two new meantones and paultones (cash value: 1/20 of 1¢)

One of my favorite websites,

http://www.kees.cc/tuning/perbl.html,

makes use, for evaluating an ET's approximation to each JI ratio, of a
weighting inversely proportional to the logarithm of the ratio's odd limit.

Applying this to meantone within the 5-limit yields the following optimal
meantone "narrow fifth" generators:

least sum of squared weighted errors: 696.4093 cents

least weighted sum of squared errors: 696.2666 cents

So it appears that Kees's weighting implies an optimal meantone fifth
somewhere in the range of 696.26 to 696.41 cents. 31-tET is close at 696.77
cents; 50-tET brackets the other side at 696 cents.

For RI fans like Jacky, these optimal fifths are within 1/20th of 1 cent of
151/101 and 157/105, respectively.

As you might have guessed, a "paultone" scale has two chains of _wide_
fifths ("decsevenths") a half-octave apart, with all the 5- and 7-limit
approximations of the decatonic scale. Applying Kees's weighting:

least sum of squared weighted errors: 707.8676 cents

least weighted sum of squared errors: 708.3646 cents

So it appears that Kees's weighting implies an optimal paultone fifth
somewhere in the range of 707.86 to 708.37 cents. 22-tET is close at 709.09
cents; 56-tET brackets the other side at 707.14 cents.

For RI fans like Jacky, these optimal decsevenths are within 1/10th of 1
cent of 146/97 and 134/89, respectively. And the half-octave is within
1/10th of 1 cent of 99/70.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> One of my favorite websites,
>
> http://www.kees.cc/tuning/perbl.html,
>
> makes use, for evaluating an ET's approximation to each JI ratio,
of a
> weighting inversely proportional to the logarithm of the ratio's
odd limit.
>
>
> Applying this to meantone within the 5-limit yields the following
optimal
> meantone "narrow fifth" generators:
>
> least sum of squared weighted errors: 696.4093 cents

Paul,

This is fascinating! These are known on the Prime Continuum too.
1087/727 @ 696.390.

>
> least weighted sum of squared errors: 696.2666 cents

Got it:
761/509 696.277

>
> As you might have guessed, a "paultone" scale has two chains of
_wide_
> fifths ("decsevenths") a half-octave apart, with all the 5- and 7-
limit
> approximations of the decatonic scale. Applying Kees's weighting:
>
> least sum of squared weighted errors: 707.8676 cents

I detect that we've got "impostor Paul-Tones" here along the Prime
road:
No problem(!):
733/487 707.870

>
> least weighted sum of squared errors: 708.3646 cents

Please(!):
929/617 708.490

>
> So it appears that Kees's weighting implies an optimal paultone
fifth
> somewhere in the range of 707.86 to 708.37 cents. 22-tET is close
at 709.09
> cents; 56-tET brackets the other side at 707.14 cents.

Wake me up when this is over:
733/487 707.870
929/617 708.490
857/569 709.040
1163/773 707.173

Looks like we've found much common ground here Paul! Justly I say!
It's the "audible quality" that makes brothers of us all.

}: )

Jacky Ligon