[Fwd: [tuning] Re: optimal tuning for diatonic scale]

I sent this message on Tue, 06 Apr 1999 14:47:25 -0400, but somehow, it never
got posted.

Brett Barbaro wrote:

> > From: manuel.op.de.coul@ezh.nl
> >
> > > (Harry Partch's Observation One: the field of attraction is inversely
> > proportional
> > > to the limit of the interval).
> >
> > Unfortunately the tolerance for mistuning works the other way.
>
> If that's so, then the equation becomes
>
> x/3=1/5*(comma-3x)
> x/3=1/5*comma-3/5*x
> x*(1/3+3/5)=1/5*comma
> x*14/15=1/5*comma
> x=3/14*comma
>
> In 3/14-comma meantone temperament,
> the fifths are 4.6 cents flat,
> the major thirds are 3.1 cents sharp,
> and the minor thirds are 7.7 cents flat.
>
> The ranking changes to this:
>
> Tuning Worst Error (cents/limit)
> 3/14-comma meantone 1.54
> 2/9-comma meantone 1.59
> 1/5-comma meantone 1.72
> 31-equal 1.73
> 43-equal 1.73
> 1/4-comma meantone 1.79
> Kornerup's Golden 1.91
> 50-Equal 1.99
> 55-equal 2.04
> 2/7-comma meantone 2.05
> 1/6-comma meantone 2.15
> LucyTuning 2.15
> 1/3-comma meantone 2.39
> 19-equal 2.41
> 12-equal 3.13
> 26-equal 3.42
> Pythagorean 4.30
>
> But if that is so, that would mean that intervals like 11/10 and 12/11,
> which Harry Partch called consonant, can be mistuned more than twice as much
> as thirds. If minor thirds can be tolerated with an error of 7 cents, then
> by Manuel Op de Coul's reckoning 11/10 can be tolerated if tuned to 12/11
> and 12/11 can be tolerated if tuned to 11/10. But if they are consonant,
> shouldn't there be a dissonant region between them?