Even-regular MV3 scale: Difference between revisions

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A '''~~diregular~~ scale''' is a type of [[ternary scale]] with an even number of notes. ~~A diregular~~ 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 ~~diregular~~ 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 ~~diregular~~ scales ==

== Notable even-regular MV3 scales ==

* Achiral [[diachrome]] (dia5s)

* [[Penslen]] (slen5m)

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== Properties ==

Another characterization of ~~diregular~~ 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 ~~diregular~~ 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 [[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.

* 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).

In terms of [[guide frame]]s and interleaved scales, in diregular 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.

Diregular 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 diregular and has [[step signature]] aXaYbZ where a is odd and b is even.

~~Diregular scales always satisfy all 3 of the [[monotone-MOS scale|monotone-MOS]] conditions.~~

== Terminology ==

The term ''~~diregular~~'' has been coined by ~~akselai and~~ Inthar.

The term ''even-regular MV3'' has been coined by Inthar.

== See also ==

* [[Odd-regular MV3 scale]]

* [[Ternary scale theorems]]

[[Category:Aberrismic theory]]

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-A '''diregular scale''' is a type of [[ternary scale]] with an even number of notes. A diregular 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 diregular 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 diregular scales ==
+== Notable even-regular MV3 scales ==
 * Achiral [[diachrome]] (dia5s)
 * [[Penslen]] (slen5m)
@@ Line 7: / Line 7: @@
 == Properties ==
-Another characterization of diregular 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 diregular 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 [[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.
+* 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).
-In terms of [[guide frame]]s and interleaved scales, in diregular 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.
-Diregular 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 diregular and has [[step signature]] aXaYbZ where a is odd and b is even.
-Diregular scales always satisfy all 3 of the [[monotone-MOS scale|monotone-MOS]] conditions.
 == Terminology  ==
-The term ''diregular'' has been coined by akselai and Inthar.
+The term ''even-regular MV3'' has been coined by Inthar.
+== See also ==
+* [[Odd-regular MV3 scale]]
+* [[Ternary scale theorems]]
 [[Category:Aberrismic theory]]