19-tone Equal Octave Stretch

Paul Ehrlich said:
> >My calculations indicate that 2-3 cents is indeed the optimal amount of
> >stretching for the 19-tone octave in order to maximize the overall consonance
> >of intervals which are approximations of ratios using numbers no higher than
> >6 (including the octave and double octave).

Very good. No matter what one's particular bias, matters such as the
optimal temperature of the octave are determinable using objective
criteria. Your figures for relative consonance, in particular for those
of the 22-tone equal, are however utterly incorrect, for, as I have
tried to demonstrate to you - and this is by no means a difficult matter
to understand - that temperament is noncyclic. You cannot blithely
assign the closest values to all consonances when evaluating the
consonance of a noncyclic temperament, for in such systems these
consonances do not exist together either in diatonic music (which you
have admitted) nor yet in chromatic or enharmonic music, which also
require a modicum of consonance and tonal coherence. I formerly made the
same error which you are making, but had no one to tell me so; I was
forced to discover this for myself.

In the present case, it is _not_ all the consonances that are
_ultimately_ relevant to the determination of the amount by which the
19-tone equal octave can or should be tempered, even though I have
considered them all, as a kind of courtesy to Fokker, and a
demonstration that all of them have to be considered initially. For,
finally, the fourth cannot be tempered by much more than 8.4 cents and
remain satisfactory, i.e no more deteriorated than the more sensitive
fifth at 5.7 cents deviation, and no less consonant than 5:3 or 5:4.
This puts a limit on the permissible temperature of the octave, as does
the fifteenth, I need not tell you why.

You should be very chary of reducing congruence with consonances to
percentages in the facile manner you have done. This is a habit of
thought that too often leads one to overlook many important factors. A
rigorous mathematical calculation of all the important factors is
certainly beyond you; it is also as yet beyond anyone. But it is
unnecessary to spend 300 years with the equations to know a good or bad
temperament when one hears and sees one.


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From: William Sethares
Subject: melodic lemons
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