The 1/20 as a 28/27(?)

Earlier this week I posted this in a thread dealing with categorizing
step sizes:

"Well not too long ago I posted this (ear derived)

0 7 12
8 15 20
[11] 18 3

+3 +4 +1 +4 +3 +3 +2 scale in 20e... and while it has four different
step sizes, I still see it as (some variety of) a "two step size" 5L &
2s "diatonic" scale..."

This inspired me to go back and attempt to give this scale some type
of analytical attention... What I was mostly interested in was trying
to flesh out why I had had such a strong ear attraction to this scale,
and after giving it some thought and careful listening, I really feel
that this scales most defining (and inveigling) characteristic is the
s=1/20 step - and that this is perhaps best understood here as a
28/27... It seems to me that this scale could be described as a
(heavily 20e colored) chain of fifths with a tetrad formed on the 2nd
and 3rd fifths:

15/8
/ \
/ \
/21/16\
/ \
I) 1/1-------3/2-------9/8-------27/16------81/64

5/3
/ \
/ \
/ 7/6 \
/ \
II) 16/9------4/3-------1/1-------3/2------9/8

40/27
/ \
/ \
/28/27\
/ \
III) 128/81-----32/27-----16/9-----4/3------1/1

10/7
/ \
/ \
/ 1/1 \
/ \
IV) 32/21------8/7------12/7------9/7-----27/14

5/4
/ \
/ \
/ 7/4 \
/ \
V) 4/3-------1/1-------3/2-------9/8------27/16

10/9
/ \
/ \
/14/9 \
/ \
VI) 32/27-------16/9-------4/3--------1/1--------3/2

1/1
/ \
/ \
/ 7/5 \
/ \
VII) 16/15-------8/5-------6/5-------9/5------27/20

Of course it is readily apparent that consistency (20e is only
consistent through the 3-limit) will play havoc on (not surprisingly)
the consistency of any lattice representation (this represents the
1/1, 9/8, 81/64, 21/16, 3/2, 27/16, 15/8, 2/1 cross set shown as best
[(LOG(N)-LOG(D))*(n/LOG(2)] appproximaxtions where n=20):

11----03----15~~~~06----18
/' /\ '\ 10 /' /\
/ ' / \ ' \/'/ ' / \
/ 01-'/-13~\'~04/\/'16-'/-08 \
13----05----17~~~~08--'-20-'--12~~~~03----15----07
\ 12-/'-04'/\/16~'\~07-/'-19 /
\ / ' /'/\ ' \ / ' /
\/ '/ 10 \' \/ '/
02----14~~~~05----17----09

But if the +3 +4 +1 +4 +3 +3 +2 is seen as the rotations of a 1/1,
9/8, 81/64, 21/16, 4/3, 27/16, 15/8, 2/1, the degrees (or fractions)
of 20e would have to be recognized as the following intervals:

I II III IV V VI VII
0 1/1...
1 28/27
2 16/15
3 9/8 [9/8] 10/9
4 9/8 8/7
5 7/6 32/27 [32/27] 6/5
6 5/4
7 81/64 9/7
8 21/16 4/3 [4/3] [4/3]
9 4/3 27/20
10 10/7 7/5
11 40/27 3/2
12 3/2 [3/2] 32/21 [3/2]
13 128/81 14/9
14 8/5
15 27/16 5/3 12/7 [27/16]
16 16/9 7/4
17 16/9 [16/9] 9/5
18 15/8
19 27/14
20 2/1...

or:

11------03------15~~~~~~06------18
/\ /\ /\ 10 /\ /\
/ \ / \ / \ /\ / \ / \
/ 01-\--/-13-\--/05~~\~~/~~16\--/-08 \
/ 16/ 08/ \ / \/ \ / 11/ 03
13--05-----17------09----X--20--X--12------04-------15--07
16\ 08\ / \ /\ / \ 11\ 03\ /
\ 12-/--\04~~/~~\~~15/--\-07-/--\-19 /
\ / \ / \/ \ / \ / \ /
\/ \/ 10 \/ \/ \/
02------14~~~~~~05------17------09

The one consistent factor here is the step structure, all the modes
maintain the same 5L & 2s where 1 = 28/27, 2 = 16/15, 3 = 10/9 & 9/8
twice, and 4 = 9/8 & 8/7.

Dan