Generator-offset property: Difference between revisions
m →Theorems: mv3 > unconditionally mv3 |
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'''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''': 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 of Thm 1 === | === Proof of Thm 1 === | ||
Assuming both AG and unconditionally MV3, we have two chains of generator g0 (going right). The two cases are: | |||
O-O-...-O (m notes) | O-O-...-O (m notes) | ||
O-O-...-O (m notes) | O-O-...-O (m notes) | ||