The Envelopes of Consonant Intervals and Chords in Just Intonation and Equal Temperament

Abstract

[EN] Musical consonances are most times assumed to be associated with the absence of beats or roughness, which can only be achieved, in a strict sense, in just intonation. In this paper, signals representing consonant intervals and chords are analyzed, both in just intonation and in an arbitrary slight deviation from it. Analytical approximate formulas for their envelopes are obtained and then applied to the particular case of equal temperament. It was found that, in just intonation, the envelopes are flat but, in the other cases, the envelopes have a ripple which corresponds to beats or roughness, thus indicating a loss of consonance. Both the amplitudes and periodicities of the ripples are obtained for all types of consonances.