User:Contribution/Exploring Selected Modes in 12-EDO

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Modes

Modes of limited transposition

Period Modes
1\12 1
2\12 2
3\12 3 ; 1 2
4\12 4 ; 1 3 ; 1 1 2
6\12 6 ; 1 5 ; 2 4 ; 1 1 4 ; 1 2 3 ; 1 3 2 ; 1 1 1 3 ; 1 1 2 2 ; 1 1 1 1 2
12\12 12
All commas tempered out throughout series of 5-odd-limit intervals with all notes distinct and played
Period Mode 5-limit commas tempered out
1\12 1 1 1 1 1 1 1 1 1 1 1 1 All commas (see below)
2\12 2 2 2 2 2 2 None (128/125 for its truncation)
3\12 1 2 1 2 1 2 1 2 648/625
4\12 3 1 3 1 3 1 128/125
2 1 1 2 1 1 2 1 1
6\12 1 4 1 1 4 1 2048/2025
1 2 3 1 2 3 648/625
1 3 2 1 3 2
1 1 3 1 1 1 3 1 81/80, 128/125, 2048/2025
1 2 2 1 1 2 2 1 81/80, 648/625, 2048/2025
1 1 2 1 1 1 1 2 1 1 81/80, 128/125, 648/625, 2048/2025, 6561/6250, 82944/78125, 10616832/9765625
Period Mode Perfect circle Ratio Plagal circle
3\12 1 2 3-4 • 54 4 5 4 5 4 5 4 5 648/625 7 8 7 8 7 8 7 8 34 • 5-4
9 7 9 5 9 7 9 5 7 3 5 3 7 3 5 3
4 9 8 9 4 9 8 9 3 4 3 8 3 4 3 8
4 5 4 5 9 7 9 5 7 3 5 3 7 8 7 8
4 9 8 9 4 5 4 5 7 8 7 8 3 4 3 8
4 9 8 9 9 7 9 5 7 3 5 3 3 4 3 8
5 9 7 9 9 8 9 4 8 3 4 3 3 5 3 7
4\12 1 3 5-3 3 5 3 5 3 5 128/125 7 9 7 9 7 9 53
8 7 8 8 9 8 4 3 4 4 5 4
3 5 3 8 9 8 4 3 4 9 7 9
8 7 8 5 3 5 7 9 7 4 5 4
1 1 2 3 8 8 9 4 9 8 8 3 9 4 4 3 8 3 4 4 9
7 8 8 5 4 5 8 8 7 5 4 4 7 8 7 4 4 5
3 8 8 9 4 5 8 8 7 5 4 4 7 8 3 4 4 9
7 8 8 5 4 9 8 8 3 9 4 4 3 8 7 4 4 5
5 8 8 7 4 7 8 8 5 7 4 4 5 8 5 4 4 7
9 8 8 3 4 3 8 8 9 3 4 4 9 8 9 4 4 3
5 8 8 7 4 3 8 8 9 3 4 4 9 8 5 4 4 7
9 8 8 3 4 7 8 8 5 7 4 4 5 8 9 4 4 3
7 8 8 9 8 9 8 8 7 5 4 4 3 4 3 4 4 5
9 8 8 7 8 7 8 8 9 3 4 4 5 4 5 4 4 3
7 8 8 9 5 3 5 8 7 5 4 7 9 7 3 4 4 5
9 8 8 7 3 5 3 8 9 3 4 9 7 9 5 4 4 3
7 8 5 3 5 9 8 8 7 5 4 4 3 7 9 7 4 5
9 8 3 5 3 7 8 8 9 3 4 4 5 9 7 9 4 3
6\12 1 1 4 3-4 • 5-2 5 8 5 5 8 5 2048/2025 7 4 7 7 4 7 34 • 52
1 2 3 3-4 • 54 9 4 5 9 4 5 648/625 7 8 3 7 8 3 34 • 5-4
1 3 2 4 9 5 4 9 5 7 3 8 7 3 8
1 1 1 3 3-4 • 51 8 5 8 5 4 9 4 5 81/80 7 8 3 8 7 4 7 4 34 • 5-1
5-3 8 7 3 7 8 5 5 5 128/125 7 7 7 4 5 9 5 4 53
3-4 • 5-2 5 3 5 5 5 3 5 5 2048/2025 7 7 9 7 7 7 9 7 34 • 52
8 9 8 5 8 9 8 5 7 4 3 4 7 4 3 4
8 9 8 5 5 3 5 5 7 7 9 7 7 4 3 4
1 1 2 2 3-4 • 51 5 8 8 9 5 4 4 5 81/80 7 8 8 7 3 4 4 7 34 • 5-1
5 4 4 5 9 8 8 5 7 4 4 3 7 8 8 7
3-4 • 54 5 4 4 5 5 4 4 5 648/625 7 8 8 7 7 8 8 7 34 • 5-4
3-4 • 5-2 5 8 8 9 9 8 8 5 2048/2025 7 4 4 3 3 4 4 7 34 • 52
5 8 8 9 5 8 8 9 3 4 4 7 3 4 4 7
9 8 8 5 9 8 8 5 7 4 4 3 7 4 4 3
1 1 1 1 2 Too many solutions (130 perfect circles, 130 plagal circles)

Modes based on the circle of 3-odd-limit

Alteration Modes 5-limit commas tempered out
Schisma 2 2 1 1 1 2 1 1 1 81/80, 128/125, 648/625, 2048/2025, 6561/6250, 32805/32768
Ion 2 2 1 2 2 2 1 81/80
Ion b3 2 1 2 2 2 2 1 81/80, 648/625
Ion b6 2 2 1 2 1 3 1 81/80, 128/125, 648/625
Ion b3 b6 2 1 2 2 1 3 1 81/80, 128/125, 648/625
Ion b2 1 3 1 2 2 2 1 128/125
Ion b2 b3 1 2 2 2 2 2 1 None
Ion b2 b6 1 3 1 2 1 3 1 128/125
Ion b2 b3 b6 1 2 2 2 1 3 1 128/125
Penta MOS 2 2 3 2 3 81/80
Penta b7 2 2 3 3 2 None
Penta #4 b7 2 2 2 4 2 None
Mode Perfect circle Ratio Plagal circle
2 2 1 2 2 2 1 3-4 • 51 5 4 5 8 5 4 5 81/80 7 8 7 4 7 8 7 34 • 5-1
5 4 5 9 5 3 5 7 9 7 3 7 8 7
5 3 5 9 5 4 5 7 8 7 3 7 9 7
9 8 9 8 9 8 9 3 4 3 4 3 4 3
9 8 5 4 5 8 9 3 4 7 8 7 4 3
9 8 9 5 3 5 9 3 7 9 7 3 4 3
9 5 3 5 9 8 9 3 4 3 7 9 7 3
9 8 5 4 5 9 8 4 3 7 8 7 4 3
8 9 5 4 5 8 9 3 4 7 8 7 3 4
2 1 2 2 2 2 1 3-4 • 51 8 9 9 8 9 9 8 81/80 4 3 3 4 3 3 4 34 • 5-1
8 9 5 4 5 9 8 4 3 7 8 7 3 4
3-4 • 54 4 4 9 5 4 5 5 648/625 7 7 8 7 3 8 8 34 • 5-4
5 5 4 5 9 4 4 8 8 3 7 8 7 7
4 4 5 9 8 9 9 3 3 4 3 7 8 8
9 9 8 9 5 4 4 8 8 7 3 4 3 3
2 2 1 2 1 3 1 3-4 • 51 8 8 9 9 9 8 9 81/80 3 4 3 3 3 4 4 34 • 5-1
5 5 9 9 9 8 3 9 4 3 3 3 7 7
5 8 9 9 9 5 3 9 7 3 3 3 4 7
9 9 5 4 5 8 8 4 4 7 8 7 3 3
8 8 5 9 5 4 9 3 8 7 3 7 4 4
5-3 7 3 5 5 5 3 8 128/125 4 9 7 7 7 9 5 53
8 5 3 3 3 5 9 3 7 9 9 9 7 4
3-4 • 54 4 4 9 9 9 8 5 648/625 7 4 3 3 3 8 8 34 • 5-4
4 7 9 9 9 5 5 7 7 3 3 3 5 8
5 9 5 4 5 4 4 8 8 7 8 7 3 7
5 9 5 9 7 9 4 8 3 5 3 7 3 7
4 5 9 9 8 9 4 8 3 4 3 3 7 8
2 1 2 2 1 3 1 3-4 • 51 9 8 9 9 9 8 8 81/80 4 4 3 3 3 4 3 34 • 5-1
3 8 9 9 9 5 5 7 7 3 3 3 4 9
3 5 9 9 9 8 5 7 4 3 3 3 7 9
8 8 5 4 5 9 9 3 3 7 8 7 4 4
9 4 5 9 5 8 8 4 4 7 3 7 8 3
5-3 8 3 5 5 5 3 7 128/125 5 9 7 7 7 9 4 53
9 5 3 3 3 5 8 4 7 9 9 9 7 3
3-4 • 54 5 8 9 9 9 4 4 648/625 8 8 3 3 3 4 7 34 • 5-4
5 5 9 9 9 7 4 8 5 3 3 3 7 7
4 4 5 4 5 9 5 7 3 7 8 7 8 8
4 9 7 9 5 9 5 7 3 7 3 5 3 8
4 9 8 9 9 5 4 8 7 3 3 4 3 8
1 3 1 2 2 2 1 5-3 8 7 7 8 5 5 8 128/125 4 7 7 4 5 5 4 53
8 3 7 8 5 9 8 4 3 7 4 5 9 4
1 2 2 2 2 2 1 None
1 3 1 2 1 3 1 5-3 5 8 7 8 7 8 5 128/125 7 4 5 4 5 4 7 53
7 8 5 8 9 8 3 9 4 3 4 7 4 5
3 8 9 8 5 8 7 5 4 7 4 3 4 9
5 8 3 4 3 8 5 7 4 9 8 9 4 7
5 5 8 7 3 5 3 9 7 9 5 4 7 7
3 5 3 7 8 5 5 7 7 4 5 9 7 9
1 2 2 2 1 3 1 5-3 8 5 5 8 7 7 8 128/125 4 5 5 4 7 7 4 53
8 9 5 8 7 3 8 4 9 5 4 7 3 4
2 2 3 2 3 3-4 • 51 5 5 4 5 5 81/80 7 7 8 7 7 34 • 5-1
5 9 8 9 5 7 3 4 3 7

Blues scales

Added notes Modes 5-limit commas tempered out
#1 b3 #5/b6 1 1 1 1 3 1 1 3 81/80, 648/625, 2048/2025
#1 b3 1 1 1 1 3 2 3 81/80
#1 1 1 2 3 2 3 81/80
b3 2 1 1 3 2 3 81/80
None 2 2 3 2 3 81/80

MOS series of 5-odd-limit intervals tempering out 5-limit commas

Perfect circle Ratio Plagal circle
3-4 • 51 9 5 5 5 81/80 7 7 7 3 34 • 5-1
5 5 4 5 5 7 7 8 7 7
9 8 9 8 9 8 9 3 4 3 4 3 4 3
5-3 8 8 8 128/125 4 4 4 53
3 5 3 5 3 5 7 9 7 9 7 9
3-4 • 54 9 9 9 9 648/625 3 3 3 3 34 • 5-4
4 5 4 5 4 5 4 5 7 8 7 8 7 8 7 8
3-4 • 5-2 5 8 5 5 8 5 2048/2025 7 4 7 7 4 7 34 • 52
3 5 5 5 3 5 5 5 7 7 7 9 7 7 7 9
8 9 8 9 8 8 9 8 9 8 4 3 4 3 4 4 3 4 3 4
3-8 • 55 9 9 5 9 9 5 9 5 6561/6250 7 3 7 3 3 7 3 3 38 • 5-5
9 8 9 9 9 8 9 9 9 8 9 3 4 3 3 3 4 3 3 3 4 3
3-8 • 5-1 5 5 5 5 8 5 5 5 5 32805/32768 7 7 7 7 4 7 7 7 7 38 • 51
3 5 5 5 5 5 5 5 5 5 7 7 7 7 7 7 7 7 7 9
3-4 • 57 9 4 9 4 9 4 9 82944/78125 3 8 3 8 3 8 3 34 • 5-7
9 9 9 7 9 9 7 9 9 7 5 3 3 5 3 3 5 3 3 3
4 5 4 4 5 4 5 4 4 5 4 8 7 8 8 7 8 7 8 8 7 8
3-4 • 5-5 8 5 8 5 8 5 8 5 8 262144/253125 4 7 4 7 4 7 4 7 4 34 • 55
3-12 • 56 9 5 9 5 9 5 9 5 9 5 9 5 531441/500000 7 3 7 3 7 3 7 3 7 3 7 3 312 • 5-6
3-12 5 5 5 5 5 5 5 5 5 5 5 5 531441/524288 7 7 7 7 7 7 7 7 7 7 7 7 312
3-12 • 59 9 9 9 5 9 9 9 5 9 9 9 5 2125764/1953125 7 3 3 3 7 3 3 3 7 3 3 3 312 • 5-9
3-4 • 510 4 9 4 9 4 4 9 4 9 4 10616832/9765625 8 3 8 3 8 8 3 8 3 8 34 • 5-10
3-4 • 5-8 8 8 5 8 8 5 8 8 5 8 8 5 33554432/31640625 7 4 4 7 4 4 7 4 4 7 4 4 34 • 58
3-8 • 511 9 4 9 9 9 4 9 9 9 4 9 53747712/48828125 3 8 3 3 3 8 3 3 3 8 3 38 • 5-11

Tempered commas

All commas tempered out in 12-tet throughout series of 5-odd-limit intervals with all notes distinct
Ratio Factorization Cents Limit - Cents 1 / Factorization 1 / Ratio
81/80 2-4 • 34 • 5-1 21.506 5 -21.506 24 • 3-4 • 51 80/81
128/125 27 • 5-3 41.059 5 -41.059 2-7 • 53 125/128
648/625 23 • 34 • 5-4 62.565 5 -62.565 2-3 • 3-4 • 54 625/648
2048/2025 211 • 3-4 • 5-2 19.553 5 -19.553 2-11 • 34 • 52 2025/2048
6561/6250 2-1 • 38 • 5-5 84.071 5 -84.071 21 • 3-8 • 55 6250/6561
32805/32768 2-15 • 38 • 51 1.954 5 -1.954 215 • 3-8 • 5-1 32768/32805
82944/78125 210 • 34 • 5-7 103.624 5 -103.624 2-10 • 3-4 • 57 78125/82944
262144/253125 218 • 3-4 • 5-5 60.611 5 -60.611 2-18 • 34 • 55 253125/262144
531441/500000 2-5 • 312 • 5-6 105.578 5 -105.578 25 • 3-12 • 56 500000/531441
531441/524288 2-19 • 312 23.460 3 -23.460 219 • 3-12 524288/531441
2125764/1953125 22 • 312 • 5-9 146.637 5 -146.637 2-2 • 3-12 • 59 1953125/2125764
10616832/9765625 217 • 34 • 5-10 144.683 5 -144.683 2-17 • 3-4 • 510 9765625/10616832
33554432/31640625 225 • 3-4 • 5-8 101.670 5 -101.670 2-25 • 34 • 58 31640625/33554432
53747712/48828125 213 • 38 • 5-11 166.189 5 -166.189 2-13 • 3-8 • 511 48828125/53747712
7-limit commas tempered out in 12-tet with Benedetti height < 2**16
Ratio Factorization Cents Limit - Cents 1 / Factorization 1 / Ratio
36/35 22 • 32 • 5-1 • 7-1 48.770 7 -48.770 2-2 • 3-2 • 51 • 71 35/36
50/49 21 • 52 • 7-2 34.976 7 -34.976 2-1 • 5-2 • 72 49/50
64/63 26 • 3-2 • 7-1 27.264 7 -27.264 2-6 • 32 • 71 63/64
126/125 21 • 32 • 5-3 • 71 13.795 7 -13.795 2-1 • 3-2 • 53 • 7-1 125/126
225/224 2-5 • 32 • 52 • 7-1 7.712 7 -7.712 25 • 3-2 • 5-2 • 71 224/225
256/245 28 • 5-1 • 7-2 76.034 7 -76.034 2-8 • 51 • 72 245/256
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