Rank 3 scale: Difference between revisions
m Added Mike's definition of a rank-3 scale (reworded slightly) and changed subheadings to headings, getting rid of heading 'rank 3 scales' |
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A rank-''n'' scale is a finite set of notes of a rank-''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats. | A rank-''n'' scale a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank-3 group. Alternatively, a rank-3 scale is a finite set of notes of a rank-''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats. | ||
Rank-1 tunings and scales are [[ET]]<nowiki/>s. | Rank-1 tunings and scales are [[ET]]<nowiki/>s. | ||
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''Multi-MOS'' scales are MOS scales that are multiple periods of a WF scale. The interval class represented by any multiple of a period of a WF scale comes in only a single size, hence multi-MOS scales do not possess Myhill’s property. | ''Multi-MOS'' scales are MOS scales that are multiple periods of a WF scale. The interval class represented by any multiple of a period of a WF scale comes in only a single size, hence multi-MOS scales do not possess Myhill’s property. | ||
Rank-3 scales are introduced from here, as generalizations of MOS scales. | |||
==[[User:Inthar/MV3|MV3 scales]]== | |||
Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3. | Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3. | ||
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'''Conjecture:''' The only symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below). | '''Conjecture:''' The only symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are 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. | 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 word|Product words]]== | |||
Two MOS scales can be combined into a rank-3 scale as a ''Product word'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated. | Two MOS scales can be combined into a rank-3 scale as a ''Product word'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated. | ||
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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. | 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 (PWF) scales == | |||
Pairwise well-formed (PWF) scales, another 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. | ||
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PWF scales can only have odd numbers of notes. | PWF scales can only have odd numbers of notes. | ||
== Pairwise DE/MOS scales == | |||
A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise-DE 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 scales. | A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise-DE 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 scales. | ||
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There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales. | There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales. | ||
== 3-SN scales == | |||
The scales a...ba...c and abacaba are [[SN scales]], which are symmetric, and can be uniquely defined by a signature. | The scales a...ba...c and abacaba are [[SN scales]], which are symmetric, and can be uniquely defined by a signature. | ||
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The only SN scales that are MV3, and have mean variety < 3 are a…ba…c | The only SN scales that are MV3, and have mean variety < 3 are a…ba…c | ||
== SN scales, OTC scales, and MOS Cradle Scales == | |||
It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|MOS Cradle Scale]]) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal (OTC), and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly OTC. SN scales not generated from an approximation of the Pythagorean trichord may also be OTC, including the scale (2/1, 3/2, 5/4: 225/224)[19] with step signature 10L+2M+7s (LsLsLMLsLsLsLMLsLsL), mapped to (135/128~21/20, 25/24~28/27, 64/63~50/49), for which 9/8 ~ LsL, and 32/27 ~ sLMLs. All SN scales generated from OTC scales are OTC, and all SN scales generated from strongly OTC scales are strongly OTC. Further, the episturmium morphism that generates SN scales (in which an instance of a new or the existing smallest step is added to the top or bottom of every larger step, see [[SN scales]]) can be applied to generate larger OTC and strongly OTC scales from OTC and strongly OTC scales, or a more general morphism, in which an instance of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size. Even more broadly, any [[MOS Cradle Scales|MOS Cradle Scale]] generated from an OTC scale is OTC, and any MOS Cradle Scale generated from a strongly OTC scale is strongly OTC. | It follows from the proof of [[omnitetrachordality]] that any SN scale (or any [[MOS Cradle Scales|MOS Cradle Scale]]) generated from an approximation of the Pythagorean trichord 4/3 4/3 9/8 is omnitetrachordal (OTC), and any SN scale generated from an approximation of the Pythagorean pentatonic is strongly OTC. SN scales not generated from an approximation of the Pythagorean trichord may also be OTC, including the scale (2/1, 3/2, 5/4: 225/224)[19] with step signature 10L+2M+7s (LsLsLMLsLsLsLMLsLsL), mapped to (135/128~21/20, 25/24~28/27, 64/63~50/49), for which 9/8 ~ LsL, and 32/27 ~ sLMLs. All SN scales generated from OTC scales are OTC, and all SN scales generated from strongly OTC scales are strongly OTC. Further, the episturmium morphism that generates SN scales (in which an instance of a new or the existing smallest step is added to the top or bottom of every larger step, see [[SN scales]]) can be applied to generate larger OTC and strongly OTC scales from OTC and strongly OTC scales, or a more general morphism, in which an instance of a new or the existing smallest step is added to the top or bottom of every instance of any larger step size. Even more broadly, any [[MOS Cradle Scales|MOS Cradle Scale]] generated from an OTC scale is OTC, and any MOS Cradle Scale generated from a strongly OTC scale is strongly OTC. | ||
== Theorem, Proofs and Conjectures on 3-SN scales == | |||
'''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4)/(''N''-1) | '''Theorem:''' Scales of the form a...ba...c have mean variety (3''N''-4)/(''N''-1) | ||