Fokker block: Difference between revisions

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==== Fourth definition of a Fokker block ====

The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a product ~~word~~, and the process can be reversed, so that ~~product words~~ of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.

The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word|step pattern product]] taken. This entails that every Fokker block leads to a step pattern product, and the process can be reversed, so that step pattern products of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.

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

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is the periodic scale with which we began this analysis.

==== ~~Product words~~ and the fourth definition of a Fokker block ====

==== Step pattern products and the fourth definition of a Fokker block ====

Starting from our example 22-note-per-octave scale, we can produce a list of 22 steps: {{nowrap|steps[''i''] {{=}} 33/32}}, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap| '''a''' {{=}} −{{val| 10 16 23 28 34 }} }} and {{nowrap| '''b''' {{=}} {{val| 12 19 28 34 42 }} }}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap| '''c''' {{=}} −{{val| 3 5 7 9 10 }} }} and {{nowrap| '''d''' {{=}} {{val| 19 30 44 53 66 }} }}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap| '''e''' {{=}} {{val| 9 14 21 25 31 }} }} and {{nowrap| '''f''' {{=}} −{{val| 13 21 30 37 45 }} }}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap| '''g''' {{=}} {{val| 7 11 16 20 24 }} }} and {{nowrap| '''h''' {{=}} −{{val| 15 24 35 42 52 }} }}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word]]s, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge ~~product words~~.

Starting from our example 22-note-per-octave scale, we can produce a list of 22 steps: {{nowrap|steps[''i''] {{=}} 33/32}}, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap| '''a''' {{=}} −{{val| 10 16 23 28 34 }} }} and {{nowrap| '''b''' {{=}} {{val| 12 19 28 34 42 }} }}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap| '''c''' {{=}} −{{val| 3 5 7 9 10 }} }} and {{nowrap| '''d''' {{=}} {{val| 19 30 44 53 66 }} }}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap| '''e''' {{=}} {{val| 9 14 21 25 31 }} }} and {{nowrap| '''f''' {{=}} −{{val| 13 21 30 37 45 }} }}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap| '''g''' {{=}} {{val| 7 11 16 20 24 }} }} and {{nowrap| '''h''' {{=}} −{{val| 15 24 35 42 52 }} }}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word|step pattern product]]s, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge step pattern products.

As noted above, pajara, magic, orwell, and porcupine correspond to the commas 385/384, 176/175, 100/99, and 225/224. For example, if we take 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[the dual|dual]] we obtain the wedgie for zeus, which is {{multival|rank=3| 2 -3 1 -1 -1 2 11 3 -10 4 }}. Taking the interior product of this with the steps of our scale gives '''wxwwyzywzywxwwxwyzwyzy''', where {{nowrap| '''w''' {{=}} {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''z''' {{=}} {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }} }}. If we set {{nowrap|Orw[''i''] {{=}} orwell ∨ steps[''i'']}} and {{nowrap|Por[''i''] {{=}} porcupine ∨ steps[''i'']}}, then {{nowrap|Zeus[''i''] {{=}} Orw[''i''] ∧ Por[''i'']}}, which exhibits the scale tempered in zeus as a product ~~word~~ of the orwell mos with the porcupine mos. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.

As noted above, pajara, magic, orwell, and porcupine correspond to the commas 385/384, 176/175, 100/99, and 225/224. For example, if we take 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[the dual|dual]] we obtain the wedgie for zeus, which is {{multival|rank=3| 2 -3 1 -1 -1 2 11 3 -10 4 }}. Taking the interior product of this with the steps of our scale gives '''wxwwyzywzywxwwxwyzwyzy''', where {{nowrap| '''w''' {{=}} {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''z''' {{=}} {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }} }}. If we set {{nowrap|Orw[''i''] {{=}} orwell ∨ steps[''i'']}} and {{nowrap|Por[''i''] {{=}} porcupine ∨ steps[''i'']}}, then {{nowrap|Zeus[''i''] {{=}} Orw[''i''] ∧ Por[''i'']}}, which exhibits the scale tempered in zeus as a step pattern product of the orwell mos with the porcupine mos. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.

==== The tempered scales of a Fokker block ====

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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{{!}}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.

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 step pattern product 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 54: / Line 54: @@
 ==== Fourth definition of a Fokker block ====
-The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.
+The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word|step pattern product]] taken. This entails that every Fokker block leads to a step pattern product, and the process can be reversed, so that step pattern products of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.
 === Determining if a scale is a Fokker block ===
@@ Line 89: / Line 89: @@
 is the periodic scale with which we began this analysis.
-==== Product words and the fourth definition of a Fokker block ====
+==== Step pattern products and the fourth definition of a Fokker block ====
-Starting from our example 22-note-per-octave scale, we can produce a list of 22 steps: {{nowrap|steps[''i''] {{=}} 33/32}}, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap| '''a''' {{=}} −{{val| 10 16 23 28 34 }} }} and {{nowrap| '''b''' {{=}} {{val| 12 19 28 34 42 }} }}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap| '''c''' {{=}} −{{val| 3 5 7 9 10 }} }} and {{nowrap| '''d''' {{=}} {{val| 19 30 44 53 66 }} }}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap| '''e''' {{=}} {{val| 9 14 21 25 31 }} }} and {{nowrap| '''f''' {{=}} −{{val| 13 21 30 37 45 }} }}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap| '''g''' {{=}} {{val| 7 11 16 20 24 }} }} and {{nowrap| '''h''' {{=}} −{{val| 15 24 35 42 52 }} }}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word]]s, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge product words.
+Starting from our example 22-note-per-octave scale, we can produce a list of 22 steps: {{nowrap|steps[''i''] {{=}} 33/32}}, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap| '''a''' {{=}} −{{val| 10 16 23 28 34 }} }} and {{nowrap| '''b''' {{=}} {{val| 12 19 28 34 42 }} }}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap| '''c''' {{=}} −{{val| 3 5 7 9 10 }} }} and {{nowrap| '''d''' {{=}} {{val| 19 30 44 53 66 }} }}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap| '''e''' {{=}} {{val| 9 14 21 25 31 }} }} and {{nowrap| '''f''' {{=}} −{{val| 13 21 30 37 45 }} }}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap| '''g''' {{=}} {{val| 7 11 16 20 24 }} }} and {{nowrap| '''h''' {{=}} −{{val| 15 24 35 42 52 }} }}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word|step pattern product]]s, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge step pattern products.
-As noted above, pajara, magic, orwell, and porcupine correspond to the commas 385/384, 176/175, 100/99, and 225/224. For example, if we take 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[the dual|dual]] we obtain the wedgie for zeus, which is {{multival|rank=3| 2 -3 1 -1 -1 2 11 3 -10 4 }}. Taking the interior product of this with the steps of our scale gives '''wxwwyzywzywxwwxwyzwyzy''', where {{nowrap| '''w''' {{=}} {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''z''' {{=}} {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }} }}. If we set {{nowrap|Orw[''i''] {{=}} orwell ∨ steps[''i'']}} and {{nowrap|Por[''i''] {{=}} porcupine ∨ steps[''i'']}}, then {{nowrap|Zeus[''i''] {{=}} Orw[''i''] ∧ Por[''i'']}}, which exhibits the scale tempered in zeus as a product word of the orwell mos with the porcupine mos. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.
+As noted above, pajara, magic, orwell, and porcupine correspond to the commas 385/384, 176/175, 100/99, and 225/224. For example, if we take 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[the dual|dual]] we obtain the wedgie for zeus, which is {{multival|rank=3| 2 -3 1 -1 -1 2 11 3 -10 4 }}. Taking the interior product of this with the steps of our scale gives '''wxwwyzywzywxwwxwyzwyzy''', where {{nowrap| '''w''' {{=}} {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''z''' {{=}} {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }} }}. If we set {{nowrap|Orw[''i''] {{=}} orwell ∨ steps[''i'']}} and {{nowrap|Por[''i''] {{=}} porcupine ∨ steps[''i'']}}, then {{nowrap|Zeus[''i''] {{=}} Orw[''i''] ∧ Por[''i'']}}, which exhibits the scale tempered in zeus as a step pattern product of the orwell mos with the porcupine mos. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.
 ==== The tempered scales of a Fokker block ====
@@ Line 134: / Line 134: @@
 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{{!}}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.
+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 step pattern product 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.