Generator-offset property: Difference between revisions

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 == Theorems ==
 === Theorem 1 ===
-If a 3-step-size scale word ''S'' in L, M, and s is both alt-gen and unconditionally [[MV3]] (i.e. MV3 regardless of tuning), then the scale is of the form ''ax by bz'' for some permutation (''x'', ''y'', ''z'') of (L, M, s); and the scale's cardinality (size) is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-cardinality alt-gen scale is unconditionally MV3.
+Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is alt-gen. Then:
+# ''S'' is unconditionally MV3 (i.e. MV3 regardless of tuning).
+# ''S'' is of the form ''ax by bz'' for some permutation (''x'', ''y'', ''z'') of (L, M, s).
+# The cardinality (size) of ''S'' is either odd, or 4 (and ''S'' is of the form ''xyxz'').
 ==== Proof ====
-Assuming both alt-gen and unconditionally MV3, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:
+Assuming alt-gen, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:
   CASE 1: EVEN CARDINALITY
   O-O-...-O (n/2 notes)