Rank-3 scale theorems: Difference between revisions

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==== AG + unconditionally MV3 implies "ax by bz" and that the scale's ~~cardinality is odd or 4~~ ====

==== AG + unconditionally MV3 implies "ax by bz" and that the scale's ====

'''Assuming both AG and unconditional MV3''', we have two chains of generator g0 (going right). The two cases are:

O-O-...-O (m notes)

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Label the notes (1,k) and (2,k), 1 ≤ k ≤ m or m-1, for notes in the upper and lower chain respectively.

In case 1 (even scale size), let g1 = (2,1) - (1,1) and g2 = (1,2) - (2,1). We have the chain g1 g2 g1 g2... g1 g3. Consider the sizes of the n/2-step (which is an odd number of generator steps):

In case 1 (singly even scale size), let g1 = (2,1) - (1,1) and g2 = (1,2) - (2,1). We have the chain g1 g2 g1 g2... g1 g3. Consider the sizes of the n/2-step (which is an odd number of generator steps):

# from g1 ... g1, get a1 = (n/2-1)*g0 + g1 = n/2 g1 + (n/2-1) g2

# from g2 ... g2, get a2 =(n/2-1)*g0 + g2 = (n/2-1) g1 + n/2 g2

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(The above holds for any odd n >= 3.)

This proof shows that AG and unconditionally-MV3 scales must have odd ~~size~~ or ~~size 4~~.

This proof shows that AG and unconditionally-MV3 scales must have odd or doubly even cardinality.

==== An AG scale is unconditionally MV3 iff its cardinality is odd or 4 ====

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-==== AG + unconditionally MV3 implies "ax by bz" and that the scale's cardinality is odd or 4 ====
+==== AG + unconditionally MV3 implies "ax by bz" and that the scale's  ====
 '''Assuming both AG and unconditional MV3''', we have two chains of generator g0 (going right). The two cases are:
   O-O-...-O (m notes)
@@ Line 112: / Line 112: @@
 Label the notes (1,k) and (2,k), 1 ≤ k ≤ m or m-1, for notes in the upper and lower chain respectively.
-In case 1 (even scale size), let g1 = (2,1) - (1,1) and g2 = (1,2) - (2,1). We have the chain g1 g2 g1 g2... g1 g3. Consider the sizes of the n/2-step (which is an odd number of generator steps):
+In case 1 (singly even scale size), let g1 = (2,1) - (1,1) and g2 = (1,2) - (2,1). We have the chain g1 g2 g1 g2... g1 g3. Consider the sizes of the n/2-step (which is an odd number of generator steps):
 # from g1 ... g1, get a1 = (n/2-1)*g0 + g1 = n/2 g1 + (n/2-1) g2
 # from g2 ... g2, get a2 =(n/2-1)*g0 + g2 = (n/2-1) g1 + n/2 g2
@@ Line 134: / Line 134: @@
 (The above holds for any odd n >= 3.)
-This proof shows that AG and unconditionally-MV3 scales must have odd size or size 4.
+This proof shows that AG and unconditionally-MV3 scales must have odd or doubly even cardinality.
 ==== An AG scale is unconditionally MV3 iff its cardinality is odd or 4 ====