Generator-offset property: Difference between revisions

Line 1:

A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following ~~equivalent~~ 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.

# 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 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 [[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.

More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''GO''' if it satisfies the following ~~equivalent~~ 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.

# ''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.

# ''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''.

~~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).

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 stacked 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.) 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''.

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.

(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''.

== Other definitions ==

@@ Line 1: / Line 1: @@
-A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following equivalent 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.
+# 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 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 [[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.
-More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''GO''' if it satisfies the following equivalent 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''&nbsp;&minus;&nbsp;1)/2.
+# ''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>.
-# ''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''.
-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).
+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 stacked 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.) 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>.
+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.
+(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>.
 == Other definitions ==