Generator-offset property: Difference between revisions
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A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following two properties: | A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following two properties: | ||
# The scale is generated by two chains of stacked copies of a ''generator'', the two chains are separated by an ''offset'', and the lengths of the chains differ by at most one. Equivalently, the scale can be built by stacking two alternating generators (called ''alternants''), which do not necessarily take up the same number of steps. | # The scale is generated by two chains of stacked copies of a ''generator'', the two chains are separated by an ''offset'', and the lengths of the chains differ by at most one. Equivalently, the scale can be built by stacking two alternating generators (called ''alternants''), which do not necessarily take up the same number of steps. | ||
# The generator always occurs as the same number of steps. | # The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step. | ||
The [[Zarlino]] (3L 2M 2S) JI scale is an example of a GO 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 a GO 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. | ||
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More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''GO''' if it satisfies the following properties: | 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''<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 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''<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>. | ||
# All occurrences of the generator ''g'' are ''k'-steps for a fixed ''k''. | # All occurrences of the generator ''g'' are ''k''-steps for a fixed ''k''. | ||
This doesn't imply that ''g''<sub>1</sub> and ''g''<sub>2</sub> are the same number of scale steps. For example, 5-limit [[blackdye]] has ''g''<sub>1</sub> = 9/5 (a 9-step) and ''g''<sub>2</sub> = 5/3 (a 7-step). | This doesn't imply that ''g''<sub>1</sub> and ''g''<sub>2</sub> are the same number of scale steps. For example, 5-limit [[blackdye]] has ''g''<sub>1</sub> = 9/5 (a 9-step) and ''g''<sub>2</sub> = 5/3 (a 7-step). | ||