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Rank-3 scale: Difference between revisions

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→List of named rank-3 scales: These are all ternary scales.
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"SV3" is preferred over "trivalent"
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|-
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|Myhill's property (implies MV2)
|Myhill's property (implies MV2)
|Trivalent scales (subset of MV3 scales)
|SV3 scales (subset of MV3 scales)
|-
|-
|Well-formed (implies Myhill's property and Generated) (only strict MOS scales are well-formed)
|Well-formed (implies Myhill's property and Generated) (only strict MOS scales are well-formed)
|Pairwise well-formed scales (intersection of trivalent scales and Pairwise DE/MOS scales)
|Pairwise well-formed scales (intersection of SV3 scales and Pairwise DE/MOS scales)
|-
|-
|Can tessellate the entire lattice of pitch classes that it lives in
|Can tessellate the entire lattice of pitch classes that it lives in
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== MV3 scales ==
== MV3 scales ==
[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has ''exactly'' three sizes.
[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has ''exactly'' three sizes. SV3 scales are also called '''trivalent'''.


'''Conjecture:''' For all odd-cardinality SV3 scales apart from the scales ''abacaba'', and its repetitions ''abacabaabacaba'' etc., at least two of the three steps must occur the same number of times.
'''Conjecture:''' For all odd-cardinality SV3 scales apart from the scales ''abacaba'', and its repetitions ''abacabaabacaba'' etc., at least two of the three steps must occur the same number of times.
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'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form ''a…ba…c'' (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form ''a…ba…c'' (and their repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
== 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. Trivalent scales can only have odd numbers of notes.


== Product words ==
== Product words ==
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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) 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 trivalent.
PWF scales are MV3, and even SV3.


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.
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.


Not all trivalent scales are PWF. Only a single scale - ''abcba'' - is trivalent and not PWF.
Not all SV3 scales are PWF. Only a single scale - ''abcba'' - is SV3 and not PWF.


Only a single PWF scale is mirror-symmetric - ''abacaba''.
Only a single PWF scale is mirror-symmetric - ''abacaba''.
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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 3-DE scales, or equivalently, MOS scales. We may also call these ''pairwise MOS'' (PMOS) scales.


Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not trivalent; 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 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''.
PWF and pairwise-DE scales include the same number of instances of steps of 2 of the 3 different step sizes, apart from ''abacaba''.


The scale ''abacaba'' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric trivalent scale.
The scale ''abacaba'' is the only mirror-symmetric PWF / PDE / PMOS, and the only mirror-symmetric SV3 scale.


The scales ''a…ba…c'', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.
The scales ''a…ba…c'', and the scale abacaba are the only mirror-symmetric pairwise-DE scales, and the only mirror-symmetric MV3 scales.
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3-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.
3-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.


'''Conjecture''': The only SN scale that is trivalent is ''abacaba''.
'''Conjecture''': The only SN scale that is SV3 is ''abacaba''.


'''Conjecture''': The only SN scales that are MV3 are ''abacaba'', and scales of the form ''a…ba…c''.
'''Conjecture''': The only SN scales that are MV3 are ''abacaba'', and scales of the form ''a…ba…c''.
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