Generator-offset property: Difference between revisions

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[[Diasem]] is an example of an AG scale.

More formally, a cyclic word S (representing a [[periodic scale]]) 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 m, or one chain has size m and the second has size ''m'' - 1.

# ''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.

These are equivalent, since the separating interval can be taken to be ''g''1 and the generator of each chain = ''g''1 + ''g''2.

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

CASE 1: EVEN CARDINALITY

O-O-...-O (m notes)

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

O-O-...-O (m notes)

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

and

CASE 2: ODD CARDINALITY

O-O-O-...-O (m notes)

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

O-O-...-O (m-1 notes).

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

Label the notes (1, ''k'') and (2, ''k''), 1 ≤ ''k'' ≤ ~~''m'' or ''m'' − 1~~, for notes in the upper and lower chain respectively.

Label the notes (1, ''k'') and (2, ''k''), 1 ≤ ''k'' ≤ (chain length), for notes in the upper and lower chain respectively.

In case 1, let ''g''1 = (2, 1) − (1, 1) and ''g''2 = (1, 2) − (2, 1). We have the chain ''g''1 ''g''2 ''g''1 ''g''2 ... ''g''1 ''g''3.

@@ Line 5: / Line 5: @@
 [[Diasem]] is an example of an AG scale.
-More formally, a cyclic word S (representing a [[periodic scale]]) 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 m, or one chain has size m and the second has size ''m'' - 1.
+# ''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.
 These are equivalent, since the separating interval can be taken to be ''g''<sub>1</sub> and the generator of each chain = ''g''<sub>1</sub> + ''g''<sub>2</sub>.
@@ Line 16: / Line 16: @@
 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 CARDINALITY
-  O-O-...-O (m notes)
+  O-O-...-O (n/2 notes)
-  O-O-...-O (m notes)
+  O-O-...-O (n/2 notes)
 and
   CASE 2: ODD CARDINALITY
-  O-O-O-...-O (m notes)
+  O-O-O-...-O ((n+1)/2 notes)
-  O-O-...-O (m-1 notes).
+  O-O-...-O ((n-1)/2 notes).
-Label the notes (1, ''k'') and (2, ''k''), 1 ≤ ''k'' ≤ ''m'' or ''m'' &minus; 1, for notes in the upper and lower chain respectively.
+Label the notes (1, ''k'') and (2, ''k''), 1 ≤ ''k'' ≤ (chain length), for notes in the upper and lower chain respectively.
 In case 1, let ''g''<sub>1</sub> = (2, 1) &minus; (1, 1) and ''g''<sub>2</sub> = (1, 2) &minus; (2, 1). We have the chain ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>3</sub>.