User:Contribution/Exploring Selected Modes in 12-EDO

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
4\12	1 3	5-3	3 5 3 5 3 5	128/125		53
			8 7 8 5 3 5
			3 5 3 8 9 8
			8 7 8 8 9 8
6\12	1 1 4	3-4 • 5-2	5 8 5 5 8 5	2048/2025		34 • 52
	1 2 3	3-4 • 54	9 4 5 9 4 5	648/625		34 • 5-4
	1 3 2		4 9 5 4 9 5
	1 1 1 3	3-4 • 51	5 8 5 8 5 4 9 4	81/80		34 • 5-1
		5-3	8 7 3 7 8 5 5 5	128/125		53
		3-4 • 5-2	5 3 5 5 5 3 5 5	2048/2025	7 7 7 9 7 7 7 9	34 • 52
			8 9 8 5 8 9 8 5
			8 9 8 5 5 3 5 5
	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

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

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