Generator-offset property: Difference between revisions

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[[Diasem]] is an example of an AG scale, because it is built by stacking alternating 7/6 and 8/7 for [[chirality|left-handed]] diasem, or 8/7 and 7/6 for right-handed diasem.

More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of ~~length~~ ''n'' is '''AG''' if it satisfies the following equivalent properties:

More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of size ''n'' is '''AG''' if it satisfies the following equivalent properties:

# ''S'' can be built by stacking a single chain of alternating generators ''g''1 and ''g''2, resulting in a circle of the form either ''g''1 ''g''2 ... ''g''1 ''g''2 ''g''1 ''g''3 or ''g''1 ''g''2 ... ''g''1 ''g''2 ''g''3.

# ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''n''/2, or one chain has size (''n'' + 1)/2 and the second has size (''n'' − 1)/2.

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== Theorems ==

=== 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 some permutation (''x'', ''y'', ''z'') of (L, M, s); and the scale's ~~length~~ is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-~~length~~ AG scale is unconditionally MV3.

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 (size) is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-cardinality AG scale is unconditionally MV3.

==== Proof ====

Assuming both AG and unconditionally MV3, we have two chains of generator ''g''0 (going right). The two cases are:

CASE 1: EVEN ~~LENGTH~~

CASE 1: EVEN CARDINALITY

O-O-...-O (n/2 notes)

and

CASE 2: ODD ~~LENGTH~~

CASE 2: ODD CARDINALITY

O-O-O-...-O ((n+1)/2 notes)

O-O-...-O ((n-1)/2 notes).

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# ''a''4 − ''a''2 = ''g''1 − 2 ''g''2 + ''g''3 = (''g''3 − ''g''2) + (''g''1 − ''g''2) = (chroma ± ε) != 0 by choice of tuning.

By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''1 and ''g''2 must themselves be step sizes. Thus we see that an even-~~length~~, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and

By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''1 and ''g''2 must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and

''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.

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(The above holds for any odd ''n'' ≥ 3.)

Now we only need to see that AG + odd ~~length~~ => 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 AG + 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 ===

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

# unconditionally MV3,

# has odd ~~length~~,

# has odd cardinality,

# is not of the form ''mx my mz'',

# and is not of the form ''xyxzxyx'',

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 [[Diasem]] is an example of an AG scale, because it is built by stacking alternating 7/6 and 8/7 for [[chirality|left-handed]] diasem, or 8/7 and 7/6 for right-handed diasem.
-More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of length ''n'' is '''AG''' if it satisfies the following equivalent properties:
+More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of size ''n'' is '''AG''' if it satisfies the following equivalent properties:
 # ''S'' can be built by stacking a single chain of alternating generators ''g''<sub>1</sub> and ''g''<sub>2</sub>, resulting in a circle of the form either ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>3</sub> or ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>3</sub>.
 # ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''n''/2, or one chain has size (''n'' + 1)/2 and the second has size (''n''&nbsp;&minus;&nbsp;1)/2.
@@ Line 12: / Line 12: @@
 == Theorems ==
 === 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 some permutation (''x'', ''y'', ''z'') of (L, M, s); and the scale's length is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-length AG scale is unconditionally MV3.
+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 (size) is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-cardinality AG scale is unconditionally MV3.
 ==== Proof ====
 Assuming both AG and unconditionally MV3, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are:
-  CASE 1: EVEN LENGTH
+  CASE 1: EVEN CARDINALITY
   O-O-...-O (n/2 notes)
   O-O-...-O (n/2 notes)
 and
-  CASE 2: ODD LENGTH
+  CASE 2: ODD CARDINALITY
   O-O-O-...-O ((n+1)/2 notes)
   O-O-...-O ((n-1)/2 notes).
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 # ''a''<sub>4</sub> &minus; ''a''<sub>2</sub> = ''g''<sub>1</sub> &minus; 2 ''g''<sub>2</sub> + ''g''<sub>3</sub> = (''g''<sub>3</sub> &minus; ''g''<sub>2</sub>) + (''g''<sub>1</sub> &minus; ''g''<sub>2</sub>) = (chroma ± ε) != 0 by choice of tuning.
-By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''<sub>1</sub> and ''g''<sub>2</sub> must themselves be step sizes. Thus we see that an even-length, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and
+By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''<sub>1</sub> and ''g''<sub>2</sub> must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and
 ''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.
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 (The above holds for any odd ''n'' ≥ 3.)
-Now we only need to see that AG + odd length => 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 AG + 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 ===
 If a non-multiperiod 3-step size scale word is
 # unconditionally MV3,
-# has odd length,
+# has odd cardinality,
 # is not of the form ''mx my mz'',
 # and is not of the form ''xyxzxyx'',