Generator-offset property: Difference between revisions
m →Proof |
|||
| Line 11: | Line 11: | ||
== 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 (x,y,z) | 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 g0 (going right). The two cases are: | Assuming both AG and unconditionally MV3, we have two chains of generator g0 (going right). The two cases are: | ||