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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 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 (2/1, 2/T, A), or equivalently as SNS (2/1, T, A), where 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 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-limit ABACABA scales with steps > 20c ==
225 is chosen as the 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.   


=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
Line 244: Line 242:
|}
|}


== 729-limit ABACABA scales with steps > 20c ==
== 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 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.  
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 ===
=== Tetrachord to 4/3 -> C = 9/8 ===
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|}
|}


=== Tetrachord to 5/4 -> C = 32/25 ===
=== Tetrachord to 23/20 -> C = 800/529 ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,434: Line 1,432:
|23/22
|23/22
|550/529
|550/529
|1/1 23/22 25/23 25/22 44/24 46/25 44/23 2/1
|1/1 23/22 25/23 25/22 44/25 46/25 44/23 2/1
|625
|625
|}
|}
Line 1,634: Line 1,632:
|}
|}


== 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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| 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
|}
=== Tetrachord to 10/9 -> C = 243/200 (~337.15c) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| 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
|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)===
{| 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 10/9 -> C = 27/25 (~133.24c) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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
|}
{{Navbox scale gallery}}
[[Category: Just intonation scales]]
[[Category: Just intonation scales]]
[[Category: Step-nested scales]]
[[Category: Step-nested scales]]
[[Category:Wakalixes]]
[[Category:Lists of scales]]
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