Generator-offset property: Difference between revisions
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The [[Zarlino]] (3L 2M 2S) JI scale is an example of an AG scale, because it is built by stacking alternating 5/4 and 6/5 generators. 7-limit [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7. | The [[Zarlino]] (3L 2M 2S) JI scale is an example of an AG scale, because it is built by stacking alternating 5/4 and 6/5 generators. 7-limit [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7. | ||
More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of size ''n'' is '''AG''' if it satisfies the following equivalent properties: | More formally, a cyclic word ''S'' (representing the steps of 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 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'' 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 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. | # ''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. | ||