ABACABA JI scales - Revision history

BudjarnLambeth: {{Navbox scale gallery}}

2025-09-28T02:31:06Z

← Older revision		Revision as of 02:31, 28 September 2025
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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]]

Lhearne: edited first sentence to highlight equivalence of Fraenkel words to power SNS

2024-02-19T02:33:40Z

edited first sentence to highlight equivalence of Fraenkel words to power SNS

← Older revision		Revision as of 02:33, 19 February 2024
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	ABACABA is the ternary [[Fraenkel word]] and the singular [[Rank-3 scale#Pairwise well-formed scales\|pairwise well-formed]] generalized step ~~pattern, and the (4, 2, 1) [[SN scale\|SNS]]~~ 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.		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.

Inthar at 16:34, 15 February 2024

2024-02-15T16:34:51Z

← Older revision		Revision as of 16:34, 15 February 2024
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	ABACABA is the singular [[Rank-3 scale#Pairwise well-formed scales\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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		ABACABA is the ternary [[Fraenkel word]] and the singular [[Rank-3 scale#Pairwise well-formed scales\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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.

Lhearne at 23:56, 1 April 2023

2023-04-01T23:56:14Z

← Older revision		Revision as of 23:56, 1 April 2023
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	ABACABA is the singular [[Rank-3 scale#Pairwise well-formed scales\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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, A), or equivalently as SNS (P, T, A), 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		ABACABA is the singular [[Rank-3 scale#Pairwise well-formed scales\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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.

Lhearne: /* 729-limit (675-limit) ABACABA scales with period 3/2, with steps > 20c */

2023-04-01T23:37:11Z

729-limit (675-limit) ABACABA scales with period 3/2, with steps > 20c

← Older revision		Revision as of 23:37, 1 April 2023
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	== 729-limit ~~(675-limit)~~ ABACABA scales with period 3/2, with steps > 20c ==		== 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.

Lhearne: /* 729-limit (675-limit) ABACABA scales with period 4/3, with steps > 20c */

2023-04-01T23:36:51Z

729-limit (675-limit) ABACABA scales with period 4/3, with steps > 20c

← Older revision		Revision as of 23:36, 1 April 2023
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	== 729-limit ~~(675-limit)~~ ABACABA scales with period 4/3, with steps > 20c ==		== 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.

Lhearne: /* Tetrachord to 15/13 -> C = 169/150 (~206.47c) */

2023-04-01T06:02:07Z

Tetrachord to 15/13 -> C = 169/150 (~206.47c)

← Older revision		Revision as of 06:02, 1 April 2023
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	ABACABA is the singular [[Rank-3 ~~scales#https://en.xen.wiki/w/Rank-3%20scale~~#Pairwise~~%20well~~-formed~~%20scales~~\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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, A), or equivalently as SNS (P, T, A), 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		ABACABA is the singular [[Rank-3 scale#Pairwise well-formed scales\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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, A), or equivalently as SNS (P, T, A), 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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	\|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
	\|~~243~~		\|729
	\|}		\|}

Lhearne at 06:01, 1 April 2023

2023-04-01T06:01:17Z

← Older revision		Revision as of 06:01, 1 April 2023
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	ABACABA is the singular pairwise well-formed ~~generalised~~ step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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, A), or equivalently as SNS (P, T, A), 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		ABACABA is the singular [[Rank-3 scales#https://en.xen.wiki/w/Rank-3%20scale#Pairwise%20well-formed%20scales\|pairwise well-formed]] generalized step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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, A), or equivalently as SNS (P, T, A), 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.

Lhearne at 05:59, 1 April 2023

2023-04-01T05:59:38Z

← Older revision		Revision as of 05:59, 1 April 2023
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	ABACABA is the singular pairwise well-formed generalised step pattern, and the (4, 2, 1) [[SN scale\|SNS]] pattern~~. Scales over a 2/1 period 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 singular pairwise well-formed generalised step pattern, and the (4, 2, 1) [[SN scale\|SNS]] 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, A), or equivalently as SNS (P, T, A), 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.

Lhearne: corrected limits of scales (with 3/2 and 4/3 period), changed order accordingly

2023-04-01T05:38:55Z

corrected limits of scales (with 3/2 and 4/3 period), changed order accordingly

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