ABACABA JI scales: Difference between revisions
corrected limits of scales (with 3/2 and 4/3 period), changed order accordingly |
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ABACABA is the | 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 [[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 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. | ||
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== 729-limit | == 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. | 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. | ||
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|260/243 (~117.07c) | |260/243 (~117.07c) | ||
|1/1 27/26 10/9 15/13 39/30 27/20 13/9 3/2 | |1/1 27/26 10/9 15/13 39/30 27/20 13/9 3/2 | ||
| | |729 | ||
|} | |} | ||
== 729-limit | == 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. | 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. | ||
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{{Navbox scale gallery}} | |||
[[Category: Just intonation scales]] | [[Category: Just intonation scales]] | ||
[[Category: Step-nested scales]] | [[Category: Step-nested scales]] | ||
[[Category:Wakalixes]] | [[Category:Wakalixes]] | ||
[[Category:Lists of scales]] | [[Category:Lists of scales]] | ||