Toward a Theory of Intervals

Abstract

This paper is intended to start inquiry into the question: Do chords, considered as fused klang effects, have individual “tone colors” analogous to the timbres of different instruments? As a first step, only the qualities of two-tone intervals are considered. Sine waves representing simple tones (without overtones) are superposed, by graphic means, so as to produce characteristic composite wave patterns for each of the commonly used intervals. Study of the resultant curves would seem to suggest that: (1) Each interval has a characteristic composite sound-wave-pattern. (2) The length of the composite wave pattern is equal to a number of cycles of the lower frequency expressed by its number in the interval ratio. (3) The composite wave patterns of dissonant intervals are longer and more complex than those of consonant intervals. (4) Comparison of the curves of an interval and its inversion shows little outward resemblance. (5) Intervals may be grouped together into “families,” according to the lengths of the composite wave patterns. These groupings differ somewhat from those usually recognized by harmonic theory. (6) An interval may be said to have a “frequency” equal to the number of complete cycles per second of the composite wave pattern characteristic of that interval. The lower interval frequencies are more dissonant. (7) A study of the configuration of the composite wave patterns shows how it is still possible for the ear to identify the component pitches in a given interval. (8) There seems to be graphic evidence of the physical existence, in the composite wave patterns, of frequencies corresponding to the difference tones.