Triad’s consonance and dissonance: combining roughness and compactness models

Theories of consonance and dissonance based on the “roughness” approach are those that explain these perceptions as due to the primary beatings between harmonics. Originally proposed by Helmholtz, this approach has been very popular in the last century, being naturally associated to continuous functions of the frequency ratios, on the contrary of theories based on the “compactness” approach. In a previous work, we focused on the roughness consonance and dissonance indicators for dyads, showing the importance of including weight functions and especially secondary beatings. Here, we generalize the roughness indicators to describe the consonance and dissonance for triads. We compare our model predictions with perceptual data from a recent psychoacoustic test by means of a Chi-square analysis. The result is that roughness indicators provide a quite effective, but not fully satisfactory, description of consonance and dissonance for triads. We then study the effect of combining roughness and compactness models for triads: in this case, a very satisfactory agreement with perceptual data is achieved.