Rank-3 scale: Difference between revisions

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

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

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).''

PWF scales are MV3~~, and even SV3~~.

PWF scales are SV3 (and therefore MV3).

When mappings are considered, PWF scales are rank-3 ''[[Gallery of wakalixes|wakalix]]es'' - Fokker blocks which are Fokker blocks in more than one way.

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Only a single PWF scale is mirror-symmetric - '''''abacaba'''''.

Apart from '''''abacaba''''', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.

Apart from '''''abacaba''''', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period, i.e., apart from '''''abacaba''''', all PWF scales are GO scales.

PWF scales can only have odd numbers of notes.

'''Conjecture''': The GO scales of odd cardinality are the PWF scales that are not '''''abacaba'''''.

== Pairwise DE/MOS scales ==

@@ Line 77: / Line 77: @@
 Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of product words.
-If equating any pair of step sizes (tempering out their difference, if we involve mappings) or a rank-3 scale leads to 3 WF scales, the rank-3 scale is ''pairwise well-formed (PWF).''
+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).''
-PWF scales are MV3, and even SV3.
+PWF scales are SV3 (and therefore MV3).
 When mappings are considered, PWF scales are rank-3 ''[[Gallery of wakalixes|wakalix]]es'' - Fokker blocks which are Fokker blocks in more than one way.
@@ Line 87: / Line 87: @@
 Only a single PWF scale is mirror-symmetric - '''''abacaba'''''.
-Apart from '''''abacaba''''', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.
+Apart from '''''abacaba''''', PWF scales can be generated by an alternating generator sequence of two generators, modulo the period, i.e., apart from '''''abacaba''''', all PWF scales are GO scales.
 PWF scales can only have odd numbers of notes.
+'''Conjecture''': The GO scales of odd cardinality are the PWF scales that are not '''''abacaba'''''.
 == Pairwise DE/MOS scales ==