Generator-offset property: Difference between revisions

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# ''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.

These are equivalent, since the ~~separating interval~~ can be taken to be ''g''1 and the generator ~~of each chain =~~ ''g''1 + ''g''2. This doesn't imply that ''g''1 and ''g''2 are the same number of scale steps. For example, 5-limit [[blackdye]] has ''g''1 = 9/5 (a 9-step) and ''g''2 = 5/3 (a 7-step).

These are equivalent, since the offset can be taken to be ''g''1 and the generator then equals ''g''1 + ''g''2. This doesn't imply that ''g''1 and ''g''2 are the same number of scale steps. For example, 5-limit [[blackdye]] has ''g''1 = 9/5 (a 9-step) and ''g''2 = 5/3 (a 7-step).

More generally, we say that a scale is ''m'''''-GO''' if the scale consists of ''m'' + 1 chains of generator ''g'' (implying ''m'' offsets δ1, ...,δ''m'' from a fixed chain), each chain having either ''k'' or ''k'' + 1 notes. (Thus 1-GO is the same thing as GO.)

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 # ''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>.
-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>. 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).
+These are equivalent, since the offset can be taken to be ''g''<sub>1</sub> and the generator then equals ''g''<sub>1</sub> + ''g''<sub>2</sub>. 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).
 More generally, we say that a scale is ''m'''''-GO''' if the scale consists of ''m'' + 1 chains of generator ''g'' (implying ''m'' offsets δ<sub>1</sub>, ...,δ<sub>''m''</sub> from a fixed chain), each chain having either ''k'' or ''k'' + 1 notes. (Thus 1-GO is the same thing as GO.)