Re: Wonder Scale and Scala optimization -- thank you, Manuel

Thank you, Manuel, both for a very accessible tutorial on the
CALCULATE/LEASTSQUARE feature of Scala, and for suggesting that
"Wonder Scale" might be the best name for the 3/2^1/3 generator.

Your point that a 31-note version contains no less than 48 complete
sonorities of 12:14:18:21 and 14:18:21:24 -- when we count these two
inversion forms as distinct -- nicely illustrates the similarity to
the 3/2^1/6 or "Miracle" scale, as you say.

From this viewpoint, I'd consider your 31-note scale with pure fifths
(temp31g3.scl) as one version of "Wonder Scale," a very attractive one
for all the just "Pythagorean-like" sonorities.

Other members of the "Wonder Scale" family would be optimizations
closer to 36-tET, with least sum of squares around 233.536 cents.

When I run CALCULATE/LEASTSQUARE and implement Paul's refinement of
the optimization by giving a weight of 2.0 to the 3/2 and 12/7, by the
way, it doesn't change the result I get giving everything a default
weight of 1.0. This is the same result I got with some crude calculus,
if giving an interval a weight of 2.0 is the same as counting it
twice.

This is a really great feature of Scala, and can be used for some
complex optimizations involving more than one sonority or chord.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
> This is the same result I got with some crude calculus,
> if giving an interval a weight of 2.0 is the same as counting it
> twice.

It is. But I think you could have used _only_ the 3:2 and the 7:6, or
_only_ the 7:4 and 7:6, or any relative weighting of the two pairs,
and you'd still get the same result.