Fokker block: Difference between revisions
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Consider the periodic scale ''S''[''i''] with quasiperiod {{nowrap|''P'' {{=}} 22}} whose values for ''i'' from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that {{nowrap|'''V''' {{=}} {{val| 22 35 51 62 76 }}}} sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {{{multival| 1 9 -2 -6 12 -6 -13 -30 -45 -10 }}, {{multival| 2 -4 -4 -12 -11 -12 -26 2 -14 -20 }}, {{multival| 6 10 10 8 2 -1 -8 -5 -16 -12 }}, {{multival| 2 -4 -4 10 -11 -12 9 2 37 42 }}}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, which are pajara, {{nowrap|magic {{=}} pajara + hedgehog − suprapyth − pajarous}}, {{nowrap|orwell {{=}} pajara + hedgehog − suprapyth}}, and {{nowrap|porcupine {{=}} suprapyth + pajarous}}; hence, ''S'' is a Fokker block, in the {{dash|pajara, magic, orwell, porcupine|med}} arena. | Consider the periodic scale ''S''[''i''] with quasiperiod {{nowrap|''P'' {{=}} 22}} whose values for ''i'' from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that {{nowrap|'''V''' {{=}} {{val| 22 35 51 62 76 }}}} sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {{{multival| 1 9 -2 -6 12 -6 -13 -30 -45 -10 }}, {{multival| 2 -4 -4 -12 -11 -12 -26 2 -14 -20 }}, {{multival| 6 10 10 8 2 -1 -8 -5 -16 -12 }}, {{multival| 2 -4 -4 10 -11 -12 9 2 37 42 }}}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, which are pajara, {{nowrap|magic {{=}} pajara + hedgehog − suprapyth − pajarous}}, {{nowrap|orwell {{=}} pajara + hedgehog − suprapyth}}, and {{nowrap|porcupine {{=}} suprapyth + pajarous}}; hence, ''S'' is a Fokker block, in the {{dash|pajara, magic, orwell, porcupine|med}} arena. | ||
If {{nowrap|Q(''a'', ''b'', ''c'', ''d'')}} is the {{nowrap|∑(''T''[''i''] − ''μ'')<sup>2</sup>}} quadratic form on {{nowrap|''a'' | If {{nowrap|Q(''a'', ''b'', ''c'', ''d'')}} is the {{nowrap|∑(''T''[''i''] − ''μ'')<sup>2</sup>}} quadratic form on {{nowrap|''a'' · suprapyth + ''b'' · pajara + ''c'' · hedgehog + ''d'' · pajarous}}, then explicitly we have | ||
<math>Q = 2205.5 a^2 + 880 b^2 + 2904 c^2 + 1254 d^2 + 264ab + 2992 ac − 2574ad − 1848bc − 440bd − 880cd</math> | <math>Q = 2205.5 a^2 + 880 b^2 + 2904 c^2 + 1254 d^2 + 264ab + 2992 ac − 2574ad − 1848bc − 440bd − 880cd</math> | ||
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By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank-''r'' Fokker block, meaning one which generates a group of rank ''r'', has {{nowrap|''r'' − 1}} abstract mos scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the {{nowrap|''r'' − 1}} abstract mosses, that means each interval class in the scale has at most 2<sup style="white-space: nowrap;">(''r'' − 1)</sup> possible values; in other words, it has maximum variety less than or equal to 2<sup style="white-space: nowrap;">(''r'' − 1)</sup>. | By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank-''r'' Fokker block, meaning one which generates a group of rank ''r'', has {{nowrap|''r'' − 1}} abstract mos scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the {{nowrap|''r'' − 1}} abstract mosses, that means each interval class in the scale has at most 2<sup style="white-space: nowrap;">(''r'' − 1)</sup> possible values; in other words, it has maximum variety less than or equal to 2<sup style="white-space: nowrap;">(''r'' − 1)</sup>. | ||
The reconstitution can be obtained as follows: for every note of ''S''[''i''] except ''S''[0], S[''i''] will be either the rational number obtained by finding the monzo of the wedge products of the {{nowrap|''r'' − 1}} abstract mos vals for ''i'', taking the dual, and dividing by ''i''<sup style="white-space: nowrap;">(''r'' − 1)</sup>, or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[''i''] by {{nowrap|('''v'''<sub>1</sub> ∧ '''v'''<sub>2</sub> ∧ … ∧ '''v'''<sub>(''r'' − 1)</sub>)°/''i''<sup>(''r'' − 1)</sup>}}. | The reconstitution can be obtained as follows: for every note of ''S''[''i''] except ''S''[0], S[''i''] will be either the rational number obtained by finding the monzo of the wedge products of the {{nowrap|''r'' − 1}} abstract mos vals for ''i'', taking the dual, and dividing by ''i''<sup style="white-space: nowrap;">(''r'' − 1)</sup>, or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[''i''] by {{nowrap|('''v'''<sub>1</sub> ∧ '''v'''<sub>2</sub> ∧ … ∧ '''v'''<sub>(''r'' − 1)</sub>)°/''i'' <sup>(''r'' − 1)</sup>}}. | ||
=== The Fokblock function and modal UDP notation === | === The Fokblock 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'', ''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. | 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'', ''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 (pajara 7 | 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)) · (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''<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. | 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. | ||