The Pythagorean comma and preference for a stretched octave

Abstract

Pythagorean tuning, which derives all musical intervals of a Western chromatic scale from repeated application of a fifth and subsequent reduction (collapse) into a single octave, results in an octave interval slightly larger than 2:1, and the difference between a Pythagorean octave and a 2:1 octave is referred to as the Pythagorean comma. Empirical studies suggest that listeners prefer a stretched octave, that is, an octave interval slightly larger than 2:1. The magnitudes of the Pythagorean comma and of the stretch from a 2:1 ratio in preferred octave tuning are similar, and it is suggested these phenomena might be connected. Implications of such a similarity that are addressed include (a) how the Pythagorean comma predicts preference for a stretched octave, (b) rejection of uncertainties in tuning as a cause of preference for a stretched octave, (c) how the Pythagorean comma and preference for a stretched octave might be related to tension and musical aesthetics, (d) the nature of “scales” and “intervals” in musical and psychophysical senses, (e) the role of motion through auditory pitch space in the Pythagorean comma and preference for a stretched octave, and (f) incorporation of elements of Pythagorean tuning into the representation of the octave.