ABACABADABACABA JI scales
ABACABADABACABA is the quaternary Fraenkel word or the rank-4 power SNS, i.e., the (8,4,2,1) SNS pattern. When covering a period of 2/1, such scales are known as Cantor-3 scales. ABACABADABACABA scales can be conceived of as two equivalent ABACABA scales and a remaining step D, akin to how ABACABA scales can be conceived of as two equivalent ABA tetrachords and a remaining step 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, O 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.
1225-limit ABACABADABACABA JI scales with period 2/1, with steps > 20c
'Form' is added to the tables below, following the above section on 17-form. The 'form' is the also the smallest edo to approximate the scale.
Octochord to 4/3 -> D = 9/8 (~203.91c)
| A | B | C | Scale | odd-limit of scale intervals | Form |
|---|---|---|---|---|---|
| 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 | 17 |
| 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 | 17 |
| 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 | 17 |
| 34/33 (~51.68c) | 297/289 (~47.27c) | 121/108 (~196.77c) | 1/1 34/33 18/17 12/11 11/9 34/27 22/17 4/3 3/2 17/11 27/17 18/11 11/6 17/9 33/17 2/1 | 1089 | 24 |
| 34/33 (~51.68c) | 363/340 (~113.32c) | 300/289 (~64.67c) | 1/1 34/33 11/10 17/15 20/17 40/33 22/17 4/3 3/2 17/11 33/20 17/10 30/17 20/11 33/17 2/1 | 1089 | 17 |
| 36/35 (~48.77c) | 175/162 (~133.63c) | 49/48 (~35.7c) | 1/1 36/35 10/9 8/7 7/6 6/5 35/27 4/3 3/2 54/35 5/3 12/7 7/4 9/5 35/18 2/1 | 1225 | 17 |
| 36/35 (~48.77c) | 1225/1188 (~53.1c) | 121/81 (~196.77c) | 1/1 36/35 35/33 12/11 11/9 44/35 35/27 4/3 3/2 54/35 35/22 18/11 11/6 66/35 35/18 2/1 | 1225 | 24 |
Octochord to 7/5 -> D = 50/49 (~34.98c)
| A | B | C | Scale | odd-limit of scale intervals | Form |
|---|---|---|---|---|---|
| 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 | 19 |
| 26/25 (~67.9c) | 175/169 (~60.4c) | 125/112 (~190.12c) | 1/1 26/25 14/13 28/25 5/4 13/10 35/26 7/5 10/7 52/35 20/13 8/5 25/14 13/7 25/13 2/1 | 1225 | 19 |
| 28/27 (~62.96c) | 729/700 (~70.28c) | 125/112 (~190.12c) | 1/1 28/27 27/25 28/25 5/4 35/27 27/20 7/5 10/7 40/27 54/35 8/5 25/14 50/27 27/14 2/1 | 1225 | 19 |
| 16/15 (~111.73c) | 525/512 (~43.41c) | 36/35 (~48.77c) | 1/1 16/15 35/32 7/6 6/5 32/25 21/16 7/5 10/7 32/21 25/16 5/3 12/7 64/35 15/8 2/1 | 1225 | 15 |
| 31/30 (~56.77c) | 1050/961 (~153.34c) | 36/35 (~48.77c) | 1/1 31/30 35/31 7/6 6/5 31/25 42/31 7/5 10/7 31/21 50/31 5/3 12/7 62/35 40/31 2/1 | 1225 | 15 |
| 35/34 (~50.18c) | 126/121 (~166.50c) | 36/35 (~48.77c) | 1/1 35/34 17/15 7/6 6/5 21/17 34/25 7/5 10/7 25/17 34/21 5/3 12/7 30/17 68/35 2/1 | 1225 | 15 |
Octochord to 5/4 -> D = 32/25 (~427.37c)
| A | B | C | Scale | odd-limit of scale intervals | Form |
|---|---|---|---|---|---|
| 33/32 (~53.27c) | 1120/1089 (48.59c) | 256/245 (~76.03c) | 1/1 33/32 35/33 35/32 8/7 33/28 40/33 5/4 8/5 33/20 48/33 7/4 64/35 66/35 64/33 2/1 | 1225 | 22 |
| 35/34 (~50.18c) | 289/280 (~54.77c) | 256/245 (~76.03c) | 1/1 35/34 17/16 35/32 8/7 20/17 17/14 5/4 8/5 28/17 17/10 7/4 64/35 32/17 35/17 2/1 | 1225 | 22 |
Octochord to 6/5 -> D = 25/18 (~568.72c)
| A | B | C | Scale | odd-limit of scale intervals | Form |
|---|---|---|---|---|---|
| 36/35 (~48.77c) | 49/48 (~35.70c) | 250/243 (~49.17c) | 1/1 36/35 21/20 27/25 10/9 8/7 7/6 6/5 5/3 12/7 7/4 9/5 50/27 40/21 35/18 2/1 | 1225 | 26/27 |
1225-limit ABACABADABACABA JI scales with period 3/2, with steps > 20c
Octochord to 6/5 -> D = 25/24 (~70.67c)
| A | B | C | Scale | odd-limit of scale intervals |
|---|---|---|---|---|
| 36/35 (~48.77c) | 49/48 (~35.70c) | 250/243 (~49.17c) | 1/1 36/35 21/20 27/25 10/9 8/7 7/6 6/5 5/4 9/7 21/16 27/20 25/18 10/7 35/24 3/2 | 1225 |
Noticing that A and C are almost exactly the same size, we temper them together without much loss of accuracy, tempering out 4375/4374, the Ragisma. The tempered scale then has a scale pattern of ABAAABACABAAABA (relabeling so the most frequent step is A and the least frequent is C).