new (?) method for tempered circular tuning: irregular mapping

We are familiar with methods of finding 'best fit' scales that
minimise some weighted function of deviations of fifths and thirds
from 5-limit JI. (Also for regular temperaments that minimise some
function treating all fifths and thirds equally...)

In some kind of extension of this, what I recently did was to choose a
set of 19-limit JI intervals, each of which is to be mapped from a
specific pair of notes from a 12-tone scale. Most of the major thirds
are mapped to 5/4, but some 'modified' ones are mapped to 24/19, or
19/15, or 14/11; the 'modified' minor thirds are mapped to 19/16 or
13/11; 'regular' tritones are mapped to 7/5, some 'modified' ones are
mapped to 24/17.

Then assigning some weights to all these intervals, and to each of 12
fifths, we can minimize overall and get to a scale that tempers not
only 81/80 and the Pythagorean comma, but also a group of much more
complicated 'commas' associated with 19-limit intervals where they
occur in the mapping.

One might think of it as assigning a given JI 'character' to specific
intervals in the 12-note scale and then seeing what tuning produces
the best possible realization of these.

My best effort so far here is:

F-A, C-E, G-B, D-F# -> 5/4
E-G#, Eb-G -> 24/19
B-D#, Ab-C -> 19/15
F#-A#, Db-F -> 14/11

C-Eb, G#-B -> 19/16
D#-F#, F-Ab -> 13/11

C-F#, F-B -> 7/5
Eb-A, D-G# -> 24/17

(& all fifths to 3/2)

One curious comma here is 4693/4675, which I've never seen before...
entering as the difference between 30/19 . 11/13 . 5/4 and 19/16 . 24/17.

Numerical results come later.
There may be some relation with Margo's double-sided circular tuning
combining meantone and neo-medieval features.
~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> My best effort so far here is:
>
> F-A, C-E, G-B, D-F# -> 5/4
> E-G#, Eb-G -> 24/19
> B-D#, Ab-C -> 19/15
> F#-A#, Db-F -> 14/11
>
> C-Eb, G#-B -> 19/16
> D#-F#, F-Ab -> 13/11
>
> C-F#, F-B -> 7/5
> Eb-A, D-G# -> 24/17
>
> (& all fifths to 3/2)
>

- In addition I find that C#-Bb -> 13/11 is quite reasonable.

Using this mapping and setting *all* weights to 1, but forcing the
fifths F-C-G-D-A-E-B-F# to be equal, I get the following results for
the deviations from just intervals.

Fifths are
(-4.8, -4.8, -4.8, -4.8, -4.8, -4.8, 1.8, 2.2, 1.8, 2.2, 1.8, -4.8)
round the circle

Deviations from (5/4, 24/19, 19/15, 14/11) are (2.4, -2.1, 0.3, -1.6)
resp.

Deviations from (19/16, 13/11) are (-2.6, -0.9 or -1.3(Bb-C#)) resp.

Deviations from (7/5, 24/17) are (0.6, -0.3) resp.

For reference the scale is
0
697.2
194.4
891.6
388.8
1086.0
583.1
86.9
791.1
294.9
999.0
502.8

- previously I had 'guessed' by hand 0, 698, 195, 892, 389, 1087, 584,
86, 791, 296, 1001, 503.
~~~T~~~