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 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.

== 225-limit ABACABA scales with period 2/1, with steps > 20c ==

== 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 — [[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]].

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== 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.

=== Tetrachord to 4/3 -> C = 9/8 ===

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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 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.
+== 225-limit ABACABA scales with period 2/1, with steps > 20c ==
-== 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 — [[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]].
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 |}
-== 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.
 === Tetrachord to 4/3 -> C = 9/8 ===