source file: m1568.txt Date: Thu, 29 Oct 1998 16:46:16 -0500 Subject: RE: consonance definition? From: "Paul H. Erlich" >Can anyone offer a comprehensive mathematical explanation of consonance? I hope not! Consonance is not a mathematical phenomenon, it is a psychoacoustic one. Once the ear-brain system is understood, quantifying consonance would thus seem possible; in fact there are at least three distinct psychoacoustic phenomena which can be modelled mathematically and contribute to the perception of consonance: (1) critical band roughness, on which Kameoka & Kuriyagawa and Sethares have done work on fleshing out the quantitative implications; (2) fundamental tracking / virtual pitch / fusion, which motivates Parncutt's whole quantitative approach; (3) nonlinear combination tones, which are the basis of Bohlen's view of consonance and have been studied extensively by psychoacousticians whose names I don't remember. Unfortunately, these three different phenomena are not independent so formulating a unified model of consonance seems impossibly difficult at the present. However, all three of these phenomena lead to the following conclusion, at least for tones with harmonic timbres: Approximate small-integer frequency ratios = Consonant intervals. But beyond these psychoacoustic factors, there are psychological ones that relate to way music unfolds in time. A diminished fourth is dissonant in 12-tET even though it sounds exactly like a major third out of context. As Ernst Toch pointed out, even an octave can be dissonant in certain contexts. So even a unified model as suggested above would have rather limited applicability. If you would like an elaboration of any of the above points, just ask! (Sethares is around too).