La nascita delle teorie 'continue' della consonanza. La ignorata curva di Draghetti e Foderà, poi di Helmholtz (1771-1837)

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.

Theories of consonance and dissonance based on the “compactness” approach include the two sub-categories of periodicity and harmonicity. In a previous work, we discussed the related consonance and dissonance indicators for dyads; as they are given by discontinuous functions of the dyad frequency ratio, we proposed a method to extend them to the continuum, based on the auditory discrimination limen. Here, we generalize the compactness indicators to describe the consonance and dissonance for triads and discuss their extension to the continuum. We compare our model predictions with perceptual data from a recent psychoacoustic test by means of a Chi-square analysis. The result is that compactness indicators provide a quite effective, but not fully satisfactory, description of consonance, and dissonance for triads.

Much of Western classical music relies on instruments based on acoustic resonance, which produce harmonic or quasi-harmonic sounds. In contrast, since the mid-twentieth century, popular music has increasingly been produced in recording studios, where it is not bound by the constraints of harmonic sounds. In this study, we use modified MPEG-7 features to explore and characterise the evolution of noise and inharmonicity in popular music since 1961. We place this evolution in the context of other broad categories of music, including Western classical piano music, orchestral music, and musique concrète. We introduce new features that distinguish between inharmonicity caused by noise and that resulting from interactions between discrete partials. Our analysis reveals that the history of popular music since 1961 can be divided into three phases. From 1961 to 1972, inharmonicity in popular music, initially only slightly higher than in orchestral music, increased significantly. Between 1972 and 1986, this rise in inharmonicity was accompanied by an increase in noise, but since 1986, both inharmonicity and noise have moderately decreased. In recent years (up to 2020), popular music has remained much more inharmonic than popular music from the 1960s or orchestral music involving acoustic resonance instruments. However, it has become less noisy, with noise levels comparable to those of orchestral music. We relate these trends to the evolution of music production techniques. In particular, the use of multi-tracking may explain the higher inharmonicity in popular music compared to orchestral music. We illustrate these trends with analyses of key artists and tracks.

At present, there are two approaches that aim at explaining on physical grounds the psychoacoustic perception of consonance and dissonance for dyads, whose pioneers have been, respectively, Galilei and Helmholtz: One is based on the “compactness” of the waveform of the combined signal, while the other on the absence of “roughness” due to possible beats. We perform a detailed study of each approach and find that none of the associated model versions, not even the more refined ones, is fully satisfactory when faced to perceptual data on dyads. We show that combining the two approaches results instead in a surprisingly successful agreement with perceptual data: This demonstrates that compactness and roughness are both necessary ingredients for a phenomenological description of consonance and dissonance.Graphical abstract