Generator-offset property: Difference between revisions

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A scale satisfies the '''~~alternating~~ generator property''' (also '''~~alt-gen~~''' or '''AG''') if it satisfies the following equivalent properties:

A scale satisfies the '''generator-offset property''' (also '''alternating generator''' or '''AG''') if it satisfies the following equivalent properties:

* the scale can be built by stacking alternating generators

* the scale can be built by stacking two alternating generators, which do not necessarily take up the same number of steps

* the scale is generated by two chains of generators separated by ~~a fixed interval~~, and the lengths of the chains differ by at most one.

* the scale is generated by two chains of generators separated by an offset, and the lengths of the chains differ by at most one.

The [[Zarlino]] (3L 2M 2S) JI scale is an example of an alt-gen 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 a [[periodic scale]]) of size ''n'' is '''alt-gen''' if it satisfies the following equivalent properties:

# ''S'' can be built by stacking a single chain of ~~alternating generators~~ ''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 generator-offsets ''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 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.

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== Related properties ==

* A strengthening of the AG property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''1 and ''g''2 can be taken to be 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). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASGC. [[Blackdye]] is AG but not ASGC.

* A strengthening of the generator-offset property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''1 and ''g''2 can be taken to be 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). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASGC. [[Blackdye]] is AG but not ASGC.

== Theorems ==

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By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''1 and ''g''2 must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, alt-gen scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and ''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.

In case 2, let (2, 1) − (1, 1) = ''g''1, (1, 2) − (2, 1) = ''g''2 be the two ~~alternating generators~~. Let ''g''3 be the leftover generator after stacking alternating ''g''1 and ''g''2. Then the generator circle looks like ''g''1 ''g''2 ''g''1 ''g''2 ... ''g''1 ''g''2 ''g''3. Then the generators corresponding to a step are:

In case 2, let (2, 1) − (1, 1) = ''g''1, (1, 2) − (2, 1) = ''g''2 be the two generator-offsets. Let ''g''3 be the leftover generator after stacking alternating ''g''1 and ''g''2. Then the generator circle looks like ''g''1 ''g''2 ''g''1 ''g''2 ... ''g''1 ''g''2 ''g''3. Then the generators corresponding to a step are:

# ''kg''1 + (''k'' − 1)''g''2

# (''k'' − 1)''g''1 + ''kg''2

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== Conjectures ==

=== Conjecture 2 ===

An odd AG scale is ASGC.

An odd generator-offset scale is ASGC.

=== Conjecture 3 ===

If a non-multiperiod 3-step size scale word is

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-A scale satisfies the '''alternating generator property''' (also '''alt-gen''' or '''AG''') if it satisfies the following equivalent properties:
+A scale satisfies the '''generator-offset property''' (also '''alternating generator''' or '''AG''') if it satisfies the following equivalent properties:
-* the scale can be built by stacking alternating generators
+* the scale can be built by stacking two alternating generators, which do not necessarily take up the same number of steps
-* the scale is generated by two chains of generators separated by a fixed interval, and the lengths of the chains differ by at most one.
+* the scale is generated by two chains of generators separated by an offset, and the lengths of the chains differ by at most one.
 The [[Zarlino]] (3L 2M 2S) JI scale is an example of an alt-gen 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 a [[periodic scale]]) of size ''n'' is '''alt-gen''' if it satisfies the following equivalent properties:
-# ''S'' can be built by stacking a single chain of alternating generators ''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 generator-offsets ''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 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.
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 == Related properties ==
-* A strengthening of the AG property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to be 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). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASGC. [[Blackdye]] is AG but not ASGC.
+* A strengthening of the generator-offset property, tentatively named ''alternating split-generator-class'' (ASGC), states that ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to be 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). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASGC. [[Blackdye]] is AG but not ASGC.
 == Theorems ==
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 By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''<sub>1</sub> and ''g''<sub>2</sub> must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, alt-gen scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and ''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.
-In case 2, let (2, 1) &minus; (1, 1) = ''g''<sub>1</sub>, (1, 2) &minus; (2, 1) = ''g''<sub>2</sub> be the two alternating generators. Let ''g''<sub>3</sub> be the leftover generator after stacking alternating ''g''<sub>1</sub> and ''g''<sub>2</sub>. Then the generator circle looks like ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>3</sub>. Then the generators corresponding to a step are:
+In case 2, let (2, 1) &minus; (1, 1) = ''g''<sub>1</sub>, (1, 2) &minus; (2, 1) = ''g''<sub>2</sub> be the two generator-offsets. Let ''g''<sub>3</sub> be the leftover generator after stacking alternating ''g''<sub>1</sub> and ''g''<sub>2</sub>. Then the generator circle looks like ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>1</sub> ''g''<sub>2</sub> ... ''g''<sub>1</sub> ''g''<sub>2</sub> ''g''<sub>3</sub>. Then the generators corresponding to a step are:
 # ''kg''<sub>1</sub> + (''k'' &minus; 1)''g''<sub>2</sub>
 # (''k'' &minus; 1)''g''<sub>1</sub> + ''kg''<sub>2</sub>
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 == Conjectures ==
 === Conjecture 2 ===
-An odd AG scale is ASGC.
+An odd generator-offset scale is ASGC.
 === Conjecture 3 ===
 If a non-multiperiod 3-step size scale word is