The Science of Harmony: A Psychophysical Basis for Perceptual Tensions and Resolutions in Music

This paper attempts to establish a psychophysical basis for both stationary (tension in chord sonorities) and transitional (resolution in chord progressions) harmony. Harmony studies the phenomenon of combining notes in music to produce a pleasing effect greater than the sum of its parts. Being both aesthetic and mathematical in nature, it has baffled some of the brightest minds in physics and mathematics for centuries. With stationary harmony acoustics, traditional theories explaining consonances and dissonances that have been widely accepted are centred around two schools: rational relationships (commonly credited to Pythagoras) and Helmholtz's beating frequencies. The first is more of an attribution than a psychoacoustic explanation while electrophysiological (amongst other) discrepancies with the second still remain disputed. Transitional harmony, on the other hand, is a more complex problem that has remained largely elusive to acoustic science even today. In order to address both stationary and transitional harmony, we first propose the notion of interharmonic and subharmonic modulations to address the summation of adjacent and distant sinusoids in a chord. Based on this, earlier parts of this paper then bridges the two schools and shows how they stem from a single equation. Later parts of the paper focuses on subharmonic modulations to explain aspects of harmony that interharmonic modulations cannot. Introducing the concept of stationary and transitional subharmonic tensions, we show how it can explain perceptual concepts such as tension in stationary harmony and resolution in transitional harmony, by which we also address the five fundamental questions of psychoacoustic harmony such as why the pleasing effect of harmony is greater than that of the sum of its parts. Finally, strong correlations with traditional music theory and perception statistics affirm our theory with stationary and transitional harmony.