ABACABADABACABA JI scales
ABACABADABACABA is the (8,4,2,1) SNS pattern (the rank-4 power SNS). When covering a period of 2/1, such scales are known as Cantor-3 scales. ABACABADABACABA scales can be conceived of as two ABACABA scales separated by an interval D, akin to how ABACABA scales can be conceived of as two ABA tetrachords separated by an interval C. We will classify ABACABADABACABA scales on this page as such, grouping them by the interval subtended by ABACABA, which we will call an octochord. As step-nested scales, ABACABADABACABA scales can be described of as SNS (P, P/O, O/T, A), or equivalently as SNS (P, O, T, A) etc. where P is the period, H is the interval subtended by ABACABA, the octochord, and T is the interval subtended by ABA, the tetrachord. ABACABADABACABA scales, unlike ABA and ABACABA scales, are not pairwise-well formed, and thus their mean variety is above 4. ABACABADABACABA scales have v4454654 4564544, with mean variety ~ 4.57.
729-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c
Octochord to 4/3 -> D = 9/8 (~203.91c)
| A | B | C | Scale | odd-limit of scale intervals |
|---|---|---|---|---|
| 22/21 (~80.54c) | 126/121 (~70.10c) | 49/48 (~35.70c) | 1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1 | 441 |
| 24/23 (~73.68c) | 529/504 (~83.81c) | 49/48 (~35.70c) | 1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3 3/2 36/23 23/14 12/7 7/4 42/23 23/12 2/1 | 529 |
| 25/24 (~70.67c) | 128/125 (~41.06c) | 27/25 (~133.24c) | 1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3 3/2 25/16 8/5 5/3 9/5 15/8 48/25 2/1 | 625 |
Octochord to 7/5 -> D = 50/49 (~34.98c)
| A | B | C | Scale | odd-limit of scale intervals |
|---|---|---|---|---|
| 21/20 (~84.47c) | 64/63 (~27.26c) | 125/112 (~190.12c) | 1/1 21/20 16/15 28/25 5/4 21/16 4/3 7/5 10/7 3/2 32/21 8/5 25/14 15/8 40/21 2/1 | 625 |
17-form
The scales with an octachord to 4/3 follow 17-form, but with a gap for two notes between 4/3 and 3/2. In the case of the scale 1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1, if we wish to keep the limit at 441, a 12/11 above 4/3 and below 3/2 give 11/8 and 16/11, which are the simplest approximations of the two middle notes of 17edo in the scale's subgroup. The resulting scale,
1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 11/8 16/11 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1,
still remains within an odd-limit of 441, and has pattern ABACABADEDABACABA, with a mean variety of 6.
Alternatively we can keep the number of step intervals to 4 if we sacrifice the odd-limit. Of A, B, and C, the simplest intervals between 4/3 and 3/2 arise when we use C, resulting in
1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3 49/36 72/49 3/2 11/7 18/11 12/7 7/4 11/6 21/11 2/1,
which has an odd-limit of 2401, and a pattern of ABACABACDCABACABA, with mean variety of 5.
For the second scale, 1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3 3/2 36/23 23/14 12/7 7/4 42/23 23/12 2/1, the only way to use intervals of the scale to fill the gap whilst keeping the limit at 529 is to place A = 24/23 above 4/3 and below 3/2, resulting in
1/1 24/23 23/21 8/7 7/6 28/23 23/18 4/3 32/23 23/16 3/2 36/23 23/14 12/7 7/4 42/23 23/12 2/1,
with a step pattern of ABACABAADAABACABA, and a mean variety of 4.625.
The same true of the third scale, 1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3 3/2 25/16 8/5 5/3 9/5 15/8 48/25 2/1. We add A above 4/3 and below 3/2, resulting in the scale
1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3 25/18 36/25 3/2 25/16 8/5 5/3 9/5 15/8 48/25 2/1,
with an odd-limit of 625, a step pattern of ABACABAADAABACABA, and a mean variety of 4.625.
The final scale has gaps between 28/25 and 5/4, and between 8/5 and 25/14, where a minor third and major 6th 'should' be, which we can fill with intervals of the scale while keeping the odd-limit at 625 by adding A below 5/4 and above 8/5, leading to
1/1 21/20 16/15 28/25 25/21 5/4 21/16 4/3 7/5 10/7 3/2 32/21 8/5 42/25 25/14 15/8 40/21 2/1,
with a step pattern of ABACAABADABAACABA, with a mean variety of 4.625.