ABACABA is the ternary Fraenkel word, or the rank-3 power SNS, i.e., the (4, 2, 1) SNS pattern, and the singular pairwise well-formed generalized step pattern. Such scales can be thought of as mirror-symmetric (achiral) tetrachordal scales. As step-nested scales, all ABACABA scales can be described as SNS (P, P/T, T/A), or equivalently as SNS (P, T, A) etc., where P is the period, and T = ABA, the outer interval of the tetrachord. When they span a 2/1 period (P=2), scales with this step pattern are known as Cantor-2 scales.
225-limit ABACABA scales with period 2/1, with steps > 20c
225 is chosen as the odd-limit so that the list includes all ABACABA scales with complexity up to that of the 5-limit double harmonic major scale — 16/15 5/4 4/3 3/2 8/5 15/8 2/1 — and a lower limit of 20c for step sizes is chosen so that there are no steps smaller than 81/80. For ABACABA scales, 225-odd-limit implies 13-limit.
Tetrachord to 4/3 -> C = 9/8 (~203.91c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 8/7 (~231.17c)
|
49/48 (~35.70c)
|
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
|
49
|
| 10/9 (~182.40c)
|
27/25 (~133.24c)
|
1/1 10/9 6/5 4/3 3/2 5/3 9/5 2/1
|
81
|
| 12/11 (~150.64c)
|
121/108 (~196.77c)
|
1/1 12/11 11/9 4/3 3/2 18/11 11/6 2/1
|
121
|
| 13/12 (~138.57c)
|
192/169 (~220.90c)
|
1/1 13/12 16/13 4/3 3/2 13/8 24/13 2/1
|
169
|
| 16/15 (~111.72c)
|
75/64 (~247.74c)
|
1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
|
225
|
Tetrachord to 7/5 -> C = 50/49 (~34.98c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 7/6 (~266.87c)
|
36/35 (~48.77c)
|
1/1 7/6 6/5 7/5 10/7 5/3 12/7 2/1
|
49
|
| 11/10 (~182.40c)
|
140/121 (~252.50c)
|
1/1 11/10 14/11 7/5 10/7 11/7 20/11 2/1
|
121
|
| 14/13 (~128.30c)
|
169/140 (~325.92c)
|
1/1 14/13 13/10 7/5 10/7 20/13 13/7 2/1
|
169
|
Tetrachord to 5/4 -> C = 32/25 (~427.37c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 10/9 (~182.40c)
|
81/80 (~21.51c)
|
1/1 10/9 9/8 5/4 8/5 16/9 9/5 2/1
|
81
|
| 15/14 (~119.44c)
|
49/45 (~147.43c)
|
1/1 15/14 7/6 5/4 8/5 12/7 28/15 2/1
|
225
|
| 13/12 (~138.57c)
|
180/169 (~109.17c)
|
1/1 13/12 15/13 5/4 8/5 26/15 24/13 2/1
|
225
|
Tetrachord to 9/7 -> C = 98/81 (~329.83c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 9/8 (~203.91c)
|
64/63 (~27.26c)
|
1/1 9/8 8/7 9/7 14/9 7/4 16/9 2/1
|
81
|
| 15/14 (~119.44c)
|
28/25 (~196.20c)
|
1/1 15/14 6/5 9/7 14/9 5/3 28/15 2/1
|
225
|
Tetrachord to 11/8 -> C = 128/121 (~97.36c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 11/10 (~165.00c)
|
25/22 (~221.31c)
|
1/1 11/10 5/4 11/8 16/11 8/5 20/11 2/1
|
121
|
| 9/8 (~203.91c)
|
88/81 (~143.50c)
|
1/1 9/8 11/9 11/8 16/11 18/11 16/9 2/1
|
121
|
Tetrachord to 14/11 -> C = 121/98 (~364.98c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 12/11 (~150.64c)
|
77/72 (~116.23c)
|
1/1 12/11 7/6 14/11 11/7 12/7 11/6 2/1
|
121
|
| 14/13 (~128.30c)
|
169/154 (~160.91c)
|
1/1 14/13 13/11 14/11 11/7 22/13 13/7 2/1
|
169
|
Tetrachord to 18/13 -> C = 169/162 (~73.24c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 14/13 (~128.30c)
|
117/98 (~306.79c)
|
1/1 14/13 9/7 18/13 13/9 14/9 13/7 2/1
|
169
|
| 9/8 (~203.91c)
|
128/117 (~155.56c)
|
1/1 9/8 16/13 18/13 13/9 13/8 16/9 2/1
|
169
|
| 15/13 (~247.74c)
|
26/25 (~67.90c)
|
1/1 15/13 6/5 18/13 13/9 5/3 26/15 2/1
|
225
|
Tetrachord to 13/10 -> C = 200/169 (~291.57c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 13/12 (~138.57c)
|
72/65 (~177.07c)
|
1/1 13/12 6/5 13/10 20/13 5/3 24/13 2/1
|
169
|
| 11/10 (~165.00c)
|
130/121 (~137.47c)
|
1/1 11/10 13/11 13/10 20/13 22/13 20/11 2/1
|
169
|
Tetrachord to 16/13 -> C = 169/128 (~481.06c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 14/13 (~128.30c)
|
52/49 (~102.88c)
|
1/1 14/13 8/7 16/13 13/8 7/4 13/7 2/1
|
169
|
| 16/15 (~111.72c)
|
225/208 (~136.01c)
|
1/1 16/15 15/13 16/13 13/8 26/15 15/8 2/1
|
225
|
Tetrachord to 15/11 -> C = 243/225 (~133.24c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 12/11 (~150.64c)
|
55/48 (~235.68c)
|
1/1 12/11 5/4 15/11 22/15 8/5 11/6 2/1
|
225
|
| 15/14 (~119.44c)
|
196/165 (~298.07c)
|
1/1 15/14 14/11 15/11 22/15 11/7 28/15 2/1
|
225
|
| 15/13 (~247.74c)
|
169/165 (~41.47c)
|
1/1 15/13 13/11 15/11 22/15 22/13 26/15 2/1
|
225
|
Tetrachord to 6/5 -> C = 25/18 (~568.72c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 16/15 (~111.72c)
|
135/128 (~92.18c)
|
1/1 16/15 9/8 6/5 5/3 16/9 15/8 2/1
|
225
|
729-limit ABACABA scales with period 2/1, with steps > 20c
One scale under such constraints is a degenerate case, wherein A = C: the Pythagorean diatonic scale, where A = C = 9/8, and B = 256/243, with a rank-2 form ABABABA. This scale is well-formed, and (5, 2) SNS. Specifically it is SNS (2/1, 3/2)[7]. The most complex interval in this scale is the Pythagorean augmented fourth — 729/512 — and it's inversion, the Pythagorean diminished fifth — 1024/729. Accordingly, the scale is 729-limit. 729 is chosen as the limit so that the list includes all ABACABA scales with complexity up to that of the Pythagorean diatonic scale (with steps > 20c). As step-nested scales, all other ABACABA scales with period 2/1 can be best described as SNS (2/1, 2/T, A), or equivalently as SNS (2/1, T, A), where T = ABA, the outer interval of the tetrachord. For ABACABA scales, 729-odd-limit implies 23-limit, and a 27-odd limit for A.
Tetrachord to 4/3 -> C = 9/8
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 8/7
|
49/48
|
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
|
49
|
| 10/9
|
27/25
|
1/1 10/9 6/5 4/3 3/2 5/3 9/5 2/1
|
81
|
| 12/11
|
121/108
|
1/1 12/11 11/9 4/3 3/2 18/11 11/6 2/1
|
121
|
| 13/12
|
192/169
|
1/1 13/12 16/13 4/3 3/2 13/8 24/13 2/1
|
169
|
| 16/15
|
75/64
|
1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
|
225
|
| 17/15
|
300/289
|
1/1 17/15 20/17 4/3 3/2 17/10 30/17 2/1
|
289
|
| 19/18
|
432/361
|
1/1 19/18 24/19 4/3 3/2 19/12 36/19 2/1
|
361
|
| 20/19
|
361/300
|
1/1 20/19 19/15 4/3 3/2 30/19 19/10 2/1
|
361
|
| 22/21
|
147/121
|
1/1 22/21 14/11 4/3 3/2 11/7 21/11 2/1
|
441
|
| 14/13
|
169/147
|
1/1 14/13 26/21 4/3 3/2 21/13 13/7 2/1
|
441
|
| 23/21
|
588/529
|
1/1 23/21 28/23 4/3 3/2 23/14 42/23 2/1
|
529
|
| 25/24
|
768/625
|
1/1 25/24 32/25 4/3 3/2 25/16 48/25 2/1
|
625
|
| 28/25
|
625/588
|
1/1 28/25 25/21 4/3 3/2 42/25 25/14 2/1
|
625
|
| 9/8
|
256/243
|
1/1 9/8 32/27 4/3 3/2 27/16 16/9 2/1
|
729
|
| 28/27
|
243/196
|
1/1 28/27 9/7 4/3 3/2 14/9 27/14 2/1
|
729
|
| 18/17
|
289/243
|
1/1 18/17 34/27 4/3 3/2 27/17 17/9 2/1
|
729
|
Tetrachord to 7/5 -> C = 50/49
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 7/6
|
36/35
|
1/1 7/6 6/5 7/5 10/7 5/3 12/7 2/1
|
49
|
| 11/10
|
140/121
|
1/1 11/10 14/11 7/5 10/7 11/7 20/11 2/1
|
121
|
| 14/13
|
169/140
|
1/1 14/13 13/10 7/5 10/7 20/13 13/7 2/1
|
169
|
| 21/20
|
80/63
|
1/1 21/20 4/3 7/5 10/7 3/2 40/21 2/1
|
441
|
| 16/15
|
315/256
|
1/1 16/15 21/16 7/5 10/7 32/21 15/8 2/1
|
441
|
| 17/15
|
315/289
|
1/1 17/15 21/17 7/5 10/7 34/21 30/17 2/1
|
441
|
| 21/19
|
361/315
|
1/1 21/19 19/15 7/5 10/7 30/19 38/21 2/1
|
441
|
| 23/20
|
560/529
|
1/1 23/20 28/23 7/5 10/7 23/14 40/23 2/1
|
529
|
| 28/25
|
125/112
|
1/1 28/25 5/4 7/5 10/7 8/5 25/14 2/1
|
625
|
| 28/27
|
729/560
|
1/1 28/27 27/20 7/5 10/7 40/27 27/14 2/1
|
729
|
Tetrachord to 5/4 -> C = 32/25
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 10/9
|
81/80
|
1/1 10/9 9/8 5/4 8/5 16/9 9/5 2/1
|
81
|
| 15/14
|
49/45
|
1/1 15/14 7/6 5/4 8/5 12/7 28/15 2/1
|
225
|
| 13/12
|
180/169
|
1/1 13/12 15/13 5/4 8/5 26/15 24/13 2/1
|
225
|
| 17/16
|
320/289
|
1/1 17/16 20/17 5/4 8/5 17/10 32/17 2/1
|
289
|
| 20/19
|
361/320
|
1/1 20/19 19/16 5/4 8/5 32/19 40/19 2/1
|
361
|
| 25/24
|
144/125
|
1/1 25/24 6/5 5/4 8/5 5/3 48/25 2/1
|
625
|
| 11/10
|
125/121
|
1/1 11/10 25/22 5/4 8/5 44/25 20/11 2/1
|
625
|
| 21/20
|
500/441
|
1/1 21/20 25/21 5/4 8/5 42/25 40/21 2/1
|
625
|
| 25/23
|
529/500
|
1/1 25/23 23/20 5/4 8/5 40/23 46/25 2/1
|
625
|
Tetrachord to 9/7 -> C = 98/81
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 9/8
|
64/63
|
1/1 9/8 8/7 9/7 14/9 7/4 16/9 2/1
|
81
|
| 15/14
|
28/25
|
1/1 15/14 6/5 9/7 14/9 5/3 28/15 2/1
|
225
|
| 18/17
|
289/252
|
1/1 18/17 17/14 9/7 14/9 28/27 36/17 2/1
|
289
|
| 22/21
|
567/484
|
1/1 22/21 27/22 9/7 14/9 44/27 21/11 2/1
|
729
|
| 27/26
|
676/567
|
1/1 27/26 26/21 9/7 14/9 21/13 52/27 2/1
|
729
|
| 23/21
|
567/529
|
1/1 23/21 27/23 9/7 14/9 46/27 42/23 2/1
|
729
|
| 27/25
|
625/567
|
1/1 27/25 25/21 9/7 14/9 42/25 50/27 2/1
|
729
|
Tetrachord to 11/8 -> C = 128/121
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 11/10
|
25/22
|
1/1 11/10 5/4 11/8 16/11 8/5 20/11 2/1
|
121
|
| 9/8
|
88/81
|
1/1 9/8 11/9 11/8 16/11 18/11 16/9 2/1
|
121
|
| 17/16
|
352/289
|
1/1 17/16 22/17 11/8 16/11 17/11 32/17 2/1
|
289
|
| 22/19
|
361/352
|
1/1 22/19 19/16 11/8 16/11 32/19 19/11 2/1
|
361
|
| 22/21
|
441/352
|
1/1 22/21 21/16 11/8 16/11 32/21 21/11 2/1
|
441
|
Tetrachord to 14/11 -> C = 121/98
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 12/11
|
77/72
|
1/1 12/11 7/6 14/11 11/7 12/7 11/6 2/1
|
121
|
| 14/13
|
169/154
|
1/1 14/13 13/11 14/11 11/7 22/13 13/7 2/1
|
169
|
| 23/22
|
616/529
|
1/1 23/22 28/23 14/11 11/7 23/14 44/23 2/1
|
529
|
| 28/25
|
625/616
|
1/1 28/25 25/22 14/11 11/7 44/25 25/14 2/1
|
625
|
| 28/27
|
729/616
|
1/1 28/27 27/22 14/11 11/7 44/27 56/27 2/1
|
729
|
Tetrachord to 18/13 -> C = 169/162
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 14/13
|
117/98
|
1/1 14/13 9/7 18/13 13/9 14/9 13/7 2/1
|
169
|
| 9/8
|
128/117
|
1/1 9/8 16/13 18/13 13/9 13/8 16/9 2/1
|
169
|
| 15/13
|
26/25
|
1/1 15/13 6/5 18/13 13/9 5/3 26/15 2/1
|
225
|
| 18/17
|
289/234
|
1/1 18/17 17/13 18/13 13/9 26/17 17/9 2/1
|
289
|
| 27/26
|
104/81
|
1/1 27/26 4/3 18/13 13/9 3/2 52/27 2/1
|
729
|
Tetrachord to 13/10 -> C = 200/169
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 13/12
|
72/65
|
1/1 13/12 6/5 13/10 20/13 5/3 24/13 2/1
|
169
|
| 11/10
|
130/121
|
1/1 11/10 13/11 13/10 20/13 22/13 20/11 2/1
|
169
|
| 21/20
|
520/441
|
1/1 21/20 26/21 13/10 20/13 21/13 40/21 2/1
|
441
|
| 26/25
|
125/104
|
1/1 26/25 5/4 13/10 20/13 8/5 25/13 2/1
|
625
|
Tetrachord to 16/13 -> C = 169/128
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 14/13
|
52/49
|
1/1 14/13 8/7 16/13 13/8 7/4 13/7 2/1
|
169
|
| 16/15
|
225/208
|
1/1 16/15 15/13 16/13 13/8 26/15 15/8 2/1
|
225
|
| 27/26
|
832/729
|
1/1 27/26 32/27 16/13 13/8 27/16 52/27 2/1
|
729
|
Tetrachord to 15/11 -> C = 243/225
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 12/11
|
55/48
|
1/1 12/11 5/4 15/11 22/15 8/5 11/6 2/1
|
225
|
| 15/14
|
196/165
|
1/1 15/14 14/11 15/11 22/15 11/7 28/15 2/1
|
225
|
| 15/13
|
169/165
|
1/1 15/13 13/11 15/11 22/15 22/13 26/15 2/1
|
225
|
| 23/22
|
660/529
|
1/1 23/22 30/23 15/11 22/15 23/15 44/23 2/1
|
529
|
| 25/22
|
132/125
|
1/1 25/22 6/5 15/11 22/15 5/3 44/25 2/1
|
625
|
Tetrachord to 6/5 -> C = 25/18
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 16/15
|
135/128
|
1/1 16/15 9/8 6/5 5/3 16/9 15/8 2/1
|
225
|
| 18/17
|
289/270
|
1/1 18/17 17/15 6/5 5/3 30/17 17/9 2/1
|
289
|
| 21/20
|
160/147
|
1/1 21/20 8/7 6/5 5/3 7/4 40/21 2/1
|
441
|
| 24/23
|
529/480
|
1/1 24/25 23/20 6/5 5/3 40/23 48/25 2/1
|
529
|
| 15/14
|
392/375
|
1/1 15/14 28/25 6/5 5/3 25/14 28/15 2/1
|
625
|
| 26/25
|
373/338
|
1/1 26/25 15/13 6/5 5/3 26/15 52/25 2/1
|
625
|
| 27/25
|
250/243
|
1/1 27/25 10/9 6/5 5/3 9/5 50/27 2/1
|
729
|
Tetrachord to 17/13 -> C = 338/289
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 17/16
|
256/221
|
1/1 17/16 16/13 17/13 26/17 13/8 32/17 2/1
|
289
|
| 14/13
|
221/196
|
1/1 14/13 17/14 17/13 26/17 28/17 13/7 2/1
|
289
|
| 17/15
|
225/221
|
1/1 17/15 15/13 17/13 26/17 26/15 30/17 2/1
|
289
|
| 27/26
|
884/729
|
1/1 27/26 34/27 17/13 26/17 27/17 52/27 2/1
|
729
|
Tetrachord to 22/17 -> C = 289/242
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 18/17
|
187/162
|
1/1 18/17 11/9 22/17 17/11 18/11 17/9 2/1
|
289
|
| 11/10
|
200/187
|
1/1 11/10 20/17 22/17 17/11 17/10 20/11 2/1
|
289
|
| 19/17
|
374/361
|
1/1 19/17 22/19 22/17 17/11 19/11 34/19 2/1
|
361
|
| 22/21
|
441/374
|
1/1 22/21 21/17 22/17 17/11 34/21 21/11 2/1
|
441
|
Tetrachord to 17/14 -> C = 392/289
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 17/16
|
128/117
|
1/1 17/16 8/7 17/14 28/17 7/4 32/17 2/1
|
289
|
| 15/14
|
238/225
|
1/1 15/14 17/15 17/14 28/17 30/17 28/15 2/1
|
289
|
Tetrachord to 20/17 -> C = 289/200
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 18/17
|
85/81
|
1/1 18/17 10/9 20/17 17/10 9/5 17/9 2/1
|
289
|
| 20/19
|
361/340
|
1/1 20/19 19/17 20/17 17/10 34/19 19/10 2/1
|
361
|
Tetrachord to 19/15 -> C = 540/361
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 19/18
|
108/95
|
1/1 19/18 6/5 19/15 30/19 5/3 36/19 2/1
|
361
|
| 16/15
|
285/256
|
1/1 16/15 19/16 19/15 30/19 32/19 15/8 2/1
|
361
|
| 19/17
|
289/285
|
1/1 19/17 17/15 19/15 30/19 30/17 34/19 2/1
|
361
|
Tetrachord to 24/19 -> C = 361/288
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 20/19
|
57/50
|
1/1 20/19 6/5 24/19 19/12 5/3 19/10 2/1
|
361
|
| 12/11
|
121/114
|
1/1 12/11 22/19 24/19 19/12 19/11 11/6 2/1
|
361
|
| 21/19
|
152/147
|
1/1 21/19 8/7 24/19 19/12 7/4 38/21 2/1
|
441
|
| 24/23
|
529/456
|
1/1 24/23 23/19 24/19 19/12 38/23 23/12 2/1
|
529
|
Tetrachord to 19/16 -> C = 512/361
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 19/18
|
81/76
|
1/1 19/18 9/8 19/16 32/19 16/9 36/19 2/1
|
361
|
| 17/16
|
304/289
|
1/1 17/16 19/17 19/16 32/19 34/19 32/17 2/1
|
361
|
Tetrachord to 11/9 -> C = 162/121
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 19/18
|
396/391
|
1/1 19/18 22/19 11/9 18/11 19/11 36/19 2/1
|
361
|
| 22/21
|
49/44
|
1/1 22/21 7/6 11/9 18/11 12/7 21/11 2/1
|
441
|
Tetrachord to 22/19 -> C = 361/242
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 20/19
|
209/200
|
1/1 20/19 11/10 22/19 19/11 20/11 19/10 2/1
|
361
|
| 22/21
|
441/418
|
1/1 22/21 21/19 22/19 19/11 38/21 21/11 2/1
|
441
|
Tetrachord to 21/16 -> C = 512/441
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 9/8
|
28/27
|
1/1 9/8 7/6 21/16 32/21 12/7 16/9 2/1
|
441
|
| 21/20
|
25/21
|
1/1 21/20 5/4 21/16 32/21 8/5 40/21 2/1
|
441
|
| 17/16
|
336/289
|
1/1 17/16 21/17 21/16 32/21 34/21 32/17 2/1
|
441
|
| 21/19
|
361/336
|
1/1 21/19 19/16 21/16 32/21 32/19 38/21 2/1
|
441
|
Tetrachord to 7/6 -> C = 72/49
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 21/20
|
200/189
|
1/1 21/20 10/9 7/6 12/7 9/5 40/21 2/1
|
441
|
| 19/18
|
378/361
|
1/1 19/18 21/19 7/6 12/7 38/21 36/19 2/1
|
441
|
| 25/24
|
672/625
|
1/1 25/24 28/25 7/6 12/7 25/14 48/25 2/1
|
625
|
| 28/27
|
243/224
|
1/1 28/27 9/8 7/6 12/7 16/9 27/14 2/1
|
729
|
Tetrachord to 26/21 -> C = 441/338
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 13/12
|
96/91
|
1/1 13/12 8/7 26/21 21/13 7/4 24/13 2/1
|
441
|
| 22/21
|
273/242
|
1/1 22/21 13/11 26/21 21/13 22/13 21/11 2/1
|
441
|
| 23/21
|
546/529
|
1/1 23/21 26/23 26/21 21/13 23/13 42/23 2/1
|
529
|
| 26/25
|
625/546
|
1/1 26/25 25/21 26/21 21/13 42/25 25/13 2/1
|
625
|
Tetrachord to 21/17 -> C = 578/451
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 18/17
|
119/108
|
1/1 18/17 7/6 21/17 34/21 12/7 17/9 2/1
|
441
|
| 21/20
|
400/357
|
1/1 21/20 20/17 21/17 34/21 17/10 40/21 2/1
|
441
|
Tetrachord to 8/7 -> C = 49/32
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 22/21
|
126/121
|
1/1 22/21 12/11 8/7 7/4 11/6 21/11 2/1
|
441
|
| 24/23
|
529/504
|
1/1 24/23 23/21 8/7 7/4 42/23 23/12 2/1
|
529
|
Tetrachord to 32/23 -> C = 529/512
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23
|
23/18
|
1/1 24/23 4/3 32/23 23/16 3/2 23/12 2/1
|
529
|
| 8/7
|
49/46
|
1/1 8/7 28/23 32/23 23/16 23/14 7/4 2/1
|
529
|
| 26/23
|
184/169
|
1/1 26/23 16/13 32/23 23/16 13/8 23/13 2/1
|
529
|
| 16/15
|
225/184
|
1/1 16/15 30/23 32/23 23/16 23/15 15/8 2/1
|
529
|
| 25/23
|
736/625
|
1/1 25/23 32/25 32/23 23/16 50/32 46/25 2/1
|
625
|
| 27/23
|
736/729
|
1/1 27/23 32/27 32/23 23/16 27/16 46/27 2/1
|
729
|
Tetrachord to 30/23 -> C = 529/540
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23
|
115/96
|
1/1 24/23 5/4 30/23 23/15 8/5 23/12 2/1
|
529
|
| 25/23
|
138/125
|
1/1 25/23 6/5 30/23 23/15 5/3 46/25 2/1
|
529
|
| 26/23
|
345/338
|
1/1 26/23 15/13 30/23 23/15 26/15 23/13 2/1
|
529
|
| 15/14
|
392/345
|
1/1 15/14 28/23 30/23 23/15 23/14 28/15 2/1
|
529
|
| 10/9
|
243/230
|
1/1 10/9 27/23 30/23 23/15 46/27 9/5 2/1
|
729
|
Tetrachord to 23/18 -> C = 648/529
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 10/9
|
207/200
|
1/1 10/9 23/20 23/18 36/23 40/23 9/5 2/1
|
529
|
| 23/22
|
242/207
|
1/1 23/22 11/9 23/18 36/23 18/11 44/23 2/1
|
529
|
| 19/18
|
414/361
|
1/1 19/18 23/19 23/18 36/23 38/23 36/19 2/1
|
529
|
| 23/21
|
49/46
|
1/1 23/21 7/6 23/18 36/23 12/7 42/23 2/1
|
529
|
Tetrachord to 28/23 -> C = 529/392
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23
|
161/144
|
1/1 24/23 7/6 28/23 23/14 12/7 23/12 2/1
|
529
|
| 14/13
|
169/161
|
1/1 14/13 26/23 28/23 23/14 23/13 13/7 2/1
|
529
|
| 25/23
|
644/625
|
1/1 25/23 28/25 28/23 23/14 25/14 46/25 2/1
|
625
|
| 28/27
|
729/644
|
1/1 28/27 27/23 28/23 23/14 46/27 27/15 2/1
|
729
|
Tetrachord to 23/20 -> C = 800/529
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 23/22
|
121/115
|
1/1 23/22 11/10 23/20 40/23 20/11 44/23 2/1
|
529
|
| 21/20
|
460/441
|
1/1 21/20 23/21 23/20 40/23 42/23 40/21 2/1
|
529
|
Tetrachord to 23/19 -> C = 722/529
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 20/19
|
437/400
|
1/1 20/19 23/20 23/19 38/23 40/23 19/10 2/1
|
529
|
| 23/22
|
484/437
|
1/1 23/22 22/19 23/19 38/23 19/11 44/23 2/1
|
529
|
Tetrachord to 13/11 -> C = 242/169
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 23/22
|
572/529
|
1/1 23/22 26/23 13/11 22/13 23/13 44/23 2/1
|
529
|
| 26/25
|
625/572
|
1/1 26/25 25/22 13/11 22/13 44/25 25/13 2/1
|
625
|
Tetrachord to 26/23 -> C = 529/338
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23
|
299/288
|
1/1 24/23 13/12 26/23 23/13 24/13 23/12 2/1
|
529
|
| 26/25
|
625/598
|
1/1 26/25 25/23 26/23 23/13 46/25 25/13 2/1
|
625
|
Tetrachord to 25/18 -> C = 648/625
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 10/9
|
9/8
|
1/1 10/9 5/4 25/18 36/25 8/5 9/5 2/1
|
625
|
| 25/24
|
32/25
|
1/1 25/24 4/3 25/18 36/25 3/2 48/25 2/1
|
625
|
| 7/6
|
50/49
|
1/1 7/6 25/21 25/18 32/25 42/25 12/7 2/1
|
625
|
| 25/22
|
242/225
|
1/1 25/22 11/9 25/18 36/25 18/11 44/25 2/1
|
625
|
| 19/18
|
450/361
|
1/1 19/18 25/19 25/18 36/25 38/25 36/19 2/1
|
625
|
| 25/23
|
529/450
|
1/1 25/23 23/18 25/18 36/25 36/23 46/25 2/1
|
625
|
Tetrachord to 25/19 -> C = 722/625
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 20/19
|
19/16
|
1/1 20/19 5/4 25/19 38/25 8/5 19/10 2/1
|
625
|
| 21/19
|
475/441
|
1/1 21/19 25/21 25/19 38/25 42/25 38/21 2/1
|
625
|
| 25/24
|
567/475
|
1/1 25/24 24/19 25/19 38/25 19/12 48/25 2/1
|
625
|
| 25/22
|
484/475
|
1/1 25/22 22/19 25/19 38/25 19/11 44/25 2/1
|
625
|
| 25/23
|
529/475
|
1/1 25/23 23/19 25/19 38/25 38/23 46/25 2/1
|
625
|
Tetrachord to 34/25 -> C = 625/578
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 17/15
|
18/17
|
1/1 17/15 6/5 34/25 25/17 5/3 30/17 2/1
|
625
|
| 17/16
|
512/425
|
1/1 17/16 32/25 34/25 25/17 25/16 32/17 2/1
|
625
|
| 28/25
|
425/392
|
1/1 28/25 17/14 34/25 25/17 28/17 25/14 2/1
|
625
|
| 26/25
|
425/338
|
1/1 26/25 17/13 34/25 25/17 26/17 25/13 2/1
|
625
|
| 27/25
|
850/729
|
1/1 27/25 34/27 34/25 25/17 27/17 50/27 2/1
|
729
|
Tetrachord to 32/25 -> C = 32/25
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 16/15
|
9/8
|
1/1 16/15 6/5 32/25 25/16 5/3 15/8 2/1
|
625
|
| 28/25
|
50/49
|
1/1 28/25 8/7 32/25 25/16 7/4 25/14 2/1
|
625
|
| 26/25
|
200/169
|
1/1 26/25 16/13 32/25 25/16 13/8 25/13 2/1
|
625
|
| 27/25
|
800/729
|
1/1 27/25 32/27 32/25 25/16 27/16 50/27 2/1
|
729
|
Tetrachord to 25/21 -> C = 882/625
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 25/24
|
192/175
|
1/1 25/24 8/7 25/21 42/25 7/4 48/25 2/1
|
625
|
| 22/21
|
525/484
|
1/1 22/21 25/22 25/21 42/25 44/25 21/11 2/1
|
625
|
Tetrachord to 25/22 -> C = 968/625
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 25/24
|
288/275
|
1/1 25/24 12/11 25/22 44/25 11/6 48/25 2/1
|
625
|
| 23/22
|
550/529
|
1/1 23/22 25/23 25/22 44/25 46/25 44/23 2/1
|
625
|
Tetrachord to 28/25 -> C = 625/392
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 26/25
|
175/169
|
1/1 26/25 14/13 28/25 25/14 13/7 52/25 2/1
|
625
|
| 28/27
|
729/700
|
1/1 28/27 27/25 28/25 25/14 50/27 27/14 2/1
|
729
|
Tetrachord to 27/20 -> C = 800/729
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 9/8
|
16/15
|
1/1 9/8 6/5 27/20 40/27 5/3 16/9 2/1
|
729
|
| 21/20
|
60/49
|
1/1 21/20 9/7 27/20 40/27 14/9 40/21 2/1
|
729
|
| 27/25
|
125/108
|
1/1 27/25 5/4 27/20 40/27 8/5 50/27 2/1
|
729
|
| 11/10
|
135/121
|
1/1 11/10 27/22 27/20 40/27 44/27 20/11 2/1
|
729
|
| 27/26
|
169/135
|
1/1 27/26 13/10 27/20 40/27 20/13 52/27 2/1
|
729
|
| 23/20
|
540/529
|
1/1 23/20 27/23 27/20 40/27 46/27 40/23 2/1
|
729
|
Tetrachord to 27/22 -> C = 968/729
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 12/11
|
33/32
|
1/1 12/11 9/8 27/22 44/27 16/9 11/6 2/1
|
729
|
| 27/26
|
338/297
|
1/1 27/26 13/11 27/22 44/27 22/13 52/27 2/1
|
729
|
| 27/25
|
625/594
|
1/1 27/25 25/22 27/22 44/27 44/25 50/27 2/1
|
729
|
| 23/22
|
594/529
|
1/1 23/22 27/23 27/22 44/27 46/27 44/23 2/1
|
729
|
Tetrachord to 34/27 -> C = 729/578
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 10/9
|
51/50
|
1/1 10/9 17/15 34/27 27/17 30/17 9/5 2/1
|
729
|
| 28/27
|
459/392
|
1/1 28/27 17/14 34/27 27/17 28/17 27/14 2/1
|
729
|
| 17/16
|
512/459
|
1/1 17/16 32/27 34/27 27/17 27/16 32/17 2/1
|
729
|
Tetrachord to 32/27 -> C = 729/512
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 16/15
|
25/24
|
1/1 16/15 10/9 32/27 27/16 9/5 15/8 2/1
|
729
|
| 28/27
|
54/49
|
1/1 28/27 8/7 32/27 27/16 7/4 27/14 2/1
|
729
|
Tetrachord to 9/8 -> C = 128/81
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 25/24
|
648/625
|
1/1 25/24 27/25 9/8 16/9 50/27 48/25 2/1
|
729
|
| 27/26
|
169/162
|
1/1 27/26 13/12 9/8 16/9 24/13 52/27 2/1
|
729
|
Tetrachord to 27/23 -> C = 1058/729
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23
|
69/64
|
1/1 24/23 9/8 27/23 46/27 16/9 23/12 2/1
|
729
|
| 27/26
|
676/621
|
1/1 27/26 26/23 27/23 46/27 23/13 52/27 2/1
|
729
|
Tetrachord to 15/13 -> C = 338/225
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 27/26
|
260/243
|
1/1 27/26 10/9 15/13 26/15 9/5 52/27 2/1
|
729
|
Tetrachord to 10/9 -> C = 81/50
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 28/27
|
405/392
|
1/1 28/27 15/14 10/9 9/5 28/15 27/14 2/1
|
729
|
729-limit ABACABA scales with period 3/2, with steps > 20c
Given the scales repeat at 3/2, factors of 3 in the odd-limit vary with transposition by a period. Accordingly the odd-limit listed is the odd-limit for intervals in a single period of the scale. There are no 729-limit ABACABA scales with period 3/2, with steps > 20c. The list has an effective odd-limit of 675.
Tetrachord to 9/8 -> C = 32/27 (~294.13c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 21/20 (~84.47c)
|
50/49 (~34.98c)
|
1/1 21/20 15/14 9/8 4/3 7/5 10/7 2/1
|
147
|
| 27/26 (~65.34c)
|
169/162 (~73.24c)
|
1/1 27/26 13/12 9/8 4/3 18/13 13/9 3/2
|
243
|
| 25/24 (~70.67c)
|
648/625 (~62.57c)
|
1/1 25/24 27/25 9/8 4/3 25/18 36/25 3/2
|
625
|
Tetrachord to 17/14 -> C = 294/289 (~29.70c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 17/16 (~104.96c)
|
128/117 (~115.56c)
|
1/1 17/16 8/7 17/14 28/17 21/16 24/17 3/2
|
357
|
| 15/14 (~119.44c)
|
238/225 (~97.24c)
|
1/1 15/14 17/15 17/14 28/17 45/34 7/5 3/2
|
675
|
Tetrachord to 7/6 -> C = 54/49 (~168.21c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 19/18 (~93.60c)
|
378/361 (~79.65c)
|
1/1 19/18 21/19 7/6 9/7 19/14 27/19 3/2
|
361
|
| 21/20 (~84.47c)
|
200/189 (~97.94c)
|
1/1 21/20 10/9 7/6 9/7 27/20 10/7 3/2
|
567
|
Tetrachord to 19/16 -> C = 384/361 (~106.93c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 19/18 (~93.60c)
|
81/76 (~110.31c)
|
1/1 19/18 9/8 19/16 24/19 4/3 27/19 3/2
|
361
|
Tetrachord to 6/5 -> C = 25/24 (~70.67c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 21/20 (~84.47c)
|
160/147 (~146.71c)
|
1/1 21/20 8/7 6/5 5/4 21/16 10/7 3/2
|
441
|
| 24/23 (~73.68c)
|
529/480 (~168.28c)
|
1/1 24/23 23/20 6/5 5/4 30/23 23/16 3/2
|
529
|
| 16/15 (~111.73c)
|
135/128 (~92.18c)
|
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2
|
675
|
| 18/17 (~98.95c)
|
289/270 (~117.73c)
|
1/1 18/17 17/15 6/5 5/4 45/34 17/12 3/2
|
675
|
| 27/25 (~133.24c)
|
250/243 (~49.17c)
|
1/1 27/25 10/9 6/5 5/4 27/20 25/18 3/2
|
729
|
Tetrachord to 27/23 -> C = 529/486 (~146.77c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23 (~73.68c)
|
69/64 (~130.23c)
|
1/1 24/23 9/8 27/23 23/18 4/3 23/16 3/2
|
529
|
Tetrachord to 23/20 -> C = 600/529 (~218.03c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 23/22 (~76.96c)
|
121/115 (~88.05c)
|
1/1 23/22 11/10 23/20 30/23 15/11 33/23 3/2
|
529
|
Tetrachord to 25/22 -> C = 726/625 (~259.34c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 25/24 (~70.67c)
|
288/275 (~79.96c)
|
1/1 25/24 12/11 25/22 33/25 11/8 36/25 3/2
|
625
|
Tetrachord to 10/9 -> C = 243/200 (~337.15c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 25/24 (~70.67c)
|
128/125 (~41.06c)
|
1/1 25/24 16/15 10/9 27/20 45/32 36/25 3/2
|
675
|
Tetrachord to 15/13 -> C = 169/150 (~206.47c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 27/26 (~65.34c)
|
260/243 (~117.07c)
|
1/1 27/26 10/9 15/13 39/30 27/20 13/9 3/2
|
729
|
729-limit ABACABA scales with period 4/3, with steps > 20c
2/1 period scales with two periods of these ABACABA scales and a remaining interval of 9/8 may be built, akin to octave species scales built of two copies of a tetrachord (with a 9/8 remainder). The remaining 9/8 interval may be filled in a number of different ways. There are no 729-limit ABACABA scales with period 4/3, with steps > 20c. The list has an effective odd-limit of 675.
Tetrachord to 8/7 -> C = 49/48 (~35.70c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 22/21 (~80.54c)
|
126/121 (~70.10c)
|
1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3
|
189
|
| 24/23 (~73.68c)
|
529/504 (~83.81c)
|
1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3
|
529
|
Tetrachord to 26/23 -> C = 529/507 (~73.54c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 24/23 (~73.68c)
|
299/288 (~64.89c)
|
1/1 24/23 13/12 26/23 46/39 16/13 23/18 4/3
|
529
|
Tetrachord to 10/9 -> C = 27/25 (~133.24c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 25/24 (~70.67c)
|
128/125 (~41.06c)
|
1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3
|
625
|
Tetrachord to 28/25 -> C = 625/588 (~105.65c)
| A
|
B
|
Scale
|
odd-limit of scale intervals
|
| 26/25 (~67.90c)
|
175/169 (~60.40c)
|
1/1 26/25 14/13 28/25 25/21 26/21 50/39 4/3
|
625
|