Force Dynamics of Tempo Change in Music

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Abstract
This paper reports a study of the quantitative shape of ideally executed tempo decrease and increase (ritardando and accelerando) in music. Five expertly performed samples of real music (two ritards and two accelerandos from European classical music, and one sample of Venezuelan [Yanomami] Indian barter chant) are analyzed, and each is found to be a polynomial of degree at least two, and in some cases probably three—not simply linear as had been previously reported, or exponential, as is sometimes suspected. We formally develop a simple force model of musical "motion," in which the progression of music over time is conceived of as being controlled by a mental analog of a mechanical force. Each of the five observed tempo profiles is shown to be well accounted for as the result of one of just two types of force event: (1) a linear force, which produces a quadratic ritardando (or accelerando), and (2) a parabolic force, which produces a spline-shaped cubic tempo profile, a ritardando that joins smoothly at both ends to adjoining regions of constant tempo.

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