Even-regular MV3 scale - Revision history

Inthar: /* Properties */

2026-01-06T04:42:55Z

Properties

← Older revision		Revision as of 04:42, 6 January 2026
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	* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
	* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.		* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.
	* If an even-regular MV3 is oddly even, then it is ~~a contra-~~[[interleaving]] of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3, except in the trivial case xyxz where it is an interleaving of two 2-note MOSes (x+y)(x+z).		* If an even-regular MV3 is oddly even, then it is an [[interleaving]] of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3, except in the trivial case xyxz where it is an interleaving of two 2-note MOSes (x+y)(x+z).

	== Terminology ==		== Terminology ==

Inthar: /* Properties */

2025-12-31T23:49:15Z

Properties

← Older revision		Revision as of 23:49, 31 December 2025
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	* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
	* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.		* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.
	* If an even-regular MV3 is oddly even, then it is a contra-interleaving of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3, except in the trivial case xyxz where it is an interleaving of two 2-note MOSes (x+y)(x+z).		* If an even-regular MV3 is oddly even, then it is a contra-[[interleaving]] of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3, except in the trivial case xyxz where it is an interleaving of two 2-note MOSes (x+y)(x+z).

	== Terminology ==		== Terminology ==

Inthar: /* Properties */

2025-12-31T23:48:49Z

Properties

← Older revision		Revision as of 23:48, 31 December 2025
Line 11:		Line 11:
	* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
	* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.		* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.
	* If an even-regular MV3 is oddly even, then it is a contra-interleaving of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3.		* If an even-regular MV3 is oddly even, then it is a contra-interleaving of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3, except in the trivial case xyxz where it is an interleaving of two 2-note MOSes (x+y)(x+z).

	== Terminology ==		== Terminology ==

Inthar: /* Properties */

2025-12-31T23:48:16Z

Properties

← Older revision		Revision as of 23:48, 31 December 2025
Line 11:		Line 11:
	* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
	* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.		* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.
			* If an even-regular MV3 is oddly even, then it is a contra-interleaving of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3.

	== Terminology ==		== Terminology ==

Inthar: /* Properties */

2025-01-06T09:51:50Z

Properties

← Older revision		Revision as of 09:51, 6 January 2025
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	* Even-regular MV3 scales always satisfy all 3 of the [[monotone-MOS scale\|monotone-MOS]] conditions.		* Even-regular MV3 scales always satisfy all 3 of the [[monotone-MOS scale\|monotone-MOS]] conditions.
	* Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].		* Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].
	* Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are [[maximum variety]] 3 (MV3) but not [[strict variety]] 3 (SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
	* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.		* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.

Inthar: /* Properties */

2025-01-04T15:36:40Z

Properties

← Older revision		Revision as of 15:36, 4 January 2025
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	* Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].		* Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].
	* Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
	* Even-regular MV3 scales are [[chirality\|achiral]].		* Even-regular MV3 scales are [[chirality\|achiral]]. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.

	== Terminology ==		== Terminology ==

Inthar: /* Properties */

2025-01-04T15:23:41Z

Properties

← Older revision		Revision as of 15:23, 4 January 2025
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	== Properties ==		== Properties ==
	Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].		* Even-regular MV3 scales always satisfy all 3 of the [[monotone-MOS scale\|monotone-MOS]] conditions.
			* Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].
	Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		* Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
			* Even-regular MV3 scales are [[chirality\|achiral]].
	Even-regular MV3 scales ~~always satisfy all 3 of the~~ [[~~monotone-MOS scale~~\|~~monotone-MOS~~]] ~~conditions~~.

	== Terminology ==		== Terminology ==

Inthar at 15:12, 4 January 2025

2025-01-04T15:12:16Z

← Older revision		Revision as of 15:12, 4 January 2025
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	An '''even-regular MV3 scale''' is a type of [[ternary scale]] with an even number of notes. An even-regular MV3 scale consists of two identical generator chains, where all generators are identical and subtend the same [[interval class\|step class]]. The two chains are offset by an interval that subtends ''k'' steps in a 2''k''-note even-regular MV3 scale.		An '''even-regular MV3 scale''' is a type of [[ternary scale]] with an even number of notes per period. An even-regular MV3 scale consists of two identical generator chains, where all generators are identical and subtend the same [[interval class\|step class]]. The two chains are offset by an interval that subtends ''k'' steps in a 2''k''-note even-regular MV3 scale.
	== Notable even-regular MV3 scales ==		== Notable even-regular MV3 scales ==
	* Achiral [[diachrome]] (dia5s)		* Achiral [[diachrome]] (dia5s)

Inthar at 15:06, 4 January 2025

2025-01-04T15:06:50Z

← Older revision		Revision as of 15:06, 4 January 2025
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	== Properties ==		== Properties ==
	Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].		Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of [[diachrome]].

	In terms of [[guide frame]]s and interleaved scales, in even-regular MV3 scales the [[interleaved scale\|interleaving offset]] is generated by the guided generator sequence GS(g), and the 2-note strand scale [0, len(scale)/2-step] is the offset for the guide frame. The other type of generator-offset scale is represented by scales including bipentatonic scales (such as [[blackdye]]), where the strand is generated by GS(g) and the interleaving offset is the offset.

	Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.		Even-regular MV3 scales are MV3 (but not SV3), and by the [[ternary scale theorems\|MV3 classification theorem]] a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has [[step signature]] aXaYbZ where a is odd and b is even.
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	== See also ==		== See also ==
	* [[Odd-regular MV3 scale]]		* [[Odd-regular MV3 scale]]
			* [[Ternary scale theorems]]
	[[Category:Aberrismic theory]]		[[Category:Aberrismic theory]]

Inthar at 15:05, 4 January 2025

2025-01-04T15:05:15Z

← Older revision		Revision as of 15:05, 4 January 2025
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	The term ''even-regular MV3'' has been coined by Inthar.		The term ''even-regular MV3'' has been coined by Inthar.

			== See also ==
			* [[Odd-regular MV3 scale]]
	[[Category:Aberrismic theory]]		[[Category:Aberrismic theory]]