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

  • Luis NuñoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11502)


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.


Consonance Beat Roughness Envelope Just intonation Equal temperament 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.IUI ITACA Polytechnic University of ValenciaValenciaSpain

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