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ABACABA JI scales: Difference between revisions

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ABACABA is the singular pairwise well-formed generalised step pattern, and the (4, 2, 1) [[SN scale|SNS]] pattern. Scales over a 2/1 period with this step pattern are known as Cantor-2 scales. Such scales can be thought of as mirror-symmetrical tetrachordal scales. As [[step-nested scales]], all ABACABA scales can be described as SNS (P, P/T, A), or equivalently as SNS (P, T, A), where P is the period, and T = ABA, the outer interval of the tetrachord.
ABACABA is the ternary [[Fraenkel word]], or the rank-3 power SNS, i.e., the (4, 2, 1) [[SN scale|SNS]] pattern, and the singular [[Rank-3 scale#Pairwise well-formed scales|pairwise well-formed]] generalized step pattern. Such scales can be thought of as mirror-symmetric ([[Chirality|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-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 — [[SNS (2/1, 3/2, 5/4)-7|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.   
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 — [[SNS (2/1, 3/2, 5/4)-7|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.   
Line 1,632: Line 1,632:
|}
|}


== 729-limit (675-limit) ABACABA scales with period 3/2, with steps > 20c ==
== 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.
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 6/5 -> C = 25/24 (~70.67c) ===
=== Tetrachord to 9/8 -> C = 32/27 (~294.13c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,644: Line 1,644:
|-
|-
|21/20 (~84.47c)
|21/20 (~84.47c)
|160/147 (~146.71c)
|50/49 (~34.98c)
|1/1 21/20 8/7 6/5 5/4 21/16 10/7 3/2
|1/1 21/20 15/14 9/8 4/3 7/5 10/7 2/1
|147
|147
|-
|-
|24/23 (~73.68c)
|27/26 (~65.34c)
|529/480 (~168.28c)
|169/162 (~73.24c)
|1/1 24/23 23/20 6/5 5/4 30/23 23/16 3/2
|1/1 27/26 13/12 9/8 4/3 18/13 13/9 3/2
|529
|243
|-
|-
|27/25 (~133.24c)
|25/24 (~70.67c)
|250/243 (~49.17c)
|648/625 (~62.57c)
|1/1 27/25 10/9 6/5 5/4 27/20 25/18 3/2
|1/1 25/24 27/25 9/8 4/3 25/18 36/25 3/2
|625
|625
|-
|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
|}
|}


=== Tetrachord to 9/8 -> C = 32/27 (~294.13c) ===
=== Tetrachord to 17/14 -> C = 294/289 (~29.70c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,677: Line 1,667:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|21/20 (~84.47c)
|17/16 (~104.96c)
|50/49 (~34.98c)
|128/117 (~115.56c)
|1/1 21/20 15/14 9/8 4/3 7/5 10/7 2/1
|1/1 17/16 8/7 17/14 28/17 21/16 24/17 3/2
|147
|357
|-
|-
|27/26 (~65.34c)
|15/14 (~119.44c)
|169/162 (~73.24c)
|238/225 (~97.24c)
|1/1 27/26 13/12 9/8 4/3 18/13 13/9 3/2
|1/1 15/14 17/15 17/14 28/17 45/34 7/5 3/2
|243
|-
|33/32 (~53.27c)
|128/121 (~97.36c)
|1/1 33/32 12/11 9/8 4/3 11/8 16/11 3/2
|363
|-
|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
|-
|45/44 (~38.91c)
|242/225 (~126.10c)
|1/1 45/44 11/10 9/8 4/3 15/11 22/15 3/2
|675
|675
|}
|}
Line 1,710: Line 1,685:
!Scale
!Scale
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|21/20 (~84.47c)
|200/189 (~97.94c)
|1/1 21/20 10/9 7/6 9/7 27/20 10/7 3/2
|243
|-
|-
|19/18 (~93.60c)
|19/18 (~93.60c)
Line 1,720: Line 1,690:
|1/1 19/18 21/19 7/6 9/7 19/14 27/19 3/2
|1/1 19/18 21/19 7/6 9/7 19/14 27/19 3/2
|361
|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 15/13 -> C = 169/150 (~206.47c) ===
=== Tetrachord to 19/16 -> C = 384/361 (~106.93c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,730: Line 1,705:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|27/26 (~65.34c)
|19/18 (~93.60c)
|260/243 (~117.07c)
|81/76 (~110.31c)
|1/1 27/26 10/9 15/13 39/30 27/20 13/9 3/2
|1/1 19/18 9/8 19/16 24/19 4/3 27/19 3/2
|243
|361
|}
|}


=== Tetrachord to 17/14 -> C = 294/289 (~29.70c) ===
=== Tetrachord to 6/5 -> C = 25/24 (~70.67c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,744: Line 1,719:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|17/16 (~104.96c)
|21/20 (~84.47c)
|128/117 (~115.56c)
|160/147 (~146.71c)
|1/1 17/16 8/7 17/14 28/17 21/16 24/17 3/2
|1/1 21/20 8/7 6/5 5/4 21/16 10/7 3/2
|289
|441
|-
|-
|15/14 (~119.44c)
|24/23 (~73.68c)
|238/225 (~97.24c)
|529/480 (~168.28c)
|1/1 15/14 17/15 17/14 28/17 45/34 7/5 3/2
|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
|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 19/16 -> C = 384/361 (~106.93c) ===
=== Tetrachord to 27/23 -> C = 529/486 (~146.77c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,763: Line 1,753:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|19/18 (~93.60c)
|24/23 (~73.68c)
|81/76 (~110.31c)
|69/64 (~130.23c)
|1/1 19/18 9/8 19/16 24/19 4/3 27/19 3/2
|1/1 24/23 9/8 27/23 23/18 4/3 23/16 3/2
|361
|529
|}
|}


=== Tetrachord to 33/28 -> C = 392/363 (~133.06c) ===
=== Tetrachord to 23/20 -> C = 600/529 (~218.03c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,777: Line 1,767:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|15/14 (~119.44c)
|23/22 (~76.96c)
|77/75 (~45.56c)
|121/115 (~88.05c)
|1/1 15/14 11/10 33/28 14/11 15/11 7/5 3/2
|1/1 23/22 11/10 23/20 30/23 15/11 33/23 3/2
|363
|529
|-
|33/32 (~53.27c)
|256/231 (~177.90c)
|1/1 33/32 8/7 33/28 14/11 21/16 16/11 3/2
|363
|}
|}


=== Tetrachord to 13/12 -> C = 216/169 (~424.81c) ===
=== Tetrachord to 25/22 -> C = 726/625 (~259.34c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,796: Line 1,781:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|39/38 (~44.97c)
|25/24 (~70.67c)
|361/351 (~48.63c)
|288/275 (~79.96c)
|1/1 39/38 19/18 13/12 18/13 27/19 19/13 3/2
|1/1 25/24 12/11 25/22 33/25 11/8 36/25 3/2
|507
|625
|}
|}


=== Tetrachord to 21/19 -> C = 361/294 (~355.42c) ===
=== Tetrachord to 10/9 -> C = 243/200 (~337.15c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,810: Line 1,795:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|39/38 (~44.97c)
|25/24 (~70.67c)
|532/507 (~83.33c)
|128/125 (~41.06c)
|1/1 39/38 14/13 21/19 19/14 39/28 19/13 3/2
|1/1 25/24 16/15 10/9 27/20 45/32 36/25 3/2
|507
|675
|}
|}


=== Tetrachord to 39/34 -> C = 578/507 (~226.90c) ===
=== Tetrachord to 15/13 -> C = 169/150 (~206.47c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,824: Line 1,809:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|18/17 (~98.95c)
|27/26 (~65.34c)
|221/216 (~39.62c)
|260/243 (~117.07c)
|1/1 18/17 13/12 39/34 17/13 18/13 17/12 3/2
|1/1 27/26 10/9 15/13 39/30 27/20 13/9 3/2
|507
|729
|}
|}


=== Tetrachord to 23/20 -> C = 600/529 (~218.03c) ===
== 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) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,838: Line 1,826:
!odd-limit of scale intervals
!odd-limit of scale intervals
|-
|-
|23/22 (~76.96c)
|22/21 (~80.54c)
|121/115 (~88.05c)
|126/121 (~70.10c)
|1/1 23/22 11/10 23/20 30/23 15/11 33/23 3/2
|1/1 22/21 12/11 8/7 7/6 11/9 14/11 4/3
|529
|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 27/23 -> C = 529/486 (~146.77c) ===
=== Tetrachord to 26/23 -> C = 529/507 (~73.54c)===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,853: Line 1,846:
|-
|-
|24/23 (~73.68c)
|24/23 (~73.68c)
|69/64 (~130.23c)
|299/288 (~64.89c)
|1/1 24/23 9/8 27/23 23/18 4/3 23/16 3/2
|1/1 24/23 13/12 26/23 46/39 16/13 23/18 4/3
|529
|529
|}
|}


=== Tetrachord to 25/22 -> C = 726/625 (~259.34c) ===
=== Tetrachord to 10/9 -> C = 27/25 (~133.24c) ===
{| class="wikitable"
|+
!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
|-
|45/44 (~38.91c)
|88/81 (~143.50c)
|1/1 45/44 10/9 25/22 33/25 27/20 22/15 3/2
|675
|}
 
=== Tetrachord to 10/9 -> C = 243/200 (~337.15c) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,887: Line 1,861:
|25/24 (~70.67c)
|25/24 (~70.67c)
|128/125 (~41.06c)
|128/125 (~41.06c)
|1/1 25/24 16/15 10/9 27/20 45/32 36/25 3/2
|1/1 25/24 16/15 10/9 6/5 5/4 32/25 4/3
|675
|}
 
=== Tetrachord to 11/10 -> C = 150/121 (~371.95c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|33/32 (~53.27c)
|512/495 (~58.46c)
|1/1 33/32 16/15 11/10 15/11 45/32 16/11 3/2
|675
|}
 
=== Tetrachord to 12/11 -> C = 121/96 (~400.68c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|45/44 (~38.91c)
|704/675 (~72.83c)
|1/1 45/44 16/15 12/11 11/8 45/32 22/15 3/2
|675
|}
 
=== Tetrachord to 45/38 -> C = 722/675 (~116.53c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|15/14 (~119.44c)
|98/95 (~53.83c)
|1/1 15/14 21/19 45/38 19/15 19/14 7/5 3/2
|675
|-
|39/38 (~44.97c)
|190/169 (~202.77c)
|1/1 39/38 15/13 45/38 19/15 13/10 19/13 3/2
|675
|}
 
== 729-limit (675-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) ===
{| class="wikitable"
|+
!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
|147
|-
|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 12/11 -> C = 121/108 (~196.77c)===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|34/33 (~51.68c)
|297/289 (~47.27c)
|1/1 34/33 18/17 12/11 34/27 22/17 4/3
|363
|}
 
=== Tetrachord to 17/15 -> C = 300/289 (~64.67c)  ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|34/33 (~51.68c)
|363/340 (113.32c)
|1/1 34/33 11/10 17/15 20/17 40/33 22/17 4/3
|363
|}
 
=== Tetrachord to 10/9 -> C = 27/25 (~133.24c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|40/39 (~43.83c)
|169/160 (~94.74c)
|1/1 40/39 13/12 10/9 6/5 16/13 13/10 4/3
|507
|-
|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
|-
|28/27 (~62.96c)
|405/392 (~56.48c)
|1/1 28/27 15/14 10/9 6/5 56/45 9/7 4/3
|675
|}
 
=== Tetrachord to 44/39 -> C = 507/484 (~80.37c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|22/21 (~80.54c)
|147/143 (~47.76c)
|1/1 22/21 14/13 44/39 13/11 26/21 14/11 4/3
|507
|-
|40/39 (~43.83c)
|429/400 (~121.17c)
|1/1 40/39 11/10 44/39 13/11 40/33 13/10 4/3
|507
|}
 
=== Tetrachord to 26/23 -> C = 529/507 (~73.54c) ===
{| class="wikitable"
|+
!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 16/15 -> C = 75/64 (~274.58c) ===
{| class="wikitable"
|+
!A
!B
!Scale
!odd-limit of scale intervals
|-
|40/39 (~43.83c)
|507/500 (~24.07c)
|1/1 40/39 26/25 16/15 5/4 50/39 13/10 4/3
|625
|625
|}
|}
Line 2,073: Line 1,879:
|}
|}


{{Navbox scale gallery}}
[[Category: Just intonation scales]]
[[Category: Just intonation scales]]
[[Category: Step-nested scales]]
[[Category: Step-nested scales]]
[[Category:Wakalixes]]
[[Category:Wakalixes]]
[[Category:Lists of scales]]
[[Category:Lists of scales]]
Retrieved from "https://en.xen.wiki/w/ABACABA_JI_scales"