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A 9-tone system in 23-tET

🔗D.Stearns <STEARNS@CAPECOD.NET>

8/17/2000 10:57:01 PM

As far as theoretically obscure lower numbered equal temperaments go,
I'm pretty sure that 23-tET must rank pretty high on the list; in fact
I can't recall ever seeing anything about any possible tempered
systems for it. In any event, 23-tET happens to work wonderfully with
the ("bimodal") 9-tone system that I've been looking at in the last
few posts.

Assuming a theoretical 18:21:23:26&5/6ths bimodal identity as the
basis for this scale, 23-tET is both consistent, and strictly proper
as it recognizes the diaschisma like 285768/279841 while still hiding
the syntonic comma like 7889/7776:

I. 0 157 261 417 522 678 783 939 1043 1200
II. 0 104 261 365 522 626 783 887 1043 1200
III. 0 157 261 417 522 678 783 939 1096 1200
IV. 0 104 261 365 522 626 783 939 1043 1200
V. 0 157 261 417 522 678 835 939 1096 1200
VI. 0 104 261 365 522 678 783 939 1043 1200
VII. 0 157 261 417 574 678 835 939 1096 1200
VIII. 0 104 261 417 522 678 783 939 1043 1200
IX. 0 157 313 417 574 678 835 939 1096 1200
X. 0 157 261 417 522 678 783 939 1043 1200

23-tET would have enharmonic pairs at the "s" and enharmonically
distinct pairs at the "L," and using the LsLsLsLsL note naming
orientation that I first gave, you would have: C, C#, Bb, B, B#/Ab, A,
A#, Ib, I, I#/Hb, H, H#, Gb, G, G#/Fb, F, F#, Eb, E, E#/Db, D, D#, Cb,
C.

Using the:
_______ _______
/ \ / \
10 15 18 23 5 8 13 18 21 3
\_____/

LsLsLsLLs note naming scheme that puts the V-I-VI, three bimodal chord
coverage in a standard lattice orientation would give a 23-tET
spelled:

A
5 H 18 C
10 F 13 E
15 D 8 G
20 Bb 3 I
2 Ib 21 B
7 Gb 16 D#
12 Eb 11 F#
17 Cb 6 H#
22 Ab 1 A#
4 Hb 19 C#
9 Fb 14 E#
14 Db 9 G#
19 Bbb 4 I#
1 Ibb 22 B#
6 Gbb 17 Dx
11 Ebb 12 Fx
16 Cbb 7 Hx
21 Abb 2 Ax
3 Hbb 20 Cx
8 Fbb 15 Ex
13 Dbb 10 Gx
18 Bbbb 5 Ix
23 Ibbb 0 Bx

Dan