User:Contribution/Exploring Selected Modes in 12-EDO

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Commas

Distincly tempered out commas

12edo is distincly consistent in the 5-odd-limit. There are precisely 14 commas distinctly tempered out, which are ratios formed by sequences of intervals where every note is unique, except for the starting and ending notes, which remain the same.

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

Other tempered out commas

12edo remains consistent within the 9-odd-limit. Therefore, it's worthwhile to explore ratios tempered out in the 7-limit, particularly those with simple factorizations that facilitate quick harmonic operations.

7-limit commas tempered out in 12-tet within three 9-odd-limit intervals
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
360/343 23 • 32 • 51 • 7-3 83.746 7 -83.746 2-3 • 3-2 • 5-1 • 73 343/360
405/392 2-3 • 34 • 51 • 7-2 56.482 7 -56.482 23 • 3-4 • 5-1 • 72 392/405
729/686 2-1 • 36 • 7-3 105.252 7 -105.252 21 • 3-6 • 73 686/729
729/700 2-2 • 36 • 5-2 • 7-1 70.277 7 -70.277 22 • 3-6 • 52 • 71 700/729

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

Due to the octave equivalence principle within the odd-limit concept, the 5-odd-limit contains only two prime numbers: 3 and 5. Consequently, all distinctly tempered commas possess at least one rank-2 MOS series of intervals that tempers them out.

All MOS series of intervals tempering out 12-tet distinctly tempered 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


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
1\12 1 Too many (967 perfect circles, 967 plagal circles)
2\12 2 None
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 (130 perfect circles, 130 plagal circles)

Modes based on the circle of 3-odd-limit

Alteration Modes 5-limit commas tempered out
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
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
Schisma 2 2 1 1 1 2 1 1 1 81/80, 128/125, 648/625, 2048/2025, 6561/6250, 32805/32768
Mode Perfect circle Ratio Plagal circle
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
2 2 3 3 2 None
2 2 2 4 2
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 1 1 1 2 1 1 1 Too many (70 perfect circles, 70 plagal circles)

Blues scales

Added notes Modes 5-limit commas tempered out
#5/b6 2 2 3 1 1 3 648/625, 2048/2025
#1 1 1 2 3 2 3 81/80
b3 2 1 1 3 2 3 81/80
#1 b3 1 1 1 1 3 2 3 81/80
#1 b3 #5/b6 1 1 1 1 3 1 1 3 81/80, 648/625, 2048/2025
Mode Perfect circle Ratio Plagal circle
2 2 3 1 1 3 3-4 • 54 5 9 4 4 9 5 648/625 7 3 8 8 3 7 34 • 5-4
3-4 • 5-2 5 5 8 8 5 5 2048/2025 7 7 4 4 7 7 34 • 52
1 1 2 3 2 3 3-4 • 51 4 9 8 5 5 5 81/80 7 7 7 4 3 8 34 • 5-1
2 1 1 3 2 3 5 5 5 8 9 4 8 3 4 7 7 7
1 1 1 1 3 2 3 5 8 9 4 9 8 5 7 4 3 8 3 4 7
1 1 1 1 3 1 1 3 5 8 9 4 4 5 8 5 7 4 7 8 8 3 4 7
5 8 5 4 4 9 8 5 7 4 3 8 8 7 4 7
3-4 • 54 5 9 9 7 7 9 9 5 648/625 7 3 3 5 5 3 3 7 34 • 5-4
3-4 • 5-2 5 3 5 5 5 5 3 5 2048/2025 7 9 7 7 7 7 9 7 34 • 52
8 9 8 5 5 8 9 8 4 3 4 7 7 4 3 4
5 3 5 5 5 8 9 8 4 3 4 7 7 7 9 7
8 9 8 5 5 5 3 5 7 9 7 7 7 4 3 4
5 5 8 5 5 9 8 3 9 4 3 7 7 4 7 7
3 8 9 5 5 8 5 5 7 7 4 7 7 3 4 9
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