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 (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 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]].   
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.   


=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
Line 243: Line 243:


== 729-limit ABACABA scales with period 2/1, 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 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.  
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 ===
Line 1,186: Line 1,186:
|}
|}


=== Tetrachord to 5/4 -> C = 32/25 ===
=== Tetrachord to 23/20 -> C = 800/529 ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,432: 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 1,632: 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]]

Latest revision as of 02:31, 28 September 2025

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


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