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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). | | 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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| == Other definitions ==
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| * A ''scale'' or ''scale word'' is a circular word with a chosen size for its equave. As we're not working with scales with distinct equaves simultaneously, all three terms are effectively synonymous for our purposes.
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| * A scale is ''primitive'', or ''single-period'', if its period is the same as its equave. A ''multimos'' or ''multiperiod mos'' is a non-primitive mos. A mos aLbs is primitive iff gcd(a, b) = 1.
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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 generator-offset scales are SGA, and aside from odd generator-offset 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 generator-offset 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 (possibly length 0) substring made of X's, bounded by Y's. 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 ternary scale is the operation of equating two of its step sizes.
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| * A ternary scale is ''pairwise-well-formed'' if all its projections are well-formed (i.e. single-period mosses).
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| == Open conjectures == | | == Open conjectures == |