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: ͑1͒ the parameter q controlling the curvature, qϭ3 corresponding to the runners' deceleration, and ͑2͒ the parameter v end 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 qϭ2 and qϭ3. 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 position͒ 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.

An evaluation of linear and non-linear models of expressive dynamics in classical piano and symphonic music

“…Considerable contributions to our current knowledge on musical expression have been made by works following the paradigm of experimental psychology, in which controlled experiments are designed and executed, investigating a single aspect of performance, such as the timing of grace notes (Timmers et al 2002), or cyclic rhythms (Repp et al 2013). Local variations in tempo as a function of phrasing have also been explicitly addressed, using computational models (Todd 1992;Friberg and Sundberg 1999). Complementary to such approaches, often testing a specific hypothesis about a particular aspect of expression, data mining and machine learning paradigms set out to discover regularities in musical expression using data sets comprising musical performances (Widmer 2003;Ramirez and Hazan 2004).…”

IS THERE A PERCEPTION-BASED ALTERNATIVE TO KINEMATIC MODELS OF TEMPO RUBATO?

“…shown to produce a good fit with a variety of empirical performance data (Friberg & Sundberg, 1999; but see also alternatives proposed by Repp [1992] and Feldman, Epstein, & Richards [1992], unrelated to the laws of physical motion; see Figure 1c). Friberg & Sundberg (1999) suggested that the final ritard alluded to human movement: the pattern of runners' deceleration.…”

“…However, it is very difficult to specify-let alone validate-what is the nature of this longassumed relationship (Clarke, 2001;Honing, 2003). An important contribution to this topic is made by a family of computational theories, so-called kinematic models, that propose an explicit relation between the laws of physical motion (elementary mechanics) in the real world and expressive timing in music performance (Sundberg & Verillo, 1980;Kronman & Sundberg, 1987;Longuet-Higgins & Lisle, 1989;Feldman, Epstein, & Richards 1992;Todd, 1992;Epstein, 1994;Todd, 1995;Friberg & Sundberg, 1999). A large number of these theories focus on modeling the final ritard, the typical slowing down at the end of a music performance, especially in music from the Western Baroque and Romantic periods.…”

“…Friberg & Sundberg (1999) suggested that the final ritard alluded to human movement: the pattern of runners' deceleration. Such a deceleration pattern can be described by a well-known model (from elementary mechanics) of velocity (v) as a function of time (t):…”

“…This function can be generalized and expressed in terms of score position x, resulting in a model where tempo v is defined as a function of (normalized) score position x, with q for curvature (varying from linear to convex shapes; see Figure 1a and 1b) and w denoting a non-zero final tempo (Friberg & Sundberg, 1999):…”