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.
 == Theorems ==
-'''Theorem 1''': If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3 (i.e. MV3 regardless of tuning), 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 unconditionally MV3.
+=== Theorem 1 ===
-=== Proof of Thm 1 ===
+If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3 (i.e. MV3 regardless of tuning), 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 unconditionally MV3.
+==== Proof ====
 Assuming both AG and unconditionally MV3, we have two chains of generator g0 (going right). The two cases are:
   O-O-...-O (m notes)