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ABACABA JI scales: Difference between revisions

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Added section 225-limit
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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.
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.
== 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 — with steps > 20c, so that there are no steps smaller than 81/80. 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.
=== Tetrachord to 4/3 -> C = 9/8 (~203.91c) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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) ===
{| class="wikitable"
|+
!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 steps > 20c ==
== 729-limit ABACABA scales with steps > 20c ==
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|-
|-
|7/6
|7/6
|36/25
|36/35
|1/1 7/6 6/5 7/5 10/7 5/3 12/7 2/1
|1/1 7/6 6/5 7/5 10/7 5/3 12/7 2/1
|49
|49
Retrieved from "https://en.xen.wiki/w/ABACABA_JI_scales"