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

This investigation explores the common assumption that music and motion are closely related by comparing the stopping of running and the termination o f a piece of music. Video recordings were made of professional dancers' sto pping from running under different deceleration conditions, and instant val ues of body velocity, step frequency, and step length were estimated. In de celerations that were highly rated for aesthetic quality by a panel of chor eographers, the mean body velocity could be approximated by a square-root f unction 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 similarit y with the mean tempo pattern of final ritardandi in music performances. Th e 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 run ners' decelerations. The translation of physical motion to musical tempo wa s 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' deceleratio n, and (2) the parameter upsilon(end) for the final velocity and tempo valu e, respectively. A listening experiment was carried out presenting music ex amples 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 deceleration s. A function previously used for modeling phrase-related tempo variations (interonset duration as a quadratic function of score position) produced th e lowest ratings and the poorest fits to individual ritardandi. The results thus seem to substantiate the commonly assumed analogies between motion an d music. (C) 1999 Acoustical Society of America. [S0001-4966(99)01402-2].