Generator-offset property: Difference between revisions
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A scale satisfies the '''alternating generator property''' (also ''' | A scale satisfies the '''alternating generator property''' (also '''alt-gen''' 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 alternating generators | ||
* 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 a fixed interval, and the lengths of the chains differ by at most one. | ||
[[Diasem]] is an example of an | [[Diasem]] is an example of an alt-gen scale, because it is built by stacking alternating 7/6 and 8/7 for [[chirality|left-handed]] diasem, or 8/7 and 7/6 for right-handed diasem. | ||
More formally, a cyclic word ''S'' (representing a [[periodic scale]]) of size ''n'' is ''' | 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 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'' 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. | # ''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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== Theorems == | == Theorems == | ||
=== Theorem 1 === | === Theorem 1 === | ||
If a 3-step-size scale word ''S'' in L, M, and s is both | If a 3-step-size scale word ''S'' in L, M, and s is both alt-gen and unconditionally [[MV3]] (i.e. MV3 regardless of tuning), then the scale is of the form ''ax by bz'' for some permutation (''x'', ''y'', ''z'') of (L, M, s); and the scale's cardinality (size) is either odd, or 4 (and is of the form ''xyxz''). Moreover, any odd-cardinality alt-gen scale is unconditionally MV3. | ||
==== Proof ==== | ==== Proof ==== | ||
Assuming both | Assuming both alt-gen and unconditionally MV3, we have two chains of generator ''g''<sub>0</sub> (going right). The two cases are: | ||
CASE 1: EVEN CARDINALITY | CASE 1: EVEN CARDINALITY | ||
O-O-...-O (n/2 notes) | O-O-...-O (n/2 notes) | ||
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Choose a tuning where ''g''<sub>1</sub> and ''g''<sub>2</sub> are both very close to but not exactly 1/2*''g''<sub>0</sub>, resulting in a scale very close to the mos generated by 1/2 ''g''<sub>0</sub>. (i.e. ''g''<sub>1</sub> and ''g''<sub>2</sub> differ from 1/2*''g''<sub>0</sub> by ε, a quantity much smaller than the chroma of the ''n''/2-note mos generated by ''g''<sub>0</sub>, which is |''g''<sub>3</sub> − ''g''<sub>2</sub>|). Thus we have 4 distinct sizes for ''k''-steps: | Choose a tuning where ''g''<sub>1</sub> and ''g''<sub>2</sub> are both very close to but not exactly 1/2*''g''<sub>0</sub>, resulting in a scale very close to the mos generated by 1/2 ''g''<sub>0</sub>. (i.e. ''g''<sub>1</sub> and ''g''<sub>2</sub> differ from 1/2*''g''<sub>0</sub> by ε, a quantity much smaller than the chroma of the ''n''/2-note mos generated by ''g''<sub>0</sub>, which is |''g''<sub>3</sub> − ''g''<sub>2</sub>|). Thus we have 4 distinct sizes for ''k''-steps: | ||
# ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''a''<sub>3</sub> are clearly distinct. | # ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''a''<sub>3</sub> are clearly distinct. | ||
# ''a''<sub>4</sub> − ''a''<sub>3</sub> = ''g''<sub>1</sub> − ''g''<sub>2</sub> != 0, since the scale is a non-trivial | # ''a''<sub>4</sub> − ''a''<sub>3</sub> = ''g''<sub>1</sub> − ''g''<sub>2</sub> != 0, since the scale is a non-trivial alt-gen. | ||
# ''a''<sub>4</sub> − ''a''<sub>1</sub> = ''g''<sub>3</sub> − ''g''<sub>2</sub> = (''g''<sub>3</sub> + ''g''<sub>1</sub>) − (''g''<sub>2</sub> + ''g''<sub>1</sub>) != 0. This is exactly the chroma of the mos generated by ''g''<sub>0</sub>. | # ''a''<sub>4</sub> − ''a''<sub>1</sub> = ''g''<sub>3</sub> − ''g''<sub>2</sub> = (''g''<sub>3</sub> + ''g''<sub>1</sub>) − (''g''<sub>2</sub> + ''g''<sub>1</sub>) != 0. This is exactly the chroma of the mos generated by ''g''<sub>0</sub>. | ||
# ''a''<sub>4</sub> − ''a''<sub>2</sub> = ''g''<sub>1</sub> − 2 ''g''<sub>2</sub> + ''g''<sub>3</sub> = (''g''<sub>3</sub> − ''g''<sub>2</sub>) + (''g''<sub>1</sub> − ''g''<sub>2</sub>) = (chroma ± ε) != 0 by choice of tuning. | # ''a''<sub>4</sub> − ''a''<sub>2</sub> = ''g''<sub>1</sub> − 2 ''g''<sub>2</sub> + ''g''<sub>3</sub> = (''g''<sub>3</sub> − ''g''<sub>2</sub>) + (''g''<sub>1</sub> − ''g''<sub>2</sub>) = (chroma ± ε) != 0 by choice of tuning. | ||
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, | 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) − (1, 1) = ''g''<sub>1</sub>, (1, 2) − (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) − (1, 1) = ''g''<sub>1</sub>, (1, 2) − (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: | ||
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(The above holds for any odd ''n'' ≥ 3.) | (The above holds for any odd ''n'' ≥ 3.) | ||
Now we only need to see that | Now we only need to see that alt-gen + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning. | ||
== Conjectures == | == Conjectures == | ||
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# is not of the form ''mx my mz'', | # is not of the form ''mx my mz'', | ||
# and is not of the form ''xyxzxyx'', | # and is not of the form ''xyxzxyx'', | ||
then it is | then it is alt-gen. | ||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category: | [[Category:alt-gen scales| ]]<!--Main article--> | ||