Set mapping the 9-tET pelog

Paul Erlich wrote,

>9-tET contains a very believable pelog scale: its step sizes are 1 3 1 1 3.
The full pelog 7-note tuning would be 1 1 2 1 1 1 2.

By taking the relationship of:

F FF fF
d = -------------
f Ff ff

and applying it (by extrapolating its �architecture�) as a template for
generating scale steps within the (�tetrachordal�) fourth space, one could
view these two scales as the inverted 5 out of 8, and 7 out of 12� Where
the "d" of d = (d � O) � F/(d � O) � f, (d � O) � F = {[(d � O) � F] � (O �
F)}/{[(d � O) � F] � (O � f ), and (d � O) � f = {[(d � O) � f ] � (O �
F)}/{[(d � O) � f ] � (O � f )} is 8 and 12, a whole number relationship of

5 3 2 7 4 3
8 = ------------ and 12 = ------------
3 2 1 5 3 2

Inverting the whole and half steps of Ff and FF would convert the {(+ w + h)
+ w (+ w + h)} and {(+ w + w + h) +w (+ w + w +h)} of 0 2 3 5 7 8 and 0 2 4
5 7 9 11 12 into the {(+ h + w) + h (+ h + w)} and {(+ h + h + w) +h (+ h +
h +w)} of "1 3 1 1 3" and "1 1 2 1 1 1 2."

Mapping the step structure of these two scales would result in the following
sets of equidistant divisions of the octave:

"1 3 1 1 3"

Exterior set @ 1, 3, and 5
Perimeter set @ 2, 4, 6, 8, and 10
---------------------------------------------
Interior set @ 7 and 9:

"h"=1 "w"=2
0 1 3 4 5 7
0 171 514 686 857 1200

h=1 w=3
0 1 4 5 6 9
0 133 533 667 800 1200

"1 1 2 1 1 1 2"
Es @ 1, 3, 5, and 7
Ps @ 2, 4, 6, 8, 10, and 14
--------------------------------------------
Is @ 9, 11, and 13:

h=1 w=2
0 1 2 4 5 6 7 9
0 133 267 533 667 800 933 1200

h=1 w=3
0 1 2 5 6 7 8 11
0 109 218 545 655 764 873 1200

h=1 w=4
0 1 2 6 7 8 9 13
0 92 185 554 646 738 831 1200

While this is not meant to suggest that these subsets of 11 and 13 "contain
a very believable pelog�" perhaps this manner of set mapping can open up
potential paths of interest to the often �difficult� equidistant divisions
of the octave - specifically those seven non-12 members of the f�F right
triangle exterior set (8, 9, 11, 13,* 16, 18, 23).

Dan Stearns

*The following site has a 13-tET scherzo taken from a set of pieces in
composed in 9, 11 and 13:

http://members.xoom.com/Minor2nd/

(Near the bottom of the page.)