Generator-offset property: Difference between revisions
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== Theorems == | == Theorems == | ||
=== Theorem 1 === | === 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 some permutation (''x'', ''y'', ''z'') 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. | 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 some permutation (''x'', ''y'', ''z'') 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 ==== | ==== Proof ==== | ||
Assuming both AG and unconditionally MV3, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are: | Assuming both AG and unconditionally MV3, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are: | ||