Fokker block: Difference between revisions

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=== Determining if a scale is a Fokker block ===

The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val ''V'' with indeterminate coefficients, and finding if the linear equations {{nowrap|''V''(''S''[''i'']) {{=}} ''i''}} have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if ''r'' is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to ''V'' is a free abelian group of rank {{nowrap|''r'' − 1}}. We will assume we are working in a full ''p''-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±''W''''k'' for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of {{nowrap|''r'' − 1}} wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to {{nowrap|''S'' {{=}} ~~<nowiki>~~{~~</nowiki>~~''S''[''i'']{{~~pipe~~}} 0 ≤ ''i'' < ''P''~~<nowiki>~~}~~</nowiki>~~}} which is less than ''P'', in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.

The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val ''V'' with indeterminate coefficients, and finding if the linear equations {{nowrap|''V''(''S''[''i'']) {{=}} ''i''}} have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if ''r'' is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to ''V'' is a free abelian group of rank {{nowrap|''r'' − 1}}. We will assume we are working in a full ''p''-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±''W''''k'' for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of {{nowrap|''r'' − 1}} wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to {{nowrap|''S'' {{=}} {{(}}''S''[''i''] {{!}} 0 ≤ ''i'' < ''P''{{)}}}} which is less than ''P'', in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.

Graham complexity for ''S'' with respect to a wedgie ''W'' defines a complexity measure for the wedgies which makes the wedgies which determine if the scale ''S'' is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (''L''2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies ''W'' by setting {{nowrap|''T''[''i''] {{=}} (''W'' ∨ ''S''[''i''])(2)}}, and then taking the sum {{nowrap|∑(''T''[''i''] − ''μ'')2}} for ''i'' from 0 to {{nowrap|''P'' − 1}}, where ''μ'' is the mean (∑''T''[''i''])/''P''. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving ''W'', and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a mos with more than one period to the octave. We can make them exactly the same by modifying things slightly so that ''T''[''i''] is {{nowrap|('''W''' ∨ ''S''[''i''])(2)}} in the first period of the octave, {{nowrap|(''W'' ∨ ''S''[''i''])(2) + 1}} for the second period, and so forth. This makes all mos to result in ''P'' contiguous values, so that the resulting quadratic form returns {{nowrap|''P''(''P''2 − 1)/12}} in all cases when the wedgie results in a mos of ''P'' notes per octave, and more otherwise.

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=== The fb function and modal UDP notation ===

Using the first definition of Fokker block, since the epimorph ''V'' may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding ''a''''n'' plays no role and may be taken as 0, the block is entirely determined by the chroma basis, {{nowrap|''C'' {{=}} [''c''1, ''c''2, …, ''c''(''n'' − 1)]}} together with the offset values {{nowrap|''A'' {{=}} [''a''1, ''a''2, …, ''a''(''n'' − 1)]}}. Hence we may define a function fb(''C'', ''A'') from {{nowrap|''n'' − 1}} element listings of the chroma basis and corresponding offset values to a Fokker block within the arena defined by ''C''. If the list of wedgies ['''w'''1, '''w'''2, …, '''w'''(''n'' − 1)] is the dual Fokker group basis to the chroma basis ''C'', then the period ''P''''i'' of '''w'''''i'' may as usual be found by taking the GCD of the first {{nowrap|''n'' − 1}} elements of '''w'''''i''. If {{nowrap|''S'' {{=}} fb(''C'', ''A'')}} is a Fokker block, the smallest value of ''a''''i'' giving ''S'' is always divisble by ''P''''i'', and fixing the other elements of ''A'' there are ''P''''i'' successive values for ''a''''i'' which all give ''S''. In terms of [[modal UDP notation]], the value of ''U'' for the mos resulting from tempering ''S'' by ''W''i is ''a''''i''{{nbhsp}}/''P''''k'', where ''a''''i'' is the smallest value giving ''S'', and the value for ''D'' is {{nowrap|''V''(2)/''P''''k'' − ''U'' − 1}}. Hence, the UDP notation for the mos is {{nowrap|''U''{{~~pipe~~}}''D''(''P''''k'')}}, with these values.

Using the first definition of Fokker block, since the epimorph ''V'' may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding ''a''''n'' plays no role and may be taken as 0, the block is entirely determined by the chroma basis, {{nowrap|''C'' {{=}} [''c''1, ''c''2, …, ''c''(''n'' − 1)]}} together with the offset values {{nowrap|''A'' {{=}} [''a''1, ''a''2, …, ''a''(''n'' − 1)]}}. Hence we may define a function fb(''C'', ''A'') from {{nowrap|''n'' − 1}} element listings of the chroma basis and corresponding offset values to a Fokker block within the arena defined by ''C''. If the list of wedgies ['''w'''1, '''w'''2, …, '''w'''(''n'' − 1)] is the dual Fokker group basis to the chroma basis ''C'', then the period ''P''''i'' of '''w'''''i'' may as usual be found by taking the GCD of the first {{nowrap|''n'' − 1}} elements of '''w'''''i''. If {{nowrap|''S'' {{=}} fb(''C'', ''A'')}} is a Fokker block, the smallest value of ''a''''i'' giving ''S'' is always divisble by ''P''''i'', and fixing the other elements of ''A'' there are ''P''''i'' successive values for ''a''''i'' which all give ''S''. In terms of [[modal UDP notation]], the value of ''U'' for the mos resulting from tempering ''S'' by ''W''i is ''a''''i''{{nbhsp}}/''P''''k'', where ''a''''i'' is the smallest value giving ''S'', and the value for ''D'' is {{nowrap|''V''(2)/''P''''k'' − ''U'' − 1}}. Hence, the UDP notation for the mos is {{nowrap|''U''{{!}}''D''(''P''''k'')}}, with these values.

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.

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{{!}}3(2)) · (magic 9{{!}}12) · (orwell 4{{!}}17) · (porcupine 13{{!}}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.

@@ Line 43: / Line 43: @@
 === Determining if a scale is a Fokker block ===
-The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val ''V'' with indeterminate coefficients, and finding if the linear equations {{nowrap|''V''(''S''[''i'']) {{=}} ''i''}} have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if ''r'' is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to ''V'' is a free abelian group of rank {{nowrap|''r'' − 1}}. We will assume we are working in a full ''p''-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±''W''<sub>''k''</sub> for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of {{nowrap|''r'' − 1}} wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to {{nowrap|''S'' {{=}} <nowiki>{</nowiki>''S''[''i'']{{pipe}} 0 ≤ ''i'' < ''P''<nowiki>}</nowiki>}} which is less than ''P'', in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.
+The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val ''V'' with indeterminate coefficients, and finding if the linear equations {{nowrap|''V''(''S''[''i'']) {{=}} ''i''}} have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if ''r'' is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to ''V'' is a free abelian group of rank {{nowrap|''r'' − 1}}. We will assume we are working in a full ''p''-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±''W''<sub>''k''</sub> for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of {{nowrap|''r'' − 1}} wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to {{nowrap|''S'' {{=}} {{(}}''S''[''i''] {{!}} 0 ≤ ''i'' < ''P''{{)}}}} which is less than ''P'', in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.
 Graham complexity for ''S'' with respect to a wedgie ''W'' defines a complexity measure for the wedgies which makes the wedgies which determine if the scale ''S'' is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (''L''<sup>2</sup>) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies ''W'' by setting {{nowrap|''T''[''i''] {{=}} (''W'' ∨ ''S''[''i''])(2)}}, and then taking the sum {{nowrap|∑(''T''[''i''] − ''μ'')<sup>2</sup>}} for ''i'' from 0 to {{nowrap|''P'' − 1}}, where ''μ'' is the mean (∑''T''[''i''])/''P''. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving ''W'', and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a mos with more than one period to the octave. We can make them exactly the same by modifying things slightly so that ''T''[''i''] is {{nowrap|('''W''' ∨ ''S''[''i''])(2)}} in the first period of the octave, {{nowrap|(''W'' ∨ ''S''[''i''])(2) + 1}} for the second period, and so forth. This makes all mos to result in ''P'' contiguous values, so that the resulting quadratic form returns {{nowrap|''P''(''P''<sup>2</sup> − 1)/12}} in all cases when the wedgie results in a mos of ''P'' notes per octave, and more otherwise.
@@ Line 118: / Line 118: @@
 === The fb function and modal UDP notation ===
-Using the first definition of Fokker block, since the epimorph ''V'' may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding ''a''<sub>''n''</sub> plays no role and may be taken as 0, the block is entirely determined by the chroma basis, {{nowrap|''C'' {{=}} [''c''<sub>1</sub>, ''c''<sub>2</sub>, …, ''c''<sub>(''n'' − 1)</sub>]}} together with the offset values {{nowrap|''A'' {{=}} [''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>(''n'' − 1)</sub>]}}. Hence we may define a function fb(''C'',&nbsp;''A'') from {{nowrap|''n'' − 1}} element listings of the chroma basis and corresponding offset values to a Fokker block within the arena defined by ''C''. If the list of wedgies ['''w'''<sub>1</sub>, '''w'''<sub>2</sub>, …, '''w'''<sub style="white-space: nowrap;">(''n'' − 1)</sub>] is the dual Fokker group basis to the chroma basis ''C'', then the period ''P''<sub>''i''</sub> of '''w'''<sub>''i''</sub> may as usual be found by taking the GCD of the first {{nowrap|''n'' − 1}} elements of '''w'''<sub>''i''</sub>. If {{nowrap|''S'' {{=}} fb(''C'', ''A'')}} is a Fokker block, the smallest value of ''a''<sub>''i''</sub> giving ''S'' is always divisble by ''P''<sub>''i''</sub>, and fixing the other elements of ''A'' there are ''P''<sub>''i''</sub> successive values for ''a''<sub>''i''</sub> which all give ''S''. In terms of [[modal UDP notation]], the value of ''U'' for the mos resulting from tempering ''S'' by ''W''<sub>i</sub> is ''a''<sub>''i''</sub>{{nbhsp}}/''P''<sub>''k''</sub>, where ''a''<sub>''i''</sub> is the smallest value giving ''S'', and the value for ''D'' is {{nowrap|''V''(2)/''P''<sub>''k''</sub> − ''U'' − 1}}. Hence, the UDP notation for the mos is {{nowrap|''U''{{pipe}}''D''(''P''<sub>''k''</sub>)}}, with these values.
+Using the first definition of Fokker block, since the epimorph ''V'' may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding ''a''<sub>''n''</sub> plays no role and may be taken as 0, the block is entirely determined by the chroma basis, {{nowrap|''C'' {{=}} [''c''<sub>1</sub>, ''c''<sub>2</sub>, …, ''c''<sub>(''n'' − 1)</sub>]}} together with the offset values {{nowrap|''A'' {{=}} [''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>(''n'' − 1)</sub>]}}. Hence we may define a function fb(''C'',&nbsp;''A'') from {{nowrap|''n'' − 1}} element listings of the chroma basis and corresponding offset values to a Fokker block within the arena defined by ''C''. If the list of wedgies ['''w'''<sub>1</sub>, '''w'''<sub>2</sub>, …, '''w'''<sub style="white-space: nowrap;">(''n'' − 1)</sub>] is the dual Fokker group basis to the chroma basis ''C'', then the period ''P''<sub>''i''</sub> of '''w'''<sub>''i''</sub> may as usual be found by taking the GCD of the first {{nowrap|''n'' − 1}} elements of '''w'''<sub>''i''</sub>. If {{nowrap|''S'' {{=}} fb(''C'', ''A'')}} is a Fokker block, the smallest value of ''a''<sub>''i''</sub> giving ''S'' is always divisble by ''P''<sub>''i''</sub>, and fixing the other elements of ''A'' there are ''P''<sub>''i''</sub> successive values for ''a''<sub>''i''</sub> which all give ''S''. In terms of [[modal UDP notation]], the value of ''U'' for the mos resulting from tempering ''S'' by ''W''<sub>i</sub> is ''a''<sub>''i''</sub>{{nbhsp}}/''P''<sub>''k''</sub>, where ''a''<sub>''i''</sub> is the smallest value giving ''S'', and the value for ''D'' is {{nowrap|''V''(2)/''P''<sub>''k''</sub> − ''U'' − 1}}. Hence, the UDP notation for the mos is {{nowrap|''U''{{!}}''D''(''P''<sub>''k''</sub>)}}, with these values.
-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.
+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{{!}}3(2))&#x200A;·&#x200A;(magic 9{{!}}12)&#x200A;·&#x200A;(orwell 4{{!}}17)&#x200A;·&#x200A;(porcupine 13{{!}}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 – 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.