Generator-offset property: Difference between revisions

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== 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 ASWFG. 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 ~~alt-gen~~, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:

Assuming ASWFG, 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)

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(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 ASWFG + 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 ==

@@ Line 15: / Line 15: @@
 == 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 ASWFG. 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 alt-gen, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:
+Assuming ASWFG, 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)
@@ Line 56: / Line 56: @@
 (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 ASWFG + 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 ==