Rank 3 scale: Difference between revisions

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== Rank-3 scales ==

=== MV3 scales ===

Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.

For all MV3 scales apart from the scale abacaba, at least two of the three steps must occur the same number of times, and may be generated by an alternating generator sequence.

=== Trivalent scales ===

WF scales may be generalized into rank-3 via a generalization of Myhill's property into rank-3. We call a scale where each generic interval comes in 3 sizes ''trivalent''. Trivalent scales are clearly a subset of MV3 scales.

=== Product words ===

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Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.

'''Conjecture:''' Product words cannot be symmetric.

=== Pairwise well-formed (PWF) scales ===

Pairwise well-formed (PWF) scales, a 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 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).''

PWF scales not only have MV3, but are also ''trivalent~~'', where each generic interval comes in 3 sizes~~.

PWF scales not only have MV3, but are also trivalent.

When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|wakalixes]].

Not all ~~MV3, or~~ trivalent scales are PWF. Only ~~the~~ scale aabcb is trivalent and not PWF.

Not all trivalent scales are PWF. Only a single scale - aabcb - is trivalent and not PWF.

PWF scales can only have odd numbers of notes.

PWF ~~scales can be generated by an alternating generator sequence, modulo a period, apart from~~ abacaba.

Only a single PWF scale is symmetric - abacaba.

=== Pairwise DE/MOS scales ===

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 == Rank-3 scales ==
+=== MV3 scales ===
+Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.
+For all MV3 scales apart from the scale abacaba, at least two of the three steps must occur the same number of times, and may be generated by an alternating generator sequence.
+=== Trivalent scales ===
+WF scales may be generalized into rank-3 via a generalization of Myhill's property into rank-3. We call a scale where each generic interval comes in 3 sizes ''trivalent''. Trivalent scales are clearly a subset of MV3 scales.
 === Product words ===
@@ Line 26: / Line 34: @@
 Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.
+'''Conjecture:''' Product words cannot be symmetric.
 === Pairwise well-formed (PWF) scales ===
-Pairwise well-formed (PWF) scales, a 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 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).''
-PWF scales not only have MV3, but are also ''trivalent'', where each generic interval comes in 3 sizes.
+PWF scales not only have MV3, but are also trivalent.
 When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|wakalixes]].
-Not all MV3, or trivalent scales are PWF. Only the scale aabcb is trivalent and not PWF.
+Not all trivalent scales are PWF. Only a single scale - aabcb - is trivalent and not PWF.
 PWF scales can only have odd numbers of notes.
-PWF scales can be generated by an alternating generator sequence, modulo a period, apart from abacaba.
+Only a single PWF scale is symmetric - abacaba.
 === Pairwise DE/MOS scales ===