Generator-offset property: Difference between revisions
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== Mathematical definition == | == Mathematical definition == | ||
More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties: | More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties: | ||
# ''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<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 (implying ''n'' is even), or one chain has size (''n'' + 1)/2 and the second has size (''n'' − 1)/2 (implying ''n'' is odd). 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''. | # The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''. | ||