Rank-3 scale: Difference between revisions

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== Pairwise DE/MOS scales ==

A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to 3-DE ~~scales~~, or equivalently, MOS ~~scales~~. We may also call these ''pairwise MOS'' (PMOS) scales.

A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.

Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.

Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and at least one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.

PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from '''''abacaba'''''.

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The scales '''''a…ba…c''''', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.

There is only one ~~way~~ to arrange the steps of these scales such that they are pairwise-DE~~. This means that they~~ can be uniquely described by a signature~~, like MOS scales~~.

There are only one or two ways to arrange the steps of these scales such that they are pairwise-DE: Either a scale (if achiral) or a pair of chiral scales (equivalent by mirror-inversion and rotation) can be uniquely described by a signature.

== 3-SN scales ==

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 == Pairwise DE/MOS scales ==
-A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to 3-DE scales, or equivalently, MOS scales. We may also call these ''pairwise MOS'' (PMOS) scales.
+A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.
-Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.
+Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and at least one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.
 PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from '''''abacaba'''''.
@@ Line 104: / Line 104: @@
 The scales '''''a…ba…c''''', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.
-There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales.
+There are only one or two ways to arrange the steps of these scales such that they are pairwise-DE: Either a scale (if achiral) or a pair of chiral scales (equivalent by mirror-inversion and rotation) can be uniquely described by a signature.
 == 3-SN scales ==