Generator-offset property: Difference between revisions

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== Related properties ==

* A strengthening of the AG property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''1 and ''g''2 can be taken to be the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ~~ASWFG~~. [[Blackdye]] is AG but not ~~ASWFG~~.

* A strengthening of the AG property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''1 and ''g''2 can be taken to be the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASGC. [[Blackdye]] is AG but not ASGC.

== Theorems ==

=== Theorem 1 ===

Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is ~~ASWFG~~. Then:

Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is ASGC. 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 ~~ASWFG~~, we have two chains of generator ''g''0 (going right). The two cases are:

Assuming ASGC, we have two chains of generator ''g''0 (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 ~~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.-->

Now we only need to see that ASGC + 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 ==

=== Conjecture 2 ===

An odd AG scale is ~~ASWFG~~.

An odd AG scale is ASGC.

=== Conjecture 3 ===

If a non-multiperiod 3-step size scale word is

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 == Related properties ==
-* A strengthening of the AG property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to be the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASWFG. [[Blackdye]] is AG but not ASWFG.
+* A strengthening of the AG property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to be the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASGC. [[Blackdye]] is AG but not ASGC.
 == Theorems ==
 === Theorem 1 ===
-Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is ASWFG. Then:
+Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is ASGC. 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 ASWFG, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:
+Assuming ASGC, 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 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.-->
+Now we only need to see that ASGC + 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 ==
 === Conjecture 2 ===
-An odd AG scale is ASWFG.
+An odd AG scale is ASGC.
 === Conjecture 3 ===
 If a non-multiperiod 3-step size scale word is