Rank-3 scale: Difference between revisions

Line 75:

== Pairwise well-formed scales==

Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of ~~product words~~.

Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of MOS pattern products.

If equating any pair of step sizes (tempering out their difference, if we involve mappings) of a rank-3 scale leads to 3 WF scales, the rank-3 scale is ''pairwise well-formed (PWF).''

Line 96:

== 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 a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.

A similar generalization, a larger subset of MOS pattern products, 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 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.

@@ Line 75: / Line 75: @@
 == Pairwise well-formed scales==
-Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of product words.
+Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of MOS pattern products.
 If equating any pair of step sizes (tempering out their difference, if we involve mappings) of a rank-3 scale leads to 3 WF scales, the rank-3 scale is ''pairwise well-formed (PWF).''
@@ Line 96: / Line 96: @@
 == 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 a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.
+A similar generalization, a larger subset of MOS pattern products, 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 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.