Fokker block: Difference between revisions

Let us set ''e''_''i'' = v_''i'' (2), and also ''P'' = ''e''_''n'' = v_''n'' (2), and choose ''n'' non-negative integers ''a''₁, … , ''a''_''n'' with 0 ≤ ''a''_''k'' < ''P''. Here the choice of ''a''_''n'' doesn't matter and we can take it to be 0. Let ''t''_''i'' = log₂ (''c''_''i''), so that ''e''₁''t''₁ + ''e''₂''t''₂ + … + ''e''_''n''''t''_''n'' = 1. Now define a function on the integers by  

S[''i''] = floor ((''e''₁''i'' + ''a''₁)/''P'')''t''₁ + … + floor((e_''n''''i'' + ''a''_''n'')/''P'')''t''_''n''

Here floor (''x'') is the [[Wikipedia: Floor and ceiling functions|floor function]], the [[Wikipedia: Quasiperiodic function|quasiperiodic function]] returning the largest integer less than or equal to ''x''. When ''i'' = 0, since ''a''_''k'' < P each term is 0 and so S[0] = 0. Since for integer ''j'', floor(''x'' + ''j'') = floor(''x'') + ''j'', we have

S[''i'' + ''P''] = S[''i''] + ''e''₁''t''₁ + ''e''₂''t''₂ + … + e_''n''''t''_''n'' = S[''i''] + 1

Hence S satisfies the conditions for being a [[periodic scale]], and since our unit of measurement is the octave, i.e. we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.

By choosing various ''a''_''k'' satisfying 0 ≤ ''a''_''k'' < ''P'', for any Fokker block we may find the various [[Periodic scale #Rotations|rotations]], of the block. However, other choices lead to blocks which are not rotations of our initial block, but instead belong to [[dome]]s which are disjoint from the dome of the initial block. The collection of all Fokker blocks for any of the allowed values of the ''a''_''k'' offsets is an ''arena''; a Fokker arena is defined entirely by its chromas.

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 map, 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.