Rank-3 scale: Difference between revisions

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== MV3 and SV3 scales ==

[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset are strict-variety 3 (SV3) scales, which are a generalisation of strict MOS scales into rank-3, where-in every interval class has ''exactly'' three sizes. ~~In academic literature these~~ scales are ~~instead described as~~ ''~~'trivalent'~~''.

[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset are strict-variety 3 (SV3) scales, which are a generalisation of strict MOS scales into rank-3, where-in every interval class has ''exactly'' three sizes.

SV3 scales are sometimes called [[trivalent scale]]s.<ref>Carey, N. (2007). [https://doi.org/10.1080/17459730701376743 ''Coherence and sameness in well-formed and pairwise well-formed scales'']. Journal of Mathematics and Music, 1(2), 79–98.</ref>

'''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 SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]], and SNS wherein two step sizes occur only once.

== References ==

[[Category:Rank-3 scales| ]]

[[Category:Rank 3]]

[[Category:Pages with open problems]]

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 == MV3 and SV3 scales ==
-[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset are strict-variety 3 (SV3) scales, which are a generalisation of strict MOS scales into rank-3, where-in every interval class has ''exactly'' three sizes. In academic literature these scales are instead described as '''trivalent'''.
+[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset are strict-variety 3 (SV3) scales, which are a generalisation of strict MOS scales into rank-3, where-in every interval class has ''exactly'' three sizes.
+SV3 scales are sometimes called [[trivalent scale]]s.<ref>Carey, N. (2007). [https://doi.org/10.1080/17459730701376743 ''Coherence and sameness in well-formed and pairwise well-formed scales'']. Journal of Mathematics and Music, 1(2), 79–98.</ref>
 '''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 SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]], and SNS wherein two step sizes occur only once.
+== References ==
+<references />
 [[Category:Rank-3 scales| ]] <!--main article-->
 [[Category:Rank 3]]
 [[Category:Pages with open problems]]