Fokker block: Difference between revisions

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Let us define a new set of vals by u''k'' = ''P''v''k'' - v''k'' (2) v''n''. To apply these vals to S[''i''], note first that floor ((''e''''n''''i'' + ''a''''n'')/''P'') = floor (''i'' + ''a''''n''/''P'') = ''i'', so that v''n'' (S[''i'']) = ''i''. Hence u''n'' (S[''i'']) = ''P''v''n'' - v''n'' (2) v''n'' = 0, while for ''k'' < ''n'', u''k'' (S[''i'']) = ''P''v''k''(S[''i'']) - v''k'' (2) ''i''. Since ''x'' - 1 < floor(''x'') ≤ ''x'', we have (''e''''k''''i'' + ''a''''k'')/''P'' - 1 < floor ((''e''''k''''i'' + ''a''''k'')/''P'') ≤ (''e''''k''''i'' + ''a''''k'')/''P'', so that ''e''''k''''i'' + ''a''''k'' - ''P'' < ''P''v''k'' (S[''i'']) ≤ ''e''''k''''i'' + ''a''''k''. Since ''e''''k'' = v''k'' (2), this gives us ''a''''k'' - ''P'' < u''k'' (S[''i'']) ≤ ''a''''k''. This means that for each of the vals u''k'', the scale is mapped to a set of ''P'' integers.

The val u''k'' is a linear combination of v''k'' and v''n'', which are both vals of the rank two temperament defined by the set of chromas minus {''c''''k''}. Since u''k'' (2) = 0, u''k'' is a multiple of the generator step val of a [[Normal lists|normal val list]], or mapping, for this rank two temperament; in fact it is ±''m''G''k'', where G''k'' is the generator step val and ''m'' is the number of periods to the octave. If we take the wedge product v''n''∧G''k'' and reduce it to a [[wedgie]] W''k'', then the [[interior product]]s W''k''∨S[''i''] for ''i'' from 1 to ''P'' are ''P'' distinct vals w''i'', each of which have w''i'' (2) in a range of ''P'' successive values. The W''k'' are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of the epimorph V. It follows that the abstract [[periodic scale]] W''k''∨S represents a MOS of the temperament defined by W''k''. The Fokker block can be tempered in ''n'' - 1 distinct rank two temperament ways to ''n'' - 1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank ''n'' JI group it generates it can be tempered in ''n'' - 1 ways to ''n'' - 1 distinct MOS. The arena of the Fokker block is defined equally well by the ''n'' - 1 wedgies defining the ''n'' - 1 distinct temperings as by the ''n'' - 1 chromas introduced previously; these are dual points of view: if we take all but one of the ''n'' - 1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely.

The val u''k'' is a linear combination of v''k'' and v''n'', which are both vals of the rank two temperament defined by the set of chromas minus {''c''''k''}. Since u''k'' (2) = 0, u''k'' is a multiple of the generator step val of a [[Normal lists|normal val list]], or mapping, for this rank two temperament; in fact it is ±''m''G''k'', where G''k'' is the generator step val and ''m'' is the number of periods to the octave. If we take the wedge product v''n''∧G''k'' and reduce it to a [[wedgie]] W''k'', then the [[interior product]]s W''k''∨S[''i''] for ''i'' from 1 to ''P'' are ''P'' distinct vals w''i'', each of which have w''i'' (2) in a range of ''P'' successive values. The W''k'' are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of the epimorph V. It follows that the abstract [[periodic scale]] W''k''∨S represents a MOS of the temperament defined by W''k''. The Fokker block can be tempered in ''n'' - 1 distinct rank two temperament ways to ''n'' - 1 distinct MOS (''not'' ignoring modal rotation), and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank ''n'' JI group it generates it can be tempered in ''n'' - 1 ways to ''n'' - 1 distinct MOS. The arena of the Fokker block is defined equally well by the ''n'' - 1 wedgies defining the ''n'' - 1 distinct temperings as by the ''n'' - 1 chromas introduced previously; these are dual points of view: if we take all but one of the ''n'' - 1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely.

==== Third definition of a Fokker block ====

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 Let us define a new set of vals by u<sub>''k''</sub> = ''P''v<sub>''k''</sub> - v<sub>''k''</sub> (2) v<sub>''n''</sub>. To apply these vals to S[''i''], note first that floor ((''e''<sub>''n''</sub>''i'' + ''a''<sub>''n''</sub>)/''P'') = floor (''i'' + ''a''<sub>''n''</sub>/''P'') = ''i'', so that v<sub>''n''</sub> (S[''i'']) = ''i''. Hence u<sub>''n''</sub> (S[''i'']) = ''P''v<sub>''n''</sub> - v<sub>''n''</sub> (2) v<sub>''n''</sub> = 0, while for ''k'' &lt; ''n'', u<sub>''k''</sub> (S[''i'']) = ''P''v<sub>''k''</sub>(S[''i'']) - v<sub>''k''</sub> (2) ''i''. Since ''x'' - 1 &lt; floor(''x'') ≤ ''x'', we have (''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>)/''P'' - 1 &lt; floor ((''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>)/''P'') ≤ (''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>)/''P'', so that ''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub> - ''P'' &lt; ''P''v<sub>''k''</sub> (S[''i'']) ≤ ''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>. Since ''e''<sub>''k''</sub> = v<sub>''k''</sub> (2), this gives us ''a''<sub>''k''</sub> - ''P'' &lt; u<sub>''k''</sub> (S[''i'']) ≤ ''a''<sub>''k''</sub>. This means that for each of the vals u<sub>''k''</sub>, the scale is mapped to a set of ''P'' integers.
-The val u<sub>''k''</sub> is a linear combination of v<sub>''k''</sub> and v<sub>''n''</sub>, which are both vals of the rank two temperament defined by the set of chromas minus {''c''<sub>''k''</sub>}. Since u<sub>''k''</sub> (2) = 0, u<sub>''k''</sub> is a multiple of the generator step val of a [[Normal lists|normal val list]], or mapping, for this rank two temperament; in fact it is ±''m''G<sub>''k''</sub>, where G<sub>''k''</sub> is the generator step val and ''m'' is the number of periods to the octave. If we take the wedge product v<sub>''n''</sub>∧G<sub>''k''</sub> and reduce it to a [[wedgie]] W<sub>''k''</sub>, then the [[interior product]]s W<sub>''k''</sub>∨S[''i''] for ''i'' from 1 to ''P'' are ''P'' distinct vals w<sub>''i''</sub>, each of which have w<sub>''i''</sub> (2) in a range of ''P'' successive values. The W<sub>''k''</sub> are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of the epimorph V. It follows that the abstract [[periodic scale]] W<sub>''k''</sub>∨S represents a MOS of the temperament defined by W<sub>''k''</sub>. The Fokker block can be tempered in ''n'' - 1 distinct rank two temperament ways to ''n'' - 1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank ''n'' JI group it generates it can be tempered in ''n'' - 1 ways to ''n'' - 1 distinct MOS. The arena of the Fokker block is defined equally well by the ''n'' - 1 wedgies defining the ''n'' - 1 distinct temperings as by the ''n'' - 1 chromas introduced previously; these are dual points of view: if we take all but one of the ''n'' - 1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely.
+The val u<sub>''k''</sub> is a linear combination of v<sub>''k''</sub> and v<sub>''n''</sub>, which are both vals of the rank two temperament defined by the set of chromas minus {''c''<sub>''k''</sub>}. Since u<sub>''k''</sub> (2) = 0, u<sub>''k''</sub> is a multiple of the generator step val of a [[Normal lists|normal val list]], or mapping, for this rank two temperament; in fact it is ±''m''G<sub>''k''</sub>, where G<sub>''k''</sub> is the generator step val and ''m'' is the number of periods to the octave. If we take the wedge product v<sub>''n''</sub>∧G<sub>''k''</sub> and reduce it to a [[wedgie]] W<sub>''k''</sub>, then the [[interior product]]s W<sub>''k''</sub>∨S[''i''] for ''i'' from 1 to ''P'' are ''P'' distinct vals w<sub>''i''</sub>, each of which have w<sub>''i''</sub> (2) in a range of ''P'' successive values. The W<sub>''k''</sub> are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of the epimorph V. It follows that the abstract [[periodic scale]] W<sub>''k''</sub>∨S represents a MOS of the temperament defined by W<sub>''k''</sub>. The Fokker block can be tempered in ''n'' - 1 distinct rank two temperament ways to ''n'' - 1 distinct MOS (''not'' ignoring modal rotation), and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank ''n'' JI group it generates it can be tempered in ''n'' - 1 ways to ''n'' - 1 distinct MOS. The arena of the Fokker block is defined equally well by the ''n'' - 1 wedgies defining the ''n'' - 1 distinct temperings as by the ''n'' - 1 chromas introduced previously; these are dual points of view: if we take all but one of the ''n'' - 1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely.
 ==== Third definition of a Fokker block ====