Generator-offset property: Difference between revisions
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These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2. | These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2. | ||
== Theorems == | == Theorems == | ||
'''Theorem 1''' If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3, then the scale is of the form ax by bz for (x,y,z) some permutation of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form xyxz). Moreover, any odd-cardinality AG scale is MV3. | '''Theorem 1''': If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3, then the scale is of the form ax by bz for (x,y,z) some permutation of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form xyxz). Moreover, any odd-cardinality AG scale is MV3. | ||
=== Proof of Thm 1 === | === Proof of Thm 1 === | ||
'''Assuming both AG and unconditional MV3''', we have two chains of generator g0 (going right). The two cases are: | '''Assuming both AG and unconditional MV3''', we have two chains of generator g0 (going right). The two cases are: | ||