Analyzing Temporal Dynamics in Music: : Differential Calculus, Physics, and Functional Data Analysis Techniques

THIS ARTICLE INTRODUCES THEORETICAL and analytical tools for research involving musical emotion or musical change. We describe techniques for visualizing and analyzing data drawn from timevarying processes, such as continuous tension judgments, movement tracking, and performance tempo curves. Functional Data Analysis tools are demonstrated with real-time judgments of musical tension (a proxy for musical affect) to reveal patterns of tension and resolution in a listener's experience. The derivatives of tension judgment curves are shown to change with cycles of expectation and release in music, indexing the dynamics of musical tension. We explore notions of potential energy and kinetic energy in music and propose that affective energy is stored or released in the listener as musical tension increases and decreases. Differential calculus (and related concepts) are introduced as tools for the analysis of temporal dynamics in musical performances, and phase-plane plots are described as a means to quantify and to visualize musical change.