User:Inthar/MV3: Difference between revisions

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''Suppose we have an MV3 scale word 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. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (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).''

=== Lemma 1~~: The word made by any two of the step sizes is a MOS (except in the case "xyzyx") ===~~

=== Lemma 1: Sizes of chunks of any fixed letter form a MOS (except in the case "xyzyx") ===

~~Todo: Account for xyzyx~~

Assume the scale word S is not multiperiod. To eliminate words of the form X'Y'Z'Y'X' we manually check all words up to length 5... (todo) Henceforth we assume len(S) >= 6. [This assumption must be used somewhere!]

~~Suppose that some class C in W', the word of y's and z's formed by omitting x's, has three sizes, T1, T2, T3. Assume WOLOG that len(C) ≤ 1/2*len(W').~~

~~Assume T1, T2, T3 occur within a contiguous string of y's and z's. Then you get a 4th variant of this class (within the whole scale) by using a string of the same length including an x.~~

~~So it's pretty obvious that you have to have MV2 within a contiguous string, but what about the whole string minus the x's?~~

~~This part needs to be more careful, taking into account where X can appear:~~

First assume for simplicity that len(C) = 2. Then there are only three possible lengths: 2y, 2z, y+z. Suppose all 3 occur. By (*), 2y and 2z cannot occur within the same contiguous string. (We're assuming S has at least one x.)

~~Write down and tabulate all strings that occur as in the following chart. If a string occurs, mark the corresponding box YES. If a string doesn't occur, mark the box NO.~~

~~2y y+z 2z~~

~~0 [y y] [_ _] [z z]~~

~~1 [y x y] [_ x _] [z x z]~~

~~2 [y x x y] [_ x x _] [z x x z]~~

~~3 [y x x x y] [_ x x x _] [z x x x z]~~

~~...~~

~~=== Lemma 2~~: Sizes of chunks of any fixed letter form a MOS (except in the case "xyzyx") ===

TODO: account for case xyzyx.

Assume the scale word S is not multiperiod. To eliminate ~~XYZYX~~ we manually check all words up to length 5... (todo)

Assume the scale word S is not multiperiod. To eliminate xyzyx we manually check all words up to length 5... (todo)

Now assume len(S) >= 6.

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''To be continued...''

=== Lemma 2: The word made by any two of the step sizes is a MOS (except in the case "xyzyx") ===

Use Lemma 1?

=== Proof of the alternating-gens property (except in the case xyxzxyx) ===

@@ Line 37: / Line 37: @@
 ''Suppose we have an MV3 scale word 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. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (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).''
-=== Lemma 1: The word made by any two of the step sizes is a MOS (except in the case "xyzyx") ===
+=== Lemma 1: Sizes of chunks of any fixed letter form a MOS (except in the case "xyzyx") ===
-Todo: Account for xyzyx
-Assume the scale word S is not multiperiod. To eliminate words of the form X'Y'Z'Y'X' we manually check all words up to length 5... (todo) Henceforth we assume len(S) >= 6. [This assumption must be used somewhere!]
-Suppose that some class C in W', the word of y's and z's formed by omitting x's, has three sizes, T1, T2, T3. Assume WOLOG that len(C) ≤ 1/2*len(W').
-Assume T1, T2, T3 occur within a contiguous string of y's and z's. Then you get a 4th variant of this class (within the whole scale) by using a string of the same length including an x.
-So it's pretty obvious that you have to have MV2 within a contiguous string, but what about the whole string minus the x's?
-This part needs to be more careful, taking into account where X can appear:
-First assume for simplicity that len(C) = 2. Then there are only three possible lengths: 2y, 2z, y+z. Suppose all 3 occur. By (*), 2y and 2z cannot occur within the same contiguous string. (We're assuming S has at least one x.)
-Write down and tabulate all strings that occur as in the following chart. If a string occurs, mark the corresponding box YES. If a string doesn't occur, mark the box NO.
-y          y+z         2z
-[y y]       [_ _]       [z z]
-[y x y]     [_ x _]     [z x z]
-[y x x y]   [_ x x _]   [z x x z]
-[y x x x y] [_ x x x _] [z x x x  z]
- ...
-=== Lemma 2: Sizes of chunks of any fixed letter form a MOS (except in the case "xyzyx") ===
 TODO: account for case xyzyx.
-Assume the scale word S is not multiperiod. To eliminate XYZYX we manually check all words up to length 5... (todo)
+Assume the scale word S is not multiperiod. To eliminate xyzyx we manually check all words up to length 5... (todo)
 Now assume len(S) >= 6.
@@ Line 98: / Line 74: @@
 ''To be continued...''
+=== Lemma 2: The word made by any two of the step sizes is a MOS (except in the case "xyzyx") ===
+Use Lemma 1?
 === Proof of the alternating-gens property (except in the case xyxzxyx) ===