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 ''g'' separated by a fixed offset δ; 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'' is generated by two chains of stacked generators g separated by a fixed offset δ; 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 g1 and g2, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.

# The scale is ''well-formed'' with respect to ''g'', i.e. 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).

This doesn't imply that g1 and g2 are the same number of scale steps. For example, 5-limit [[blackdye]] has g1 = 9/5 (a 9-step) and g2 = 5/3 (a 7-step).

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

# 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]], with basis ''p'' (period), ''g'', δ1, ...,δ''m'' (though the specific tuning used may be of lower rank).

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

Note: On this page, non-italicized Latin variables refer to interval sizes.

== Other definitions ==

* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants ''g''1 and ''g''2 can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, only xyxz satisfies this property. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.

* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g1 and g2 can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, only xyxz satisfies this property. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.

== Theorems ==

~~Note: Non-italicized Latin variables refer to interval sizes.~~

=== Proposition 1 (Properties of SGA scales) ===

Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:

@@ Line 10: / Line 10: @@
 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 ''g'' separated by a fixed offset δ; 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'' is generated by two chains of stacked generators g separated by a fixed offset δ; 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>.
-# The scale is ''well-formed'' with respect to ''g'', i.e. 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).
+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 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 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 scale is well-formed with respect to ''g''.
+# 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]], with basis ''p'' (period), ''g'', δ<sub>1</sub>, ...,δ<sub>''m''</sub> (though the specific tuning used may be of lower rank).
+An ''m''-GO scale can be interpreted as a scale in a rank-(''m'' + 2) [[regular temperament]], with basis p (period), g, δ<sub>1</sub>, ...,δ<sub>''m''</sub> (though the specific tuning used may be of lower rank).
+Note: On this page, non-italicized Latin variables refer to interval sizes.
 == Other definitions ==
-* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, only xyxz satisfies this property. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.
+* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g<sub>1</sub> and g<sub>2</sub> can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, only xyxz satisfies this property. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.
 == Theorems ==
-Note: Non-italicized Latin variables refer to interval sizes.
 === Proposition 1 (Properties of SGA scales) ===
 Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then: