Generator-offset property: Difference between revisions

Line 8:

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

# ''S'' is generated by two chains of stacked 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. Equivalently, ''S'' can be built by stacking a single chain of alternants ''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 stacked generators ''g'' separated by a fixed offset δ; either both chains are of size ''n''/2, or one chain has size (''n'' + 1)/2 and the second has size (''n'' − 1)/2. Equivalently, ''S'' can be built by stacking a single chain of alternants ''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.

# The scale is ''well-formed'' with respect to ''g'', i.e. all occurrences of the generator ''g'' are ''k''-steps for a fixed ''k''.

@@ Line 8: / Line 8: @@
 More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''GO''' if it satisfies the following properties:
-# ''S'' is generated by two chains of stacked 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. Equivalently, ''S'' can be built by stacking a single chain of alternants ''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 stacked generators ''g'' separated by a fixed offset δ; 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. Equivalently, ''S'' can be built by stacking a single chain of alternants ''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>.
 # The scale is ''well-formed'' with respect to ''g'', i.e. all occurrences of the generator ''g'' are ''k''-steps for a fixed ''k''.