Generator-offset property: Difference between revisions
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== Mathematical definition == | == Mathematical definition == | ||
More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties: | More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' 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<sub>1</sub> and g<sub>2</sub>, resulting in a circle of the form either g<sub>1</sub> g<sub>2</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'' − 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''. | ||
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Label the notes (1, ''j'') and (2, ''j''), 1 ≤ ''j'' ≤ (number of notes in the chain), for notes in the upper and lower chain respectively. | Label the notes (1, ''j'') and (2, ''j''), 1 ≤ ''j'' ≤ (number of notes in the chain), for notes in the upper and lower chain respectively. | ||
===== Statement (1) ===== | ===== Statement (1) ===== | ||
In case 1, let g<sub>1</sub> = (2, 1) − (1, 1), g<sub>2</sub> = (1, 2) − (2, 1), and g<sub>3</sub> = (1, 1) − (''n''/2, 2) = (−''n''/2*g<sub>1</sub> − g<sub>1</sub> − ''n''/2*g<sub>2</sub>) mod e. We assume that g<sub>1</sub>, g<sub>2</sub> and e are '''Z'''-linearly independent. We have the chain g<sub>1</sub> g<sub>2</sub> g<sub>1</sub> g<sub>2</sub> ... | In case 1, let g<sub>1</sub> = (2, 1) − (1, 1), g<sub>2</sub> = (1, 2) − (2, 1), and g<sub>3</sub> = (1, 1) − (''n''/2, 2) = (−''n''/2*g<sub>1</sub> − g<sub>1</sub> − ''n''/2*g<sub>2</sub>) mod e. We assume that g<sub>1</sub>, g<sub>2</sub> and e are '''Z'''-linearly independent. We have the chain g<sub>1</sub> g<sub>2</sub> g<sub>1</sub> g<sub>2</sub> ... g<sub>1</sub> g<sub>3</sub> which visits every note in ''S''. | ||
Since ''S'' is generator-offset it is well-formed with respect to g = (g<sub>2</sub> + g<sub>1</sub>). Since g<sub>1</sub> and g<sub>2</sub> subtend the same number of steps, all multiples of the generator g must be even-steps, and those intervals that are "offset" by g<sub>1</sub> must be odd-steps. Letting ''M'' be the subset of all even-numbered notes (which are generated by g) and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to g, thus ''M'' (and its offset) must be a mos subset. Hence (g<sub>3</sub> + g<sub>1</sub>), the imperfect generator of the mos generated by g, subtends the same number of steps as g. Thus g<sub>2</sub> and g<sub>3</sub> subtend the same number of steps, a fact we need in order to be able to substitute one instance of g<sub>2</sub> with g<sub>3</sub> in the next part. | Since ''S'' is generator-offset it is well-formed with respect to g = (g<sub>2</sub> + g<sub>1</sub>). Since g<sub>1</sub> and g<sub>2</sub> subtend the same number of steps, all multiples of the generator g must be even-steps, and those intervals that are "offset" by g<sub>1</sub> must be odd-steps. Letting ''M'' be the subset of all even-numbered notes (which are generated by g) and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to g, thus ''M'' (and its offset) must be a mos subset. Hence (g<sub>3</sub> + g<sub>1</sub>), the imperfect generator of the mos generated by g, subtends the same number of steps as g. Thus g<sub>2</sub> and g<sub>3</sub> subtend the same number of steps, a fact we need in order to be able to substitute one instance of g<sub>2</sub> with g<sub>3</sub> in the next part. | ||
Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class (''k''-steps) reached by stacking ''r'' generators: | Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class (''k''-steps) reached by stacking ''r'' generators: | ||
# from g<sub>1</sub> ... | # from g<sub>1</sub> ... g<sub>1</sub>, we get a<sub>1</sub> = (''r'' − 1)/2*g<sub>0</sub> + g<sub>1</sub> = (''r'' + 1/2) g<sub>1</sub> + (''r'' − 1/2) g<sub>2</sub> | ||
# from g<sub>2</sub> ... | # from g<sub>2</sub> ... g<sub>2</sub>, we get a<sub>2</sub> = (''r'' − 1)/2*g<sub>0</sub> + g<sub>2</sub> = (''r'' − 1/2) g<sub>1</sub> + (''r'' + 1/2) g<sub>2</sub> | ||
# from g<sub>2</sub> (...even # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...even # of gens...) g<sub>2</sub>, we get a<sub>3</sub> = (''r'' − 1)/2 g<sub>1</sub> + (''r'' − 1)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' − ''n''/2 − 3/2)g<sub>1</sub> + (''r'' − ''n''/2 − 1/2)g<sub>2</sub> mod e. | # from g<sub>2</sub> (...even # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...even # of gens...) g<sub>2</sub>, we get a<sub>3</sub> = (''r'' − 1)/2 g<sub>1</sub> + (''r'' − 1)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' − ''n''/2 − 3/2)g<sub>1</sub> + (''r'' − ''n''/2 − 1/2)g<sub>2</sub> mod e. | ||
# from g<sub>1</sub> (...odd # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...odd # of gens...) g<sub>1</sub>, we get a<sub>4</sub> = (''r'' + 1)/2 g<sub>1</sub> + (''r'' − 3)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' − ''n''/2 − 1/2)g<sub>1</sub> + (''r'' − ''n''/2 − 3/2)g<sub>2</sub> mod e. | # from g<sub>1</sub> (...odd # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...odd # of gens...) g<sub>1</sub>, we get a<sub>4</sub> = (''r'' + 1)/2 g<sub>1</sub> + (''r'' − 3)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' − ''n''/2 − 1/2)g<sub>1</sub> + (''r'' − ''n''/2 − 3/2)g<sub>2</sub> mod e. | ||
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===== Statement (2) ===== | ===== Statement (2) ===== | ||
In case 2, let (2, 1) − (1, 1) = g<sub>1</sub>, (1, 2) − (2, 1) = g<sub>2</sub> be the two alternants. 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> ... | In case 2, let (2, 1) − (1, 1) = g<sub>1</sub>, (1, 2) − (2, 1) = g<sub>2</sub> be the two alternants. 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>. Assuming that a step is an odd number of generators, the combinations of alternants corresponding to a step come in exactly 3 sizes: | ||
# ''k''g<sub>1</sub> + (''k'' − 1)g<sub>2</sub> | # ''k''g<sub>1</sub> + (''k'' − 1)g<sub>2</sub> | ||
# (''k'' − 1)g<sub>1</sub> + ''k''g<sub>2</sub> | # (''k'' − 1)g<sub>1</sub> + ''k''g<sub>2</sub> | ||
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=== Proposition 3 (Properties of even generator-offset scales) === | === Proposition 3 (Properties of even generator-offset scales) === | ||
A primitive generator-offset scale of even size where the generator g is an even-step (i.e. | A primitive generator-offset scale of even size where the generator g is an even-step (i.e. g subtends an even number of steps) has the following properties: | ||
# It is a union of two primitive mosses of size ''n''/2 generated by g | # It is a union of two primitive mosses of size ''n''/2 generated by g | ||
# It is ''not'' SV3 | # It is ''not'' SV3 | ||