Stopping in running and in music performance Part II. A model of the final ritardando based on runners’ deceleration

A model for describing the change of tempo in final ritardandi is presented. The model was based on the previous finding that runners’ average deceleration can be characterised by a constant brake power. This implies that velocity is as a squareroot function of time or alternatively, a cubic-root function of position. The translation of physical motion to musical tempo is realised by assuming that velocity and musical tempo are equivalent. To account for the variation observed in individual measured ritardandi and in individual decelerations, two parameters were introduced; (1) the parameter q controlling the curvature with q=3 corresponding to the runners’ deceleration, and (2) the parameter vend corresponding to the final tempo. A listening experiment gave highest ratings for q=2 and q=3 and lower ratings for higher and lower q values. Out of three tempo functions, the model produced the best fit to individual measured ritardandi and individual decelerations. A commonly used function for modelling tempo variations in phrases (duration is a quadratic function of score position) produced the lowest ratings in the listening experiment and the least good fit to the measured individual ritardandi. The fact that the same model can be used for describing velocity curves in decelerations as well as tempo curves in music provides a striking example of analogies between motion and music.