Fokker block: Difference between revisions

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Returning to our pajmagorpor22 example, we have that {{nowrap|pajmagorpor22 {{=}} fb([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13])}}. It is also equal to {{nowrap|fb([385/384, 176/175, 100/99, 225/224], [15, 9, 4, 13])}}, reflecting the fact that pajara has a period of half on octave, i.e. that {{nowrap|''P''1 {{=}} 2}}. Hence the pajara mos mode is 7|3(2) in UDP notation. Finding the others by the fact that for them {{nowrap|''P''''k'' {{=}} 1}} and {{nowrap|''a''''k'' {{=}} ''U''}}, we have that the block, in product word form, is {{nowrap|(pajara 7{{pipe}}3(2)) · (magic 9{{pipe}}12) · (orwell 4{{pipe}}17) · (porcupine 13{{pipe}}8)}}. We can easily reverse this process, finding the chroma basis from the Fokker group basis, and the offset ''a''''i'' from the corresponding ''U'' and ''P''''i'' as ''P''''i''{{nbhsp}}·''U'', and so display ''S'' in terms of the function.

In terms of the rational intonation of the blocks of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values ''U'' to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the ''U'' and ''D'' found by considering the mos deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar]], a 12-note 11-limit scale which tempers to a mos in six different ways—pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for {{nowrap|fb([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8])}}, the chroma dual to pajaric, which is 245/242, is in the down direction considered as a mos, since {{nowrap|pajaric ∨ 245/242 {{=}} −'''V'''}}, where '''V''' is the epimorph, whereas 3, which can be taken as the generator, is in the up direction since {{nowrap|pajaric ∨ 3 {{=}} {{val| 2 0 11 12 7 }}}}. Note that {{nowrap|pajara ∨ 245/242 {{=}} V}}, so it is up in pajara.

In terms of the rational intonation of the blocks of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values ''U'' to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the ''U'' and ''D'' found by considering the mos deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar]], a 12-note 11-limit scale which tempers to a mos in six different ways—pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for {{nowrap|fb([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8])}}, the chroma dual to pajaric, which is 245/242, is in the down direction considered as a mos, since {{nowrap|pajaric ∨ 245/242 {{=}} −'''V'''}}, where '''V''' is the epimorph, whereas 3, which can be taken as the generator, is in the up direction since {{nowrap|pajaric ∨ 3 {{=}} {{val| 2 0 11 12 7 }}}}. Note that {{nowrap|pajara ∨ 245/242 {{=}} '''V'''}}, so it is up in pajara.

If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance [[nofives]] is {{nowrap|fb([64/63, 729/686, 5], [3, 4, 0])}}.

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 Returning to our pajmagorpor22 example, we have that {{nowrap|pajmagorpor22 {{=}} fb([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13])}}. It is also equal to {{nowrap|fb([385/384, 176/175, 100/99, 225/224], [15, 9, 4, 13])}}, reflecting the fact that pajara has a period of half on octave, i.e. that {{nowrap|''P''<sub>1</sub> {{=}} 2}}. Hence the pajara mos mode is 7|3(2) in UDP notation. Finding the others by the fact that for them {{nowrap|''P''<sub>''k''</sub> {{=}} 1}} and {{nowrap|''a''<sub>''k''</sub> {{=}} ''U''}}, we have that the block, in product word form, is {{nowrap|(pajara 7{{pipe}}3(2))&#x200A;·&#x200A;(magic 9{{pipe}}12)&#x200A;·&#x200A;(orwell 4{{pipe}}17)&#x200A;·&#x200A;(porcupine 13{{pipe}}8)}}. We can easily reverse this process, finding the chroma basis from the Fokker group basis, and the offset ''a''<sub>''i''</sub> from the corresponding ''U'' and ''P''<sub>''i''</sub> as ''P''<sub>''i''</sub>{{nbhsp}}·''U'', and so display ''S'' in terms of the function.
-In terms of the rational intonation of the blocks of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values ''U'' to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the ''U'' and ''D'' found by considering the mos deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar]], a 12-note 11-limit scale which tempers to a mos in six different ways&mdash;pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for {{nowrap|fb([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8])}}, the chroma dual to pajaric, which is 245/242, is in the down direction considered as a mos, since {{nowrap|pajaric ∨ 245/242 {{=}} &minus;'''V'''}}, where '''V''' is the epimorph, whereas 3, which can be taken as the generator, is in the up direction since {{nowrap|pajaric ∨ 3 {{=}} {{val| 2 0 11 12 7 }}}}. Note that {{nowrap|pajara ∨ 245/242 {{=}} V}}, so it is up in pajara.
+In terms of the rational intonation of the blocks of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values ''U'' to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the ''U'' and ''D'' found by considering the mos deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar]], a 12-note 11-limit scale which tempers to a mos in six different ways&mdash;pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for {{nowrap|fb([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8])}}, the chroma dual to pajaric, which is 245/242, is in the down direction considered as a mos, since {{nowrap|pajaric ∨ 245/242 {{=}} &minus;'''V'''}}, where '''V''' is the epimorph, whereas 3, which can be taken as the generator, is in the up direction since {{nowrap|pajaric ∨ 3 {{=}} {{val| 2 0 11 12 7 }}}}. Note that {{nowrap|pajara ∨ 245/242 {{=}} '''V'''}}, so it is up in pajara.
 If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance [[nofives]] is {{nowrap|fb([64/63, 729/686, 5], [3, 4, 0])}}.