minimizing m3 deviations with sine-wave weighting

Taking a least-squares minimization to a weight composed of the
deviations of minor thirds within each major triad, with a weight
sinusoidally varying between C maj maximum (ie E-G) and F# maj min (ie
A#-C#), plus a small equal weight for each fifth to stop the fifths
getting irregular, spits out some nice values.

For example a scale which is pretty near
0
88
96
297
391
501
588
698
792
893
1001
1089
(tempering of fifths -4,-4,-5,-4,-4,-3,-2,+2,+3,+2,-2,-3)

and, perhaps getting nearer Carl's ideal, a more equal one as
0
90
196
297
392
501
589
698
793
894
1000
1090
(tempering of fifths -4,-4,-4,-4,-4,-3,-1,+1,+2,+1,-1,-3)

A feature of minimising thirds with sinusoidal weight (plus some
smooth weight for fifths) is always that they give a chain of about
seven fifths with very nearly the same tempering. So the result is
practically a meantone between F and F#, with four 'irregular' notes,
here C#,G#,Eb,Bb.

Another feature (or bug) is that it's difficult to extirpate thirds
that are worse than Pythagorean in the remote keys without washing out
most of the inequality.
~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> Taking a least-squares minimization to a weight composed of the
> deviations of minor thirds within each major triad, with a weight
> sinusoidally varying between C maj maximum (ie E-G) and F# maj min (ie
> A#-C#), plus a small equal weight for each fifth to stop the fifths
> getting irregular, spits out some nice values.
>

I saved the 'best' (or at least most circular) for last:
92
197
298
393
501
591
697
794
895
1000
1092
(tempering of 5ths -3,-4,-4,-4,-3,-3,-1,0,+2,0,-1,-3)
...spot the 17/12's and 19/15's and 19/16's.

~~~T~~~