User:Inthar/MV3: Difference between revisions
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SL ML SL ML S--> | SL ML SL ML S--> | ||
== PMOS == | == Definitions and theorems == | ||
=== PMOS === | |||
Definition: ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS. | Definition: ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS. | ||
== AG == | === AG === | ||
Definition: ''S'' satisfies the ''alternating generator'' (AG) property if it satisfies the following equivalent properties: | Definition: ''S'' satisfies the ''alternating generator'' (AG) property if it satisfies the following equivalent properties: | ||
# ''S'' is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1''. | # ''S'' is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1''. | ||
# ''S'' can be built by stacking alternating generators, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | # ''S'' can be built by stacking alternating generators, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | ||
== MV3 Theorem 1== | === MV3 Theorem 1 === | ||
''The following are equivalent for a scale word S with steps x, y, z:'' | ''The following are equivalent for a scale word S with steps x, y, z:'' | ||
# ''S is MV3.'' | # ''S is MV3.'' | ||
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Below, assume that the scale word S is not multiperiod. | Below, assume that the scale word S is not multiperiod. | ||
=== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" === | ==== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ==== | ||
TODO: account for case xyzyx. | TODO: account for case xyzyx. | ||
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''To be continued...'' | ''To be continued...'' | ||
=== PMOS implies AG (except in the case xyxzxyx) === | ==== PMOS implies AG (except in the case xyxzxyx) ==== | ||
We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG. | We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG. | ||
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if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED. | if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED. | ||
=== 3-DE implies MV3 === | ==== 3-DE implies MV3 ==== | ||
We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3. | We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3. | ||
== MV3 Theorem 2 == | === MV3 Theorem 2 === | ||
''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ||