Ternary scale theorems: Difference between revisions
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** There exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the GS recipe GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps. This implies that the gap between the next higher equave and the result of stacking len(scale) − 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. | ** There exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the GS recipe GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps. This implies that the gap between the next higher equave and the result of stacking len(scale) − 1 of the generators in the recipe, called the ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. | ||
** The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale. | ** The closing generator must be distinct from all of the generators used in the generator sequence and occur only once in the scale. | ||
* The property of having a WFGS of period 2, denoted | * The property of having a WFGS of period 2, denoted AGS (''alternating generator sequence'') in this article, is important as it is equivalent to being an odd-regular MV3 scale; see below. It used to be called the "SGA property" in past versions of this article. | ||
* An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly. | * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly. | ||
* Given a linear or circular word ''s'' with a step size '''X''', define ''E''<sub>'''X'''</sub>(''s'') as the scale word resulting from deleting all instances of '''X''' from ''s''. | * Given a linear or circular word ''s'' with a step size '''X''', define ''E''<sub>'''X'''</sub>(''s'') as the scale word resulting from deleting all instances of '''X''' from ''s''. | ||
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* A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. primitive MOSes). | * A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. primitive MOSes). | ||
== Theorem 1 (Properties of | == Theorem 1 (Properties of AGS scales) == | ||
Let ''s'' be a ternary scale word in '''L''', '''M''', and '''s''' of length ''n'', and suppose ''s'' is | Let ''s'' be a ternary scale word in '''L''', '''M''', and '''s''' of length ''n'', and suppose ''s'' is AGS. Then: | ||
# The length of ''s'' is odd, or ''s'' is equivalent to ('''xy''')<sup>''r''</sup>'''xz''' for some integer {{nowrap|''r'' ≥ 1}}. | # The length of ''s'' is odd, or ''s'' is equivalent to ('''xy''')<sup>''r''</sup>'''xz''' for some integer {{nowrap|''r'' ≥ 1}}. | ||
# If ''n'' is odd, ''s'' is of the form ''a'''''x''' ''b'''''y''' ''b'''''z''' for some permutation {{nowrap|('''x''', '''y''', '''z''')}} of {{nowrap|('''L''', '''M''', '''s''')}}. | # If ''n'' is odd, ''s'' is of the form ''a'''''x''' ''b'''''y''' ''b'''''z''' for some permutation {{nowrap|('''x''', '''y''', '''z''')}} of {{nowrap|('''L''', '''M''', '''s''')}}. | ||
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In particular, odd generator-offset scales always satisfy these properties (see Proposition 2 below). | In particular, odd generator-offset scales always satisfy these properties (see Proposition 2 below). | ||
[Note: This is not true with | [Note: This is not true with AGS replaced with generator-offset; [[blackdye]] is a counterexample that is MV4.] | ||
=== Proof === | === Proof === | ||
Let '''e''' be the equave of ''s''. | Let '''e''' be the equave of ''s''. | ||
Assuming | Assuming AGS, we have two chains of the aggregate generator '''g''' (going right). In the diagrams below, O represents a note and - represents a generator '''g'''. The two cases are: | ||
<pre<includeonly />> | <pre<includeonly />> | ||
CASE 1: EVEN LENGTH | CASE 1: EVEN LENGTH | ||
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In case 1, let {{nowrap|'''g'''<sub>1</sub> {{=}} (2, 1) − (1, 1)|'''g'''<sub>2</sub> {{=}} (1, 2) − (2, 1)}}, and {{nowrap|'''g'''<sub>3</sub> {{=}} (1, 1) − ({{frac|''n''|2}}, 2)}} {{nowrap|{{=}} ((−{{frac|''n''|2}} − 1)*'''g'''<sub>1</sub> − {{frac|''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''. | In case 1, let {{nowrap|'''g'''<sub>1</sub> {{=}} (2, 1) − (1, 1)|'''g'''<sub>2</sub> {{=}} (1, 2) − (2, 1)}}, and {{nowrap|'''g'''<sub>3</sub> {{=}} (1, 1) − ({{frac|''n''|2}}, 2)}} {{nowrap|{{=}} ((−{{frac|''n''|2}} − 1)*'''g'''<sub>1</sub> − {{frac|''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 the aggregate generator {{nowrap|'''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 by the | Since ''s'' is generator-offset it is well-formed with respect to the aggregate generator {{nowrap|'''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 by the AGS assumption, each is an odd-step. All multiples of the aggregate generator '''g''' must be even-steps, and those dyads that are "offset" by '''g'''<sub>1</sub> must be odd-steps. Letting ''M'' be the subset consisting 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 {{nowrap|('''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 dyad class of ''k''-steps reached by stacking ''r'' generators: | Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the dyad class of ''k''-steps reached by stacking ''r'' generators: | ||
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# 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 {{nowrap|''a''<sub>4</sub> {{=}} {{sfrac|''r'' + 1|2}} '''g'''<sub>1</sub> + {{sfrac|''r'' − 3|2}} '''g'''<sub>2</sub> + '''g'''<sub>3</sub>}} {{nowrap|≡ {{sfrac|''r'' − ''n''|2}} − {{sfrac|1|2}}'''g'''<sub>1</sub> + {{sfrac|''r'' − ''n''|2}} − {{sfrac|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 {{nowrap|''a''<sub>4</sub> {{=}} {{sfrac|''r'' + 1|2}} '''g'''<sub>1</sub> + {{sfrac|''r'' − 3|2}} '''g'''<sub>2</sub> + '''g'''<sub>3</sub>}} {{nowrap|≡ {{sfrac|''r'' − ''n''|2}} − {{sfrac|1|2}}'''g'''<sub>1</sub> + {{sfrac|''r'' − ''n''|2}} − {{sfrac|3|2}}'''g'''<sub>2</sub> (mod '''e''')}}. | ||
Since {{nowrap|''n'' > 0}}, these are all distinct by ℤ-linear independence; hence there are at least 4 sizes for ''k''-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 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-length | Since {{nowrap|''n'' > 0}}, these are all distinct by ℤ-linear independence; hence there are at least 4 sizes for ''k''-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 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-length AGS ternary scale must be of the form (xy)<sup>''r''</sup>xz. (Note that (xy)<sup>''r''</sup>xz is not SV3, since it has only two kinds of 2-steps, '''xy''' and '''xz'''.) This proves (1). | ||
==== Statement (2) ==== | ==== Statement (2) ==== | ||
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==== Statement (3) ==== | ==== Statement (3) ==== | ||
We only need to see that if len(''s'') is odd and ''s'' is | We only need to see that if len(''s'') is odd and ''s'' is AGS, ''s'' is abstractly SV3. But the argument in case 2 above works when you substitute any odd-step dyad classes in ''s'' instead of a 1-step (abstract SV3 wasn't used). To get even-step dyad classes, we can take octave complements. Hence any dyad class in such a scale comes in (abstractly) exactly 3 sizes. | ||
==== Statement (4) ==== | ==== Statement (4) ==== | ||
Odd-numbered | Odd-numbered AGS scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale: | ||
x x x ... x | x x x ... x | ||
x x x ... x x | x x x ... x x | ||
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These two claims prove that {{nowrap|''E''<sub>'''X'''</sub>(S) {{=}} ('''YZ''')<sup>''b''</sup>}} and that the two GS generators' sizes differ by replacing one '''Y''' for a '''Z'''. {{Qed}} | These two claims prove that {{nowrap|''E''<sub>'''X'''</sub>(S) {{=}} ('''YZ''')<sup>''b''</sup>}} and that the two GS generators' sizes differ by replacing one '''Y''' for a '''Z'''. {{Qed}} | ||
== Theorem 2 (Odd generator-offset scales are | == Theorem 2 (Odd generator-offset scales are AGS) == | ||
Suppose that a periodic scale satisfies the following: | Suppose that a periodic scale satisfies the following: | ||
* is generator-offset | * is generator-offset | ||
* has odd size ''n''. | * has odd size ''n''. | ||
Then the scale is | Then the scale is AGS. | ||
=== Proof === | === Proof === | ||
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=== Proof === | === Proof === | ||
(1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator '''g''' must be even-steps ..." to "These are all distinct by ℤ-linear independence", does not rely on ''s'' having the | (1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator '''g''' must be even-steps ..." to "These are all distinct by ℤ-linear independence", does not rely on ''s'' having the AGS property). (3) and (4) are easy to check using (1). {{qed}} | ||
== Theorem 4 (Classification of pairwise well-formed scales) == | == Theorem 4 (Classification of pairwise well-formed scales) == | ||