Force Dynamics of Tempo Change in Music

Musical tempo might, in principle, be changed by any arbitrary continuous function with different starting and ending values. The performer is not obligated to follow constraints such as those suggested in this paper: no physical laws, for example, require it. Yet while change of tempo by skilled musicians may seem to be under conscious, voluntary control, the matches found here between observed profiles and theoretical ideal shapes suggest that performers actually conform in detail to mathematical constraints of which they presumably have no conscious knowledge. If this conformance actually were responsible for the aesthetically satisfying form achieved by skilled musicians, as we speculate, it suggests a paradigm for “beauty,” or at least naturalness of form, that is refreshingly concrete. Furthermore, as we have discussed, because these ideal shapes were derived from the force model, this conformance seems to reflect an underlying, unconscious conception of music as a quasi-physical thing that “moves forward” as it unfolds through time, now speeding up and now slowing down, in accord with the moment-to-moment flux in its rhythmic, harmonic, and affective character—a conception reflected in musicians’ common use of terms such as “movement,” “motion,” and “flow” to characterize the progression of music. We propose, in effect, that the relationship between tempo change (designated by the score and manipulated by the performer) and real physical movement (controlled by physical forces) is somewhat more than just a metaphor: the formal machinery is largely the same. The “forces” we invoke are neither actual physical forces, of course, nor are they literally due to physical constraints such as those that control the movement of a conductor’s baton or a drummer’s