RE: consonance definition?

>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).

- My website has a new look,

- the microtonal dictionary just keeps growing:

http://www.ixpres.com/interval/monzo/define.htm

You might want to bookmark it, and please send
me your own definitions. I'm going to cite anything
I get that's worthwhile. It's a great opportunity
to exchange links.

- and the main paper that explains my theory now
finally looks good. It should give a nice
print-out, especially in color:

http://www.ixpres.com/interval/monzo/article.htm


- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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