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.

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.

== 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.~~

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) ===

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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. 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.
 == 225-limit ABACABA scales with steps > 20c ==
-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.
+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) ===