The ripples of consonant intervals and chords arising in different tuning systems

Pure consonances are combinations of exact multiples of a fundamental frequency, particularly with the factors 2, 3, and 5. They give rise to the consonant intervals (octave, perfect fifth, and major and minor thirds) and consonant chords (major and minor). However, these kinds of consonances are only achieved in just intonation, which does not produce beats or ripples, that is, their envelopes are constant. On the contrary, other tuning systems, such as Pythagorean, meantone, or N-tone equal temperaments, which slightly differ from just intonation, actually produce beats or ripples, that is, the corresponding envelopes oscillate and, acoustically, resemble a vibrato in amplitude. In this paper, the envelopes of consonant intervals and chords are modelled mathematically by approximate formulas. They contain one variable for the intervals and two variables for the chords. The formulas include deviations of one full trigonometric period, thus being valid for any tuning system. In the case of intervals, the corresponding ripples are obtained by following a line (1D) at a specific velocity and, in the case of chords, by following a line on a graph (2D) at specific velocity and direction. The corresponding audio files considering pure tones are also obtained.