The envelopes and amplitude fluctuations of consonant intervals and chords, and their inversions, arising in different tuning systems

A key feature of musical consonances is the absence of beats or roughness. In this paper, the envelopes of consonant intervals and chords, and their inversions, are obtained considering pure tones in just intonation with appropriate phases. In just intonation, the phases are constant, while in the other tuning systems they vary with time. The corresponding envelopes, here called static and dynamic, respectively, are approximated by accurate mathematical models. They exhibit ripples with respect to the phases or time, which define the characteristics of the beats. The ripple periodicities are obtained analytically, while their amplitudes are computed numerically and have proved to be a good measure of the degree of consonance of the corresponding notes. Examples are given for Pythagorean tuning, quarter- and third-comma meantone temperaments, and 12-TET. The corresponding audios are provided as supplemental material, which can also be used for conducting auditory surveys or for educational purposes.