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Date: Tue, 7 Jul 1998.
Subject: decatonic scales in Partchian terms
Carl Lumma wrote me, privately, regarding my 22-tET paper:
>I recommend cutting the indian scale bit, and instead write about the
>just versions of the decatonic scales using standard Partchian terms.
Did you mean something like this?
Let's start with the familiar diatonic (heptatonic) scale in JI.
1. 1/1
2. 10/9 or 9/8
3. 5/4
4. 4/3
5. 3/2
6. 5/3
7. 15/8
Mode 1 is major, mode 6 is minor. The tuning of note 2 depends on the
chordal context. The following chords are complete through the 5-limit:
1,3,5 = 1/1:5/4:3/2 (1/1-O)
2,4,6 = 10/9:4/3:5/3 (5/3-U)
3,5,7 = 5/4:3/2:15/8 (15/8-U)
4,6,1 = 4/3:5/3:1/1 (4/3-O)
5,7,2 = 3/2:15/8:9/8 (3/2-O)
6,1,3 = 5/3:1/1:5/4 (5/4-U)
7,2,4 = X
A meantone tuning, including 12-, 19-, or 31-tET, allows a single pitch
to be used for note 2.
Now the pentachordal (so named because it is built of pentachords
instead of tetrachords) decatonic scale in JI:
1. 1/1
2. 21/20 or 15/14
3. 9/8
4. 5/4
5. 21/16
6. 10/7
7. 3/2
8. 5/3
9. 7/4 or 50/28
0. 15/8
Mode 1 is standard major, mode 4 is standard minor, mode 6 is alternate
major, mode 9 is alternate minor. The following chords are complete
through the 7-limit:
1,4,7,9 = 1/1:5/4:3/2:7/4 (1/1-O)
2,5,8,0 = X
3,6,9,1 = X
4,7,0,2 = 5/4:3/2:15/8:15/14 (15/8-U)
5,8,1,3 = X
6,9,2,4 = 10/7:50/28:15/14:5/4 (10/7-O)
7,0,3,5 = 3/2:15/8:9/8:21/16 (3/2-O)
8,1,4,6 = 5/3:1/1:5/4:10/7 (5/4-U)
9,2,5,7 = 7/4:21/20:21/16:3/2 (21/16-U)
0,3,6,8 = X
The symmetrical decatonic scale has at least two equally economical
representations in JI:
REPRESENTATION 1:
1. 1/1 or 63/64
2. 21/20 or 15/14
3. 9/8
4. 5/4
5. 21/16 or 75/56
6. 45/32 or 10/7
7. 3/2
8. 63/40 or 45/28
9. 7/4 or 50/28
0. 15/8
REPRESENTATION 2:
1. 1/1 or 63/64
2. 21/20 or 15/14
3. 9/8
4. 49/40 or 5/4
5. 21/16
6. 7/5 or 45/32
7. 3/2
8. 63/40 or 45/28
9. 7/4
0. 147/80 or 15/8
Modes 1 and 6 are static major, modes 2 and 7 are dynamic major, modes 5
and 0 are static minor, and modes 4 and 9 are dynamic minor. The
following chords are complete through the 7-limit:
REPRESENTATION 1:
1,4,7,9 = 1/1:5/4:3/2:7/4 (1/1-O)
2,5,8,0 = 15/14:75/56:45/28:15/8 (15/14-O)
3,6,9,1 = X
4,7,0,2 = 5/4:3/2:15/8:15/14 (15/8-U)
5,8,1,3 = 21/16:63/40:63/64:9/8 (63/64-U)
6,9,2,4 = 10/7:5/4:15/14:5/4 (10/7-O)
7,0,3,5 = 3/2:15/8:9/8:21/16 (3/2-O)
8,1,4,6 = X
9,2,5,7 = 7/4:21/20:21/16:3/2 (21/16-U)
0,3,6,8 = 15/8:9/8:45/32:45/28 (45/32-U)
REPRESENTATION 2:
1,4,7,9 = 1/1:5/4:3/2:7/4 (1/1-O)
2,5,8,0 = 21/20:21/16:63/40:147/80 (21/20-O)
3,6,9,1 = X
4,7,0,2 = 5/4:3/2:15/8:15/14 (15/8-U)
5,8,1,3 = 21/16:63/40:63/64:9/8 (63/64-U)
6,9,2,4 = 7/5:7/4:21/20:49/40 (7/5-O)
7,0,3,5 = 3/2:15/8:9/8:21/16 (3/2-O)
8,1,4,6 = X
9,2,5,7 = 7/4:21/20:21/16:3/2 (21/16-U)
0,3,6,8 = 15/8:9/8:45/32:45/28 (45/32-U)
22-tET allows a single pitch to be used for each note in the decatonic
scales.