new decatonic scales and dodecatonic scales?

Let's look at what I got by "relaxing" 10-tET:

0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3

Basically, that's C C# D E F F# G G# A B in a pseudo-meantone -- a chain of
fifths:

F--699--C-697.3-G-696.7-D--696--A-697.3-E--697--B-696.7-F#-697.3-C#--699--G#

Now let's try "relaxing" the 22-tET pentachordal decatonic scale. I get:

0 79 193 387 502 581 696 890 1003 1083 -- basically a mode of the above.

Now let's try "relaxing" the 22-tET symmetrical decatonic scale. I get:

0 109 199 385 501 599 698 814 999 1089

Rather than slipping into meantone, it retains more of its essential 22-tET
pattern: the two chains of fifths:

(999)--702--(501)--699---(0)---698--(698)--701--(199)

(385)--704-(1089)--710--(599)--710--(109)--705--(814)

But while one of the chains is hanging on to the large fifths of 22-tET, the
other one abandons them for 12-tET-like fifths. I'll have to take a closer
look at how this symmetry-breaking improves the diadic concordance . . .

So 12-tET turned out to be the only "stable" ET (other than 3-tET). What if
we start with other 12-tone scales? 12-out-of-31 leads to two notes merging
into one. The symmetrical dodecatonic scale in 22-tET relaxes right into
12-tET, while hexachordal dodecatonic scale in 22-tET relaxes to this scale:

0 108 200 305 408 502 610 699 806 904 1004 1110

A well-temperament!

(904)--704--(408)--702--(1110)--700--(610)--698--(108)--698--(806)--699--(30
5)
(305)--699--(1004)--698--(502)--698--(0)--699--(699)--701--(200)--704--(904)

This is fascinating! For some reason, this well-temperament is stable and
does not "relax" into 12-tET! It's ever-so-slightly more discordant than
12-tET, but you'd have to go over a "hump" to get from one to the other.
Anyway, if I take this series as starting on B, the resulting
well-temperament is:

C 0
C# 92
D 197
Eb 300
E 394
F 502
F# 591
G 698
G#/Ab 796
A 896
Bb 1002
B 1092

This is very similar to several historical well-temperaments (anyone with
Jorgenson handy want to provide details)?

I'm afraid I'll have to attribute that well-temperament that came out to
algorithmic instabilities -- because when I different modes in, it sometimes
remains there, but it sometimes "relaxes" to 12-tET.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Let's look at what I got by "relaxing" 10-tET:
>
> 0 78.3 194 387.3 501 581 697.3 777.3 890 1084.3
>
> Basically, that's C C# D E F F# G G# A B in a pseudo-meantone -- a
chain of
> fifths:
>

Could someone please explain to me, gently, what is meant by these
"relaxing" models... is it moving the notes of the equal temperament
to the "troughs" of the harmonic entropy model...or what??

Thanks!!!

___________ ____ __ __ _ _
Joseph Pehrson