Even-regular MV3 scale: Difference between revisions
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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. | |||
== Notable | == Notable even-regular MV3 scales == | ||
* Achiral [[diachrome]] (dia5s) | * Achiral [[diachrome]] (dia5s) | ||
* [[Penslen]] (slen5m) | * [[Penslen]] (slen5m) | ||
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== Properties == | == Properties == | ||
Another characterization of | 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 | 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 always satisfy all 3 of the [[monotone-MOS scale|monotone-MOS]] conditions. | |||
== Terminology == | == Terminology == | ||
The term '' | The term ''even-regular MV3'' has been coined by Inthar. | ||
[[Category:Aberrismic theory]] | [[Category:Aberrismic theory]] | ||
Revision as of 14:55, 4 January 2025
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 step class. The two chains are offset by an interval that subtends k steps in a 2k-note even-regular MV3 scale.
Notable even-regular MV3 scales
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.
In terms of guide frames and interleaved scales, in even-regular MV3 scales the 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 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 always satisfy all 3 of the monotone-MOS conditions.
Terminology
The term even-regular MV3 has been coined by Inthar.