Does music performance allude to locomotion? A model of final ritardandi derived from measurements of stopping runners

Abstract

This investigation explores the common assumption that music and motion are closely related by comparing the stopping of running and the termination of a piece of music. Video recordings were made of professional dancers’ stopping from running under different deceleration conditions, and instant values of body velocity, step frequency, and step length were estimated. In decelerations that were highly rated for aesthetic quality by a panel of choreographers, the mean body velocity could be approximated by a square-root function of time, which is equivalent to a cubic-root function of position. This implies a linear relationship between kinetic energy and time, i.e., a constant braking power. The mean body velocity showed a striking similarity with the mean tempo pattern of final ritardandi in music performances. The constant braking power was used as the basis for a model describing both the changes of tempo in final ritardandi and the changes of velocity in runners’ decelerations. The translation of physical motion to musical tempo was realized by assuming that velocity and musical tempo are equivalent. Two parameters were added to the model to account for the variation observed in individual ritardandi and in individual decelerations: (1) the parameter q controlling the curvature, q=3 corresponding to the runners’ deceleration, and (2) the parameter vend for the final velocity and tempo value, respectively. A listening experiment was carried out presenting music examples with final ritardandi according to the model with different q values or to an alternative function. Highest ratings were obtained for the model with q=2 and q=3. Out of three functions, the model produced the best fit to individual measured ritardandi as well as to individual decelerations. A function previously used for modeling phrase-related tempo variations (interonset duration as a quadratic function of score position) produced the lowest ratings and the poorest fits to individual ritardandi. The results thus seem to substantiate the commonly assumed analogies between motion and music.