Generator-offset property: Difference between revisions

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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:

# ''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.

# ~~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''.

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).

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More generally, we say that a scale is ''m'''''-GO''' if both of the following hold:

# the scale consists of ''m'' + 1 chains of stacked generator ''g'' (implying ''m'' offsets δ1, ...,δ''m'' from a fixed chain), each chain having either ''j'' or ''j'' + 1 notes.

# The ~~generator always subtends a fixed number~~ ''k'' ~~of steps~~.

# The scale is well-formed with respect to ''g''.

(Thus 1-GO is the same thing as GO.) An ''m''-GO scale can be interpreted as a scale in a rank-(''m'' + 2) [[regular temperament]] (though the specific tuning used may be of lower rank), with basis ''p'' (period), ''g'', δ1, ...,δ''m''.

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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:
 # ''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>.
-# 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''.
 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).
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 More generally, we say that a scale is ''m'''''-GO''' if both of the following hold:
 # the scale consists of ''m'' + 1 chains of stacked generator ''g'' (implying ''m'' offsets δ<sub>1</sub>, ...,δ<sub>''m''</sub> from a fixed chain), each chain having either ''j'' or ''j'' + 1 notes.
-# The generator always subtends a fixed number ''k'' of steps.
+# The scale is well-formed with respect to ''g''.
 (Thus 1-GO is the same thing as GO.) An ''m''-GO scale can be interpreted as a scale in a rank-(''m'' + 2) [[regular temperament]] (though the specific tuning used may be of lower rank), with basis ''p'' (period), ''g'', δ<sub>1</sub>, ...,δ<sub>''m''</sub>.