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| A ''ternary scale'' is a scale with three (positive) step sizes, with no other constraints. This page documents known properties of subtypes of ternary scales and their proofs. | | A ''ternary scale'' is a scale with three (positive) step sizes, with no other constraints. This page documents known properties of subtypes of ternary scales and their proofs. |
| == Theorems ==
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| * Every triple [[Fokker block]] is max variety 3.
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| * Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
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| * Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
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| * A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions | SpringerLink]
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| * If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
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| * Any convex object on the lattice can be converted into a hexagon.
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| * Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
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| * An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].
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| == Theorems with proofs == | | == Theorems with proofs == |
| === Conventions === | | === Conventions === |