Rank 3 scale - Revision history

Lhearne: Redirected page to Rank-3 scale

2022-02-06T12:26:06Z

Redirected page to Rank-3 scale

Inthar at 11:40, 14 September 2021

2021-09-14T11:40:00Z

← Older revision		Revision as of 11:40, 14 September 2021
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	== List of named rank-3 scales ==		== List of named rank-3 scales ==
	The following is a list of (temperament-agnostic) names that have been given to 3-step scales.		The following is a list of (temperament-agnostic) names that have been given to 3-step scales.
	* [[Zarlino]] (3L 2M 2S)
	* [[diasem]] (5L 2M 2S)		* [[diasem]] (5L 2M 2S)
	* [[blackdye]] (5L 2M 3S)		* [[blackdye]] (5L 2M 3S)
	* [[onyxwood]] (5L 3M 2S)		* [[onyxwood]] (5L 3M 2S)
	[[Category:Rank 3 scales\| ]] <!-- main article -->		[[Category:Rank 3 scales\| ]] <!-- main article -->

Inthar: /* List of named rank-3 scales */

2021-09-14T01:54:42Z

List of named rank-3 scales

← Older revision		Revision as of 01:54, 14 September 2021
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	The following is a list of (temperament-agnostic) names that have been given to 3-step scales.		The following is a list of (temperament-agnostic) names that have been given to 3-step scales.
	* [[Zarlino]] (3L 2M 2S)		* [[Zarlino]] (3L 2M 2S)
	* [[~~Diasem~~]] (5L 2M 2S)		* [[diasem]] (5L 2M 2S)
	* [[~~Blackdye~~]] (5L 2M 3S)		* [[blackdye]] (5L 2M 3S)
	* [[~~Onyxwood~~]] (5L 3M 2S)		* [[onyxwood]] (5L 3M 2S)
	[[Category:Rank 3 scales\| ]] <!-- main article -->		[[Category:Rank 3 scales\| ]] <!-- main article -->

Inthar: /* Theorems, Proofs and Conjectures on 3-SN scales */

2021-09-14T01:54:29Z

Theorems, Proofs and Conjectures on 3-SN scales

← Older revision		Revision as of 01:54, 14 September 2021
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	'''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.		'''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.
			== List of named rank-3 scales ==
			The following is a list of (temperament-agnostic) names that have been given to 3-step scales.
			* [[Zarlino]] (3L 2M 2S)
			* [[Diasem]] (5L 2M 2S)
			* [[Blackdye]] (5L 2M 3S)
			* [[Onyxwood]] (5L 3M 2S)
	[[Category:Rank 3 scales\| ]] <!-- main article -->		[[Category:Rank 3 scales\| ]] <!-- main article -->

Inthar: /* MV3 scales */

2021-08-16T06:26:01Z

MV3 scales

← Older revision		Revision as of 06:26, 16 August 2021
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	Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank 3.		Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank 3.

	For all odd-cardinality MV3 scales apart from the scales abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.		For all odd-cardinality MV3 scales apart from the scales abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1. This is the [[AG]] property.

	All [[AG]] scales of odd cardinality are MV3. The only AG scale of even cardinality is abac.		All [[AG]] scales of odd cardinality are MV3. The only AG scale of even cardinality is abac.

Inthar: /* MV3 scales */ correct claim

2021-08-16T06:25:33Z

MV3 scales: correct claim

← Older revision		Revision as of 06:25, 16 August 2021
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	Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank 3.		Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank 3.

	For all MV3 scales apart from the scales abacaba, and ~~it's~~ repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.		For all odd-cardinality MV3 scales apart from the scales abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

	All [[AG]] scales of odd cardinality are MV3~~, and it is conjectured that all odd MV3 scales satisfying certain mild restrictions are AG~~. The only AG scale of even cardinality is abac.		All [[AG]] scales of odd cardinality are MV3. The only AG scale of even cardinality is abac.

	'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).		'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

Inthar: /* MV3 scales */

2021-08-16T06:20:47Z

MV3 scales

← Older revision		Revision as of 06:20, 16 August 2021
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	For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.		For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

	All [[AG]] scales of odd cardinality are MV3, and it is conjectured that all odd MV3 scales satisfying certain ~~miled~~ restrictions are AG. The only AG scale of even cardinality is abac.		All [[AG]] scales of odd cardinality are MV3, and it is conjectured that all odd MV3 scales satisfying certain mild restrictions are AG. The only AG scale of even cardinality is abac.

	'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).		'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

Inthar: /* MV3 scales */ explain relation to alternating generator property

2021-08-16T06:19:31Z

MV3 scales: explain relation to alternating generator property

← Older revision		Revision as of 06:19, 16 August 2021
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	For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.		For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

			All [[AG]] scales of odd cardinality are MV3, and it is conjectured that all odd MV3 scales satisfying certain miled restrictions are AG. The only AG scale of even cardinality is abac.

	'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).		'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

Fredg999: Undo revision 76087 by Fredg999 (talk)

2021-08-14T03:02:55Z

Undo revision 76087 by Fredg999 (talk)

← Older revision		Revision as of 03:02, 14 August 2021
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	[[Category:Rank 3 scales\| ]] <!-- main article -->		[[Category:Rank 3 scales\| ]] <!-- main article -->
	~~[[Category:Scales by family]]~~

Fredg999: Categories

2021-08-14T02:44:02Z

← Older revision		Revision as of 02:44, 14 August 2021
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	[[Category:Rank 3 scales\| ]] <!-- main article -->		[[Category:Rank 3 scales\| ]] <!-- main article -->
			[[Category:Scales by family]]