Chords from Tones Having Stretched Partials

Abstract

Helmholtz pointed out that dissonance results when the upper partials of tones in a chord produce beats. If the musical scale and the partial structure of the tones are consistent, chords could be formed from tones having inharmonic partials, which would not produce beats. Tones have been generated on a computer, converted to analog form, and used to form chords. The tones contained partials having frequency ratios of NS, when S is the “stretch” exponent and N is the partial number. Such tones, in general, are not periodic. Scales were also constructed in which the ratio between “half steps” was (21/12)S. Chords analogous to the tonic, subdominant, and dominant, will be demonstrated for S varying from 1.001302, comparable to the smallest amount of stretch in the middle of the piano range, to 1.261859, which makes the ratio of the frequencies of the third and first partials equal to four instead of three. The sensation produced by the stretched chords is consistent with Helmholtz's criterion for dissonance. The quality of the individual tones, however, becomes more and more strange as the value of S increases. The stretch exponent may serve as a new parameter that can be manipulated by avant-garde composers.