Rank-3 scale: Difference between revisions
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|Can tessellate the entire lattice of pitch classes that it lives in | |Can tessellate the entire lattice of pitch classes that it lives in | ||
| | |MOS step pattern products = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales) | ||
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|[[Recursive structure of MOS scales|Recursive structure]], Uniquely defined by step signature and mapping (implies mirror-symmetric) | |[[Recursive structure of MOS scales|Recursive structure]], Uniquely defined by step signature and mapping (implies mirror-symmetric) | ||
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== MV3 and SV3 scales == | == 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. | [[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. | '''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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== Scale pattern product == | == Scale pattern product == | ||
Two MOS scales can be combined into a rank-3 scale as a ''[[product word| | Two MOS scales can be combined into a rank-3 scale as a ''[[product word|step pattern product]]'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated. | ||
When associated with a mapping, | When associated with a mapping, MOS step pattern products are the rank-3 ''[[Fokker blocks]]''. Fokker blocks have ''unison vectors'', which generalize the concept of the chroma of MOS scales to higher ranks. If these intervals are plotted onto a plane representing rank-3 octave equivalent pitch space, they tile the space into Fokker blocks which differ by combinations of these unison vectors. Rank-2 Fokker blocks are the MOS scales, so Fokker blocks can be considered a generalization of MOS scales into higher ranks. | ||
MOS pattern products have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''. | |||
Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS. | Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS. | ||
== Pairwise well-formed scales== | == Pairwise well-formed scales== | ||
Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of | 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).'' | 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).'' | ||
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== Pairwise DE/MOS scales == | == Pairwise DE/MOS scales == | ||
A similar generalization, a larger subset of | 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. | 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. | ||
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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. | '''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 scales| ]] <!--main article--> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
[[Category:Pages with open problems]] | [[Category:Pages with open problems]] | ||