Mathematical Model for the Perception of Redundancy and Stability in Musical Scales
David J. Rothenberg · The Journal of the Acoustical Society of America · 1963
Composers of synthetic music, although no longer bound by the constraints of fixed-tuned instruments, must still reckon with the reference structures that listeners impose on musical tones. Indeed, acoustically identical intervals will be perceived differently when these structures are altered by variations either in interval size or in timbre. Existing musical scales appear to be characterized by minimum redundancy, in the sense that few tones can be predicted from a knowledge of those already present, and by maximum stability, in the sense that acoustically equivalent intervals tend to possess equal numbers of interpolated tones. A mathematical model is proposed to measure the redundancy and stability of any scale, and a computer program has been written to implement it. The input consists of a set of integers describing the acoustic distances between consecutive tones within an octave from which the program creates a combinatorial matrix. Redundancy is calculated from the occurrence of unique combinations and stability from a count of the different contexts in which each interval occurs. When applied to existing scales, the model accounts for many of the historical rules of composition. It also permits one to evaluate new scales and predict their perceptual properties.