Generator-offset property: Difference between revisions
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== Theorems == | == Theorems == | ||
=== Theorem 1 === | <!--=== Theorem 1 === | ||
Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is alt-gen. Then: | 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 unconditionally MV3 (i.e. MV3 regardless of tuning). | ||
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(The above holds for any odd ''n'' ≥ 3.) | (The above holds for any odd ''n'' ≥ 3.) | ||
Now we only need to see that alt-gen + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning. | Now we only need to see that alt-gen + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.--> | ||
== Conjectures == | == Conjectures == | ||