________________________________________________________________________ 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.