User:Inthar/MV3: Difference between revisions

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SL ML SL ML S-->

== PMOS ==

Definition: ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.

== AG ==

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'' 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==

Theorem: ''Suppose we have an MV3 scale word S with steps x, y and z. With only one exception ("xyxzxyx"), at least two of the three steps must occur the same number of times. (The one exception to this rule is "xyxzxyx", along with their repetitions "xyxzxyx", etc.) Moreover, there always exists some "generator" interval for any max-variety-3 scale (other than two exceptions, "xyzyx" and "xyxzxyx") such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).''

''The following are equivalent for a scale word S with steps x, y, z:''

~~Restatement:~~ ''The following are equivalent for a scale word S with steps x, y, z:''

# ''S is MV3.''

# ''S is PMOS, or S is of the form x'y'z'y'x' or its repetitions.''

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=== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ===

~~A word with three step sizes x, y, z is pairwise MOS "PMOS" if the result of equating any two of the step sizes is a MOS.~~

TODO: account for case xyzyx.

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=== PMOS implies AG (except in the case xyxzxyx) ===

We now prove that except in the case xyxzxyx, if the scale is pairwise ~~well-formed~~, ~~the scale~~ is ~~generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1'' (~~AG~~) Equivalently, the scale 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~~.

We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG.

To eliminate xyxzxyx we manually check all words up to length 7... (todo)

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=== 3-DE implies MV3 ===

We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.

== 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).''

@@ Line 34: / Line 34: @@
 SL ML SL ML S-->
+== PMOS ==
+Definition: ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.
+== AG ==
+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'' 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==
-Theorem: ''Suppose we have an MV3 scale word S with steps x, y and z. With only one exception ("xyxzxyx"), at least two of the three steps must occur the same number of times. (The one exception to this rule is "xyxzxyx", along with their repetitions "xyxzxyx", etc.) Moreover, there always exists some "generator" interval for any max-variety-3 scale (other than two exceptions, "xyzyx" and "xyxzxyx") such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).''
+''The following are equivalent for a scale word S with steps x, y, z:''
-Restatement: ''The following are equivalent for a scale word S with steps x, y, z:''
 # ''S is MV3.''
 # ''S is PMOS, or S is of the form x'y'z'y'x' or its repetitions.''
@@ Line 45: / Line 49: @@
 === Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ===
-A word with three step sizes x, y, z is pairwise MOS "PMOS" if the result of equating any two of the step sizes is a MOS.
 TODO: account for case xyzyx.
@@ Line 83: / Line 85: @@
 === PMOS implies AG (except in the case xyxzxyx) ===
-We now prove that except in the case xyxzxyx, if the scale is pairwise well-formed, the scale is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1'' (AG) Equivalently, the scale 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.
+We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG.
 To eliminate xyxzxyx we manually check all words up to length 7... (todo)
@@ Line 122: / Line 124: @@
 === 3-DE implies MV3 ===
 We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.
 == 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).''