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| A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following properties: | | A scale satisfies the '''generator-offset property''' if it satisfies the following properties: |
| # The scale is generated by two chains of stacked copies of a ''generator''. | | # The scale is generated by two chains of stacked copies of an interval called the ''generator''. |
| # The two chains are separated by an ''offset'' (the difference between the first note of the second chain and the first note of the first chain). | | # The two chains are separated by a different interval called the ''offset'' (the difference between the first note of the second chain and the first note of the first chain). |
| # The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a<sub>1</sub> and a<sub>2</sub>. Note that a<sub>1</sub> does not need to [[subtend]], i.e. occur as, the same number of steps as a<sub>2</sub>.) | | # The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a<sub>1</sub> and a<sub>2</sub>. Note that a<sub>1</sub> does not need to [[subtend]], i.e. occur as, the same number of steps as a<sub>2</sub>.) |
| # The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step. | | # The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step. |
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| [[File:MV3-Labeled.png|thumb|240px|Plot of at-most-decatonic MV3 GO scale regions in the plane.]] | | [[File:MV3-Labeled.png|thumb|240px|Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.]] |
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| The [[Zarlino]] (3L 2M 2S) JI scale is an example of a GO scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7. | | The [[Zarlino]] (3L 2M 2S) JI scale is an example of a generator-offset scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7. |
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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: | | Generator-offset scales generalize the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales where the pentatonic and heptatonic are [[MOS scales]]. A related but distinct notion is [[alternating generator sequence]]. While scales produced using the generator-offset procedure can be seen as a result of an alternating generator sequence of 2 alternants, the generator-offset perspective views the sum of the two alternants as the "canonical" generator, and the alternants as rather being possible choices of the offset which are effectively equivalent up to chirality. While a well-formed AGS scale requires each alternant in the AGS to subtend the same number of steps, the generator-offset property only requires each (aggregate) generator to subtend the same number of steps. |
| # ''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>. | | |
| | Note: In Inthar's contribution to [[aberrismic theory]], this term has been superseded by [[guide frame]]s. |
| | == 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: |
| | # ''S'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size ''n''/2 (implying ''n'' is even), or one chain has size (''n'' + 1)/2 and the second has size (''n'' − 1)/2 (implying ''n'' is odd). |
| # 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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| 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).
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| More generally, we say that a scale is ''m'''''-GO''' if both of the following hold:
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| # 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.
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| # The scale is well-formed with respect to g.
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| (Thus 1-GO is the same thing as GO.)
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| 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).
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| == Conventions used on this page ==
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| * Non-italicized Latin variables refer to interval sizes, for example step sizes.
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| * Indices for all words are 1-indexed.
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| ** If ''S'' is a circular word and ''i'' < 1 or ''i'' > len(''S''), we first replace ''i'' with ''i'' % len(''S'') + 1 before using it as an argument in ''S''[-].
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| * The notation ''S''(X<sub>1</sub>, ..., X<sub>''r''</sub>) is used for an ''r''-ary scale word with variables X<sub>1</sub>, ..., X<sub>''r''</sub> possibly standing in for any sizes. If ''S''(X, Y) = XXY then ''S''(A, B) = AAB.
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| * We leave the distinction between linear and cyclic words up to context. We usually also elide the distinction between subwords and the interval sizes that subtend them.
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| == Other definitions ==
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| * A ''scale'' or ''scale word'' is a circular word with a chosen equave. Since the equave can be set to 1 (in any unit) on this page, all three terms are effectively synonymous.
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| * An ''n''-''ary'' scale is a scale with ''n'' different step sizes. ''Binary'' and ''ternary'' are used when ''n'' = 2 and 3 respectively.
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| * 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, the only ternary scales to satisfy SGA are (xy)<sup>''r''</sup>xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.
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| * An ''odd-step'' is a ''k''-step where ''k'' is odd; an ''even-step'' is defined similarly.
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| * Given a linear or cyclic 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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| * By a ''subword'', ''substring'', or ''slice'' of a word ''S'', denoted ''S''[''i'' : ''j''] (''j'' > ''i''), we mean ''S''[''i''] ''S''[''i'' + 1] ... ''S''[''j'' − 1].
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| * Given a mos aX bY, a ''chunk'' of X's is a maximal substring made of X's, bounded by Y's, possibly empty. We do not include the boundary Y's.
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| * ''Length'' is another term for a scale's size.
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| * A ''projection'' of a scale is the operation of equating some of its step sizes.
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| == Theorems ==
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| === Proposition 1 (Properties of SGA scales) ===
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| Let ''S'' be a ternary scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:
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| # The length of ''S'' is odd, or ''S'' is equivalent to (xy)<sup>''r''</sup>xz for some integer ''r'' ≥ 1.
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| # If ''n'' is odd, ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).
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| # If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).
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| # If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.
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| # If ''n'' is odd, ''S'' is pairwise-mos. That is, the following operations each result in a [[mos]]: setting L = M, setting L = s, and setting M = s.
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| # If ''n'' is odd, ''S'' is elimination-mos. That is, ''E''<sub>X</sub>(''S''), ''E''<sub>Y</sub>(''S''), ''E''<sub>Z</sub>(''S'') are all mosses.
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| In particular, odd GO scales always satisfy these properties (see Proposition 2 below).
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| [Note: This is not true with SGA replaced with GO; [[blackdye]] is a counterexample that is MV4.]
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| ==== Proof ====
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| Let e be the equave of ''S''.
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| Assuming SGA, we have two chains of generator g<sub>0</sub> (going right). The two cases are:
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| CASE 1: EVEN LENGTH
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| O-O-...-O (n/2 notes)
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| O-O-...-O (n/2 notes)
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| and
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| CASE 2: ODD LENGTH
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| O-O-O-...-O ((n+1)/2 notes)
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| O-O-...-O ((n-1)/2 notes).
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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.
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| ===== Statement (1) =====
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| 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''.
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| Since ''S'' is GO 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 or g<sub>2</sub> with g<sub>3</sub> in the next part.
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| Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class (''k''-steps) reached by stacking ''r'' generators:
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| # 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>
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| # 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>
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| # 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.
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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 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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| These are all distinct by '''Z'''-linear independence. 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-length, abstractly SV3, GO 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).
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| ===== Statement (2) =====
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| 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:
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| # ''k''g<sub>1</sub> + (''k'' − 1)g<sub>2</sub>
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| # (''k'' − 1)g<sub>1</sub> + ''k''g<sub>2</sub>
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| # (''k'' − 1)g<sub>1</sub> + (''k'' − 1) g<sub>2</sub> + g<sub>3</sub>
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| (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. This proves (2).
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| (The above holds for any odd ''n'' ≥ 3.)
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| ===== Statement (3) =====
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| For (3), we now only need to see that if len(''S'') is odd and ''S'' is SGA, ''S'' is abstractly SV3. But the argument in case 2 above works when you substitute any interval class in ''S'' instead of a 1-step (abstract SV3 wasn't used), hence any interval class comes in (abstractly) exactly 3 sizes.
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| ===== Statement (4) =====
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| For (4), assume ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1.
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| Consider the two alternants, detemperings of the generator ''i''X + ''j''W of the ''a''X 2''b''W mos ''T''(X, W) = ''S''(X, W, W) where gcd(''j'', 2''k'') = 1.
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| '''Claim 1''': Deleting X's from the alternants of ''S'' gives every ''j''-step subword in the scale ''E''<sub>X</sub>(''S'')(Y, Z), the scale word obtained by deleting all X's from ''S''.
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| Proof: Assume, possibly after inverting the generator, that the imperfect generator of ''T'' has ''j'' + 1 W's and the perfect generator has ''j'' W's. Suppose that one ''j''-step word ''R'' on index ''p'' of ''E''<sub>X</sub>(''S'') is "contained in" the corresponding "imperfect alternant" of ''S'', which is ''I'' = ''S''[''p'' : ''p'' + ''i'' + ''j'']. By this we mean that ''E''<sub>X</sub>(''I'') has ''R'' as a substring. Then ''S''[''p'' − 1: ''p'' − 1 + ''i'' + ''j''] and ''S''[''p'' + 1 : ''p'' + 1 + ''i'' + ''j''] are both detemperings of perfect generators of ''T'', and have one fewer step that is Y or Z. Thus the word ''I'' must both begin and end in a letter that is either Y or Z. Removing all the X's from ''I'' results in a word that is ''j'' + 1 letters long and is the ''j''-step we started with, with just one extra letter appended. Thus one of the two perfect generators above, namely the one that removes the extra letter, must contain this ''j''-step. The rest of the ''j''-step subwords are all contained in "perfect" alternants; take ''q'' ≠ ''p'' to be the index of the first letter of one such ''j''-step subword (as contained in ''S'') and use S[''q'' : ''q'' + ''i'' + ''j''].
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| '''Claim 2''': If a binary scale ''U'' has ''b'' Y's and ''b'' Z's, gcd(''j'', 2''b'') = 1, and consecutively stacked ''j''-steps in ''U'' occur in 2 alternating sizes, then ''U'' = (YZ)<sup>''b''</sup>.
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| Proof: Write u and v for the two sizes of ''j''-steps. Since gcd(''j'', 2''b'') = 1 there exists ''m'' such that stacking ''m''-many ''j''-steps yields scale steps of ''U'', and ''m'' is odd because gcd(''m'', 2''b'') = 1. Hence the scale steps of ''U'' are (uv)<sup>(''m''−1)/2</sup>u mod e and (vu)<sup>(''m''−1)/2</sup>v mod e, and the step sizes clearly alternate.
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| These two claims prove that ''E''<sub>X</sub>(S) = (YZ)<sup>''b''</sup> and that the two alternants' sizes differ by replacing one Y for a Z.
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| ===== Statement (5) =====
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| For (5), odd-numbered SGA 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:
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| x x x ... x
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| x x x ... x x
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| and use the vectors (-1, 2) and (ceil(n/2), 1) as the Fokker block chromas. A rank-3 Fokker block has the property that tempering out by each of the chromas gives two mosses. These correspond to two of the temperings X = Y, Y = Z and X = Z. The third tempering follows by symmetry (by taking the other chirality).
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| ===== Statement (6) =====
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| For (6), consider the mos ''a''X 2''b''W as chunks of X separated by W (tempering Y and Z together into W). Eliminating every other W (corresponding to either Y or Z) turns it into a mos:
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| # The chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos yields a mos.
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| # The sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos.
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| # If chunk sizes of a binary scale form a mos, the scale itself must be a mos; see [[Recursive structure of MOS scales]].
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| Lastly, ''E''<sub>X</sub>(''S'') is the mos ''b''Y ''b''Z; hence ''S'' is elimination-mos.
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| === Proposition 2 (Odd GO scales are SGA) ===
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| Suppose that a periodic scale satisfies the following:
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| * is generator-offset
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| * has odd size ''n''.
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| Then the scale is SGA.
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|
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| ==== Proof ====
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| Assume that the generator is a ''k''-step and ''k'' is even. (If ''k'' is not even, invert the generator.) On some note ''p'' we have a chain of (''n'' + 1)/2 notes and on ''p′'' = ''p'' + offset we'll have (''n'' − 1)/2) notes.
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| We must have gcd(''k'', ''n'') = 1. If not, since ''n'' is odd, gcd(''k'', ''n'') is an odd number at least 3, and by well-formedness with respect to the generator, the generators must form more than 2 parallel chains.
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| By modular arithmetic we have ''rk'' ≡ ''k''/2 mod ''n'' iff ''r'' ≡ (''n'' + 1)/2 mod ''n''. (Note that both 2 and ''k'' are coprime with ''n'', hence multiplicatively invertible mod ''n''.) This proves that the offset, which must be reached after (''n'' + 1)/2 ''k''-steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in (''n'' + 1)/2 steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.)
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| === Proposition 3 (Properties of even GO scales) ===
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| A GO 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:
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| # It is a union of two mosses of size ''n''/2 generated by g
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| # It is ''not'' SV3
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| # It is ''not'' chiral
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| ==== Proof ====
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| (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 SGA property). (3) is easy to check using (1).
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| === Theorem 4 (Classification of pairwise well-formed scales) ===
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| Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed (equivalently, all its projections are single-period mosses). Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either GO or equivalent to the scale word abacaba.
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| ==== Proof ====
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| ===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed =====
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| Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses.
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| index: 1 2 3 4 ... ''n''
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| Σ<sub>1</sub>: P P P P ... P I
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| Σ<sub>2</sub>: [some mos]
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| Σ<sub>3</sub>: [some mos]
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|
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| Below we write step sizes resulting from identification as a~b, b~c, and a~c.
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|
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| ===== Two sizes of ''k''-steps in ''S'' project to ''S''<sub>1</sub>'s perfect generator =====
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| We can write sizes of intervals in ''S'' as vectors (''p'', ''q'', ''r'') using the basis (a, b, c).
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|
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| Suppose for sake of contradiction that only one size of ''k''-step (α, β, γ) in ''S'' projects to P in ''S''<sub>1</sub>. Then projecting to ''S''<sub>2</sub> shows that ''S''<sub>2</sub>'s generator is the ''k''-step (α + γ)*(a~c) + βb, and Σ<sub>2</sub>'s imperfect generator is located at index ''n'', like Σ<sub>1</sub>'s imperfect generator is. Then ''S''<sub>1</sub> and ''S''<sub>2</sub> are the same mode of the same mos pattern (up to knowing which step size is the bigger one). Assume the L of ''S''<sub>1</sub> (it could be s, but it doesn't matter) is the result of identifying b and c, and all instances of s in ''S''<sub>1</sub> come from a. Then the steps of ''S''<sub>2</sub> corresponding to the L of ''S''<sub>1</sub> must be either all b's or all a~c's, thus these steps are all b's in ''S'' (otherwise they would be identified with the a, against the assumption that ''S''<sub>1</sub> and ''S''<sub>2</sub> are the same mos pattern and mode). So ''S'' has only two step sizes (a and b), contradicting the assumption that ''S'' is ternary.
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|
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| Only two sizes of ''k''-steps of ''S'' can project to P in ''S''<sub>1</sub>, for if there are three sizes of ''k''-steps (α, β, γ), (α, β′, γ′), (α, β′′, γ′′) in ''S'' that project to P, then β, β′ and β′′ are three distinct values. Thus these would project to three different ''k''-steps in ''S''<sub>3</sub>, contradicting the mos property of ''S''<sub>3</sub>.
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|
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| ===== ''n'' is odd, etc. =====
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| Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the cyclic word Λ<sub>1</sub> formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ<sub>2</sub> is a single-period mos pattern of βb + (''n'' − β)(a~c) and β′a + (''n'' − β′)(a~c), the cyclic word Λ<sub>2</sub> = the pattern of β and β′ must be a single-period mos. Similarly, Λ<sub>3</sub> = the pattern of γ and γ′ is a single-period mos.
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|
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| Suppose Λ<sub>2</sub> is the mos λβ μβ′. Then Λ<sub>3</sub> is the mos (λ ± 1)γ (μ ∓ 1)γ′. Since neither Λ<sub>2</sub> nor Λ<sub>3</sub> are multimosses, and at least one of μ and (μ ∓ 1) are even, it is now immediate that ''n'' is odd.
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|
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| Either β′′ = β or β′′ = β′. Assume β′′ = β′. Then γ′′ = γ, and Λ<sub>3</sub> is (λ + 1)γ (μ − 1)γ′. Also assume that the first ''k''-step in Σ is Q. Then we have:
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| 1 … ''n''
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| Σ = Q ''W''(Q, R) T
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| Λ<sub>1</sub> = α … α α′
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| Λ<sub>2</sub> = β ''W''(β, β′) β′
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| Λ<sub>3</sub> = γ ''W''(γ, γ′) γ
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| where ''W'' = ''W''(''x'', ''y'') is a word in two variables ''x'' and ''y'', of length ''n'' − 2.
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|
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| ===== Case analysis =====
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| Since, by our assumption, Λ<sub>3</sub> has two γ in a row, Λ<sub>3</sub> must have more γ than γ′, so μ − 1 < ''n''/2. Since Λ<sub>3</sub> is a mos, μ − 1 ≥ 1. So we have 2 ≤ μ ≤ ceil(''n''/2).
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|
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| We have three cases to consider:
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|
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| '''Case 1''': μ = 2, i.e. Λ<sub>2</sub> is the mos (''n'' − 2)β 2β′.
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|
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| For Λ<sub>2</sub> to be a mos, the first, and only, occurrence of R must be at either ''f'' = floor(''n''/2) or ceil(''n''/2). We may assume that it is at ''f''; otherwise flip the chain and reindex the words to start at 2''f''.
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|
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| 1 … ''f'' … 2''f'' ''n''
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| Σ = Q … Q R Q … Q T
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| Λ<sub>1</sub> = α … α α α … α α′
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| Λ<sub>2</sub> = β … β β′ β … β β′
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| Λ<sub>3</sub> = γ … γ γ′ γ … γ γ
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|
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| We need only consider stacks up to ''f''-many ''k''-steps. Either:
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| # the stack has only copies of Q and R; or
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| # the stack has one T and does not contain any R (since it's more than ''f'' − 1 generators away).
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| These give exactly three distinct sizes for every interval class. Hence ''S'' is SV3.
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|
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| In this case S has two chains of Q, one with floor(''n''/2) notes and and one with ceil(''n''/2) notes. Every instance of Q must be a ''k''-step, since by '''Z'''-linear independence Q = αa + βb + γc is the only way to write Q in the basis (a, b, c); so ''S'' is well-formed with respect to Q. Thus ''S'' also satisfies the generator-offset property with generator Q.
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|
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| '''Case 2:''' μ ≥ ceil(''n''/2), i.e. Λ<sub>2</sub> has fewer β than β′.
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|
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| Since Λ<sub>3</sub> has more β than β′, Λ<sub>2</sub> is floor(''n''/2)β ceil(''n''/2)β′, and Λ<sub>3</sub> is ceil(''n''/2)γ floor(''n''/2)γ′. There is a unique mode of ceil(''n''/2)γ floor(''n''/2)γ′ that both begins and ends with γ, namely γγ′γγ′…γγ′γ. Thus Λ<sub>2</sub> is ββ′ββ′…ββ′β′. It is now easy to see that if the number of ''k''-steps stacked is odd, then there are two sizes that do not contain T and one size that contains T; if the number of ''k''-steps stacked is even, then there is one size that does not contain T and two sizes that contain T. Hence ''S'' is SV3.
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|
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| In this case we have Σ = QRQR…QRT, and ''S'' is well-formed with respect to the generator Q + R, thus ''S'' satisfies the generator-offset property. By Proposition 1, ''S'' is SV3.
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|
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| '''Case 3:''' 3 ≤ μ ≤ floor(''n''/2).
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|
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| Λ<sub>2</sub> has a chunk of β (after the first β′) of size ''x'' where ''x'' = floor(''n''/μ) ≥ floor(''n''/floor(''n''/2)) = 2 or ''x'' = ceil(''n''/μ) = floor(''n''/μ) + 1. Hence Λ<sub>3</sub> has a chunk of γ of size ''x''. Λ<sub>3</sub> also has a chunk that crosses index ''n'', which must be of size ''y'', where
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|
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| <math>2 \lfloor\frac{n}{\mu}\rfloor - 1 = 2 \big(\lfloor \frac{n}{\mu} \rfloor - 1\big) + 1 \leq y \leq 2 \big(\lfloor\frac{n}{\mu}\rfloor + 1 \big) + 1 = 2\lfloor\frac{n}{\mu}\rfloor + 3.</math>
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|
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| (The lower bound is reached if Λ<sub>3</sub> has chunks of sizes floor(''n''/μ) − 1 and floor(''n''/μ), and the upper bound is reached if Λ<sub>3</sub> has chunks of sizes floor(''n''/μ) and ceil(''n''/μ).)
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|
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| The difference between the chunk sizes of Λ<sub>3</sub> is ''y'' − ''x'', which must be 0 or 1, since Λ<sub>3</sub> is a mos. We thus have the following subcases: (In the following, chunk of Λ<sub>2</sub> means chunk of β, and chunk of Λ<sub>3</sub> means chunk of γ.)
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|
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| '''Case 3.1:''' (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) − 1).
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|
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| Since ''y'' − ''x'' = floor(''n''/μ) − 1 and floor(''n''/μ) ≥ 2, we have: ''x'' = floor(''n''/μ) = 2 and ''y'' − ''x'' = 1; hence ''y'' = 2*floor(''n''/μ) − 1 = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.)
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|
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| Λ<sub>2</sub> has two consecutive chunks of size 1. Since chunk sizes of a mos themselves form a mos, Λ<sub>2</sub> has more chunks of size 1 than it has chunks of size 2.
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|
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| Λ<sub>2</sub> has only two chunks of size 1, Λ<sub>2</sub>[''n'' − 1] and Λ<sub>2</sub>[1], since otherwise Λ<sub>3</sub> would have a chunk of size 1 within Λ<sub>3</sub>[1 : ''n'']. Thus Λ<sub>2</sub> has exactly one chupnk of size 2. Thus Λ<sub>2</sub> = ββ′βββ′ββ′ and Λ<sub>3</sub> = γγ′γγγ′γγ. Thus we have:
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|
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| 1 2 3 4 5 6 7
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| Σ = Q R Q Q R Q T
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| Λ<sub>1</sub> = α α α α α α α′
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| Λ<sub>2</sub> = β β′ β β β′ β β′
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| Λ<sub>3</sub> = γ γ′ γ γ γ′ γ γ
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|
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| Suppose a step of ''S'' is reached by stacking ''t''-many ''k''-steps. We have three cases after accounting for equave complements:
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|
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| # ''t'' = 1: ''S'' is equivalent to abacaba.
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| # ''t'' = 2: ''S'' is QR QQ RQ TQ RQ QR QT => ''S'' is equivalent to abacaba.
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| # ''t'' = 3: ''S'' is QRQ QRQ TQR QQR QTQ RQQ RQT => ''S'' is equivalent to abacaba.
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|
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| (This also implies ''S'' is SV3.)
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|
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| '''Case 3.2''': (''x'', ''y'') = (floor(''n''/μ) + 1, 2*floor(''n''/μ) − 1) is impossible because ''y'' = 2*floor(''n''/μ) − 1 can only occur if Λ<sub>3</sub> has chunks of size floor(''n''/μ) − 1 and floor(''n''/μ), which contradicts the size of ''x''.
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|
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| '''Case 3.3''': (''x'', ''y'') = (floor(''n''/μ) + 1, 2*floor(''n''/μ)) is similarly impossible; ''y'' = 2*floor(''n''/μ) can only occur if Λ<sub>3</sub> has chunks of size floor(''n''/μ) − 1 and floor(''n''/μ), which contradicts the size of ''x''.
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|
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| The remaining cases are all impossible because they imply ''y'' − ''x'' ≥ 2:
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|
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| * '''Case 3.4''': (''x'', ''y'') = (floor(''n''/μ) + 1, 2*floor(''n''/μ) + 1)
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| * '''Case 3.5''': (''x'', ''y'') = (floor(''n''/μ) + 1, 2*floor(''n''/μ) + 2)
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| * '''Case 3.6''': (''x'', ''y'') = (floor(''n''/μ) + 1, 2*floor(''n''/μ) + 3)
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| * '''Case 3.7''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ))
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| * '''Case 3.8''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 1)
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| * '''Case 3.9''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 2)
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| * '''Case 3.10''': (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) + 3)
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|
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| === Theorem 5 (Classification of MV3 scales) ===
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| # A single-period MV3 is either (1) pairwise well-formed, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows:
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| ## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.
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| ## Interchange some of the Z's and X's at some (possibly none) of the borders of these copies of the mos word ''w''.
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| ## Replace every other X with Y in ''w''.
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| # Single-period MV3 scales not of type (3) are always SV3.
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| ==== Proof ====
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| Proven in Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]".
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|
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| == Open conjectures ==
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| === Conjecture ("MV3 Sequences") ===
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| Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.
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|
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| Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum ''N'', such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).
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|
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| It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large ''N'', far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.
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|
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| == Open questions ==
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| This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary)
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|
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| # Given any arbitrary MOS (or DE, etc) scale with at least three notes per period, is there *always* a MV3 GO scale which can be derived as a "detempering" of that scale? Or is this only true for some MOS's? For instance, the MOS LLsLLLs has the MV3 GO scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible DE scale with at least three notes per period, or at least for strict MOS's with one period per octave (e.g. well-formed scales)?
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| # The scale tree is a great way to analyze MOS scales. For any generator, we can compute the various MOS's it forms if we simply look at the scale tree, and indeed MOS "words" like LLsLLLs can be identified with regions on the scale tree (in this situation the interval between 4/7 and 3/5). A similar "scale plane" should exist for GO-MV3 scales, where given some word representing a GO-MV3 scale, we can look at the set of points on the generator plane which generates it; these seem to often be triangles, with the lines corresponding to MOS's and the vertices corresponding to EDOs (though is this always true?). What is the big picture of this scale plane? Can we use Viggo Brun's algorithm for this, generalizing the theory of continued fractions? Is there some simple formula we can use to predict, given some GO-MV3 scale, which region on the scale plane it corresponds to? Can we plot simple generator-size-proportions as points in this space? And so on.
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| # In the theory of MOS, there is a second [[MOS Scale Family Tree|scale tree]] that is less frequently talked about, which Erv Wilson calls the "Rabbit Sequence" ([http://www.anaphoria.com/RabbitSequence.pdf Erv Wilson's original version], [https://mikebattagliamusic.com/MOSTree/MOSTreeab.html interactive version 1], [https://mikebattagliamusic.com/MOSTree/MOSTreeLs.html interactive version 2]). This is a tree for which each MOS word has two children, depending on if the MOS is "soft" (with L/s < 2) or "hard" (with L/s > 2). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these GO-MV3 scales?
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|
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| == Falsified conjectures ==
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| === All mosses have an MV3 detempering (Counterexample: LsLsLsLsLs) ===
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| LsLsLsLsLs does ''not'' have an MV3 detempering: Wolog we “detemper” L to L and M.
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|
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| LsMsLsMsLs is not MV3: LsM LsL sMs sLs
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|
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| LsMsLsMsMs can be rotated to MsLsMsLsMs which is not MV3 by symmetry with LsMsLsMsLs.
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|
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| LsMsMsMsMs is not MV3: LsM MLM sMs sLs
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|
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| MsLsLsLsLs not MV3 for the same reason as LsMsMsMsMs
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|
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|
| [[Category:GO scales| ]]<!--Main article-->
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| [[Category:Scale]] | | [[Category:Scale]] |
