by its parameters (mass, deceleration force) and its initial condition (velocity), and therefore these models predict a simple relationship between the shapes when ritards are started at different points in the piece and at different initial tempi. Because their internal state is much more complex, models based on, for example, rhythmic expectancy predict different forms depending on the initial tempo and the starting point.

In a study on the use of expressive timing in performances of Beethoven variations WoO 70 (Desain & Honing, 1994) we showed that there was a large interaction between timing and global tempo (see Figure 1) - a realistic model of the final ritard has to account for this relation (the Beethoven data is well suited for this study as well).

5 Conclusion

While a physical motion metaphor is perfectly acceptable as a concept for a musician in talking and thinking about music, there are limitations of these models as to the explanation of music perception and production, and sometimes better alternatives exist. These alternative explanations may be complementary to the physical motion theories, because they explain properties of music performance directly from the musical material and the perceptual processes themselves. We are convinced that music is based on, plays on, and makes use of the architecture of our perceptual systems, and that perceptual theories, in principle, can go quite far in bringing forward explanations of the many subtleties involved in the performance of music. However, it is sensible to use the physical motion approach wherever it can contribute to our understanding of musical expression, and to precisely investigate the consequences of the claims made by the proponents of this approach. Finding appropriate terminology and ways of debating these issues, is another aim of this study - the issue is too important to be blurred by paradigmatic battles based on misunderstandings.

6 References

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