Consonance and dissonance

Joe Monzo considers consonance and dissonance to be opposite poles of a continuum of sensation, which he calls "sonance". However he was not the first who used the term sonance: also Wilhelm Keller distinguishes between sonanzmodal and distanzmodal aspects when analysing sounds, see his Handbuch der Tonsatzlehre from 1957.

The term sonance goes back to Giovanni Battista Benedetti

Going back to Giovanni Battista Benedetti, an Italian Renaissance mathematician and physicist, sonance can be best described as relative consonance and/or dissonance of a musical interval – a continuum of pitches encompassing consonance on one end, and dissonance on the other (Palisca, 1973).

Musical vs. sensory dissonance

Musical dissonance is a complex phenomenon depending on not only the fundamental frequency ratio but also the register, timbres, volume, spatialization, and the listener's conditioning and cultural background. Sensory dissonance or roughness, however, is a psychoacoustic effect that is much more consistent among most human listeners except for those with conditions like congenital amusia (tone deafness). In its most basic form, sensory dissonance occurs when two sine wave tones in the audible frequency range are played simultaneously, within about a critical band of each other but far apart enough that beating is not audible.

A study by Kameoka & Kuriyagawa in 1969 had listeners grade the roughness of two sine waves played simultaneously at equal intensity. They computed that if the lower sine wave has frequency f₁ Hz, then the upper frequency that maximizes roughness is about [math]\displaystyle{ f_2 = f_1 + 2.27 f_1^{0.477}\ \text{Hz} }[/math] (we will call this the maximal roughness formula). This assumes that each sine wave has intensity 57 dB SPL; the formula changes slightly depending on volume. The interval [math]\displaystyle{ f_2/f_1 }[/math] is about 1.56 semitones at 440 Hz, and narrows as frequency increases.

The same authors propose a measurement of dissonance between two arbitrary sine waves, broadly modeled as follows: two identical frequencies have a dissonance of 0, the maximal roughness formula gives a dissonance of 1, and two frequencies at an octave or greater have dissonance 0. From zero difference to maximal roughness linear interpolation is used, and from maximal roughness to an octave a gradual decay. Note, however, that this is the result of one study; subsequent replication attempts have produced somewhat different formulas, and this particular study has been critically re-evaluated such as by Mashinter in 2006.

To address some possible misconceptions, sensory dissonance is not musical dissonance, and it has nothing to do with approximation to e.g. JI intervals. It's specifically about the basilar membrane's ability to separate nearby partials. Sensory dissonance only happens when the constituent tones are played simultaneously, not necessarily in succession. Also, it's important that the partials have comparable volumes; if one is much quieter than the other, then it may be masked. Sensory dissonance models for three or more partials quickly get very complicated, and designing good experiments is challenging.

As mentioned above, psychoacoustics does not explain every aspect of musical dissonance in every culture, but there is much research dedicated to connecting the perception of sonance in specific musical traditions with the properties of the inner ear.

See also

• Tenney height
• A singular measure of dissonance
• Harmonic Entropy