Generator-offset property: Difference between revisions

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A scale satisfies the '''alternating generator property''', or the '''AG''' property for short, if it satisfies the following equivalent properties:

* the scale can be built by stacking alternating generators, for example 7/6 and 8/7.

* the scale 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.

* the scale 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.

[[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:

# S can be built by stacking a single chain of alternating generators g1 and g2, resulting in a circle of the form either ~~g1 g2~~ ... ~~g1 g2 g1 g3~~ or ~~g1 g2~~ ... ~~g1 g2 g3~~.

# ''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 m, or one chain has size m and the second has size ''m'' - 1.

These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.

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

== Theorems ==

=== Theorem 1 ===

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 A scale satisfies the '''alternating generator property''', or the '''AG''' property for short, if it satisfies the following equivalent properties:
 * the scale can be built by stacking alternating generators, for example 7/6 and 8/7.
-* the scale 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.
+* the scale 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.
 [[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:
-# S can be built by stacking a single chain of alternating generators g1 and g2, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.
+# ''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 m, or one chain has size m and the second has size ''m'' - 1.
-These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.
+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>.
 == Theorems ==
 === Theorem 1 ===