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{{Expert| Fokker block }}
{{Inacc}}
{{Todo|inline=1| intro | reduce mathslang |comment=Try explaining without wedgies. }}

== Preliminaries ==
While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[harmonic limit|''p''-limit]] situation.

Suppose ''n'' is equal to the number of primes up to and including ''p'', and that we have {{nowrap|(''n'' − 1)}} commas. Call the {{nowrap|(''n'' − 1)}} commas ''c''<sub>1</sub>, ''c''<sub>2</sub>, …, ''c''<sub>(''n'' - 1)</sub>. We can pick some uniformizing step ''c''<sub>''n''</sub> which allows us to find ''n'' vals '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, …, '''v'''<sub>''n''</sub> such that '''v'''<sub>''i''</sub> tempers out all ''c''<sub>''k''</sub> except ''c''<sub>''i''</sub>, which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as

<math>q = c_1^{\vec v_1(q)} c_2^{\vec v_2(q)} \cdots c_n^{\vec v_n(q)}.</math>.

Note that this is only possible if the lattice spanned by the commas has full rank and is [[torsion]]-free.

=== Proof of the existence of these vals ===
Having selected a step, form the ''n'' × ''n'' matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the ''n'' − 1 chromas. Because we have chosen ''c'' so that ''V''(''c'') = 1, each point ''S'' in the lattice can be found by stacking row vectors (stacking ''c'' as many times as ''S'' is mapped to by ''V'' and adjusting by the commas of ''V'') and so the determinant of this matrix will be ±1. Such a matrix is called a unimodular matrix, which is invertible in the integers. The inverse matrix is also unimodular. If we call ''c'' "''c<sub>n</sub>''", and label the chromas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>(''n'' − 1)</sub>; and if we consider the columns of the inverse matrix to be vals and call them '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, … , '''v'''<sub>''n''</sub>, then by the definition of the inverse of a matrix, '''v'''<sub>''i''</sub>(''c<sub>j</sub>'') = δ(''i'', ''j''), where δ(''i'', ''j'') is the Kronecker delta. Stated another way, '''v'''<sub>''i''</sub>(''c<sub>j</sub>'') is 0 unless ''i'' = ''j'', in which case '''v'''<sub>''i''</sub>(''c<sub>i</sub>'') = 1.

These unimodular matrices define a change of basis for the ''p''-limit JI group: just as every ''p''-limit interval can be written as a product of primes up to ''p'' with integer exponents, every such interval is a product of ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c<sub>n</sub>'' with integer exponents.

== Definitions ==
=== First definition of a Fokker block ===
Let us set {{nowrap| ''e''<sub>''i''</sub> {{=}} '''v'''<sub>''i''</sub>(2) }}, which is the number of steps 2/1 is mapped to under val ''i'', and also {{nowrap| ''P'' {{=}} ''e''<sub>''n''</sub> {{=}} '''v'''<sub>''n''</sub>(2) }}; choose ''n'' non-negative integers ''a''<sub>1</sub>, … , ''a''<sub>''n''</sub> with {{nowrap| 0 ≤ ''a''<sub>''k''</sub> < ''P'' }}. Here the choice of ''a''<sub>''n''</sub> does not matter and we can take it to be 0. Let {{nowrap| ''t''<sub>''i''</sub> {{=}} log<sub>2</sub>(''c''<sub>''i''</sub>)}}. Now {{nowrap| ''e''<sub>1</sub>''t''<sub>1</sub> + ''e''<sub>2</sub>''t''<sub>2</sub> + … + ''e''<sub>''n''</sub>''t''<sub>''n''</sub> {{=}} 1 }} as a consequence of the final equation in "Preliminaries." Now define a function on the integers by 

<math>S[i] = \bigg\lfloor \dfrac{e_1 i + a_1}{P} \bigg\rfloor t_1 + \cdots + \bigg\lfloor \dfrac{e_n i + a_n}{P} \bigg\rfloor t_n.</math>

Here &lfloor;''x''&rfloor; is the [[Wikipedia: Floor and ceiling functions|floor function]], which returns the largest integer less than or equal to ''x''. When {{nowrap|''i'' {{=}} 0}}, since {{nowrap|''a''<sub>''k''</sub> &lt; ''P''}} each term is 0 and so {{nowrap|''S''[0] {{=}} 0}}. ''S'' satisfies the conditions for being a [[periodic scale]]; note that the output of ''S'' represents pitch given in units of octaves. This gives us our first definition of a Fokker block.

By choosing various ''a''<sub>''k''</sub> satisfying {{nowrap|0 ≤ ''a''<sub>''k''</sub> < ''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''<sub>''k''</sub> offsets is an ''arena''; a Fokker arena is defined entirely by its chromas.

=== Second definition of a Fokker block ===
Let us define a new set of vals by {{nowrap|'''u'''<sub>''k''</sub> {{=}} ''P''&#x200A;'''v'''<sub>''k''</sub> − '''v'''<sub>''k''</sub>(2) '''v'''<sub>''n''</sub>}}. To apply these vals to S[''i''], note first that floor {{nowrap|((''e''<sub>''n''</sub>''i'' + ''a''<sub>''n''</sub>)/''P'')}} {{nowrap|{{=}} &lfloor;''i'' + ''a''<sub>''n''</sub>/''P''&rfloor;}} =&nbsp;''i'', so that {{nowrap|'''v'''<sub>''n''</sub>(S[''i'']) {{=}} ''i''}}. Hence {{nowrap|'''u'''<sub>''n''</sub>(S[''i'']) {{=}} ''P''&#x200A;'''v'''<sub>''n''</sub> − '''v'''<sub>''n''</sub>(2) '''v'''<sub>''n''</sub>}} =&nbsp;0, while for {{nowrap|''k'' < ''n''}}, {{nowrap|'''u'''<sub>''k''</sub>(S[''i'']) {{=}} ''P''&#x200A;'''v'''<sub>''k''</sub>(S[''i'']) − '''v'''<sub>''k''</sub>(2) ''i''}}. Since {{nowrap|''x'' − 1 < floor(''x'') ≤ ''x''}}, we have {{nowrap|(''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>)/''P'' − 1 < &lfloor;(''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>)/''P''&rfloor;}} {{nowrap|≤ (''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>)/''P''}}, so that {{nowrap|''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub> − ''P'' < ''P'''''v'''<sub>''k''</sub> (''S''[''i''])}} {{nowrap|≤ ''e''<sub>''k''</sub>''i'' + ''a''<sub>''k''</sub>}}. Since {{nowrap|''e''<sub>''k''</sub> {{=}} '''v'''<sub>''k''</sub>(2)}}, this gives us {{nowrap|''a''<sub>''k''</sub> − ''P'' < '''u'''<sub>''k''</sub>(''S''[''i'']) ≤ ''a''<sub>''k''</sub>}}. This means that for each of the vals '''u'''<sub>''k''</sub>, the scale is mapped to a set of ''P'' integers.

The val '''u'''<sub>''k''</sub> is a linear combination of '''v'''<sub>''k''</sub> and '''v'''<sub>''n''</sub>, which are both vals of the rank-2 temperament defined by the set of chromas minus {''c''<sub>''k''</sub>}. Since {{nowrap| '''u'''<sub>''k''</sub>(2) {{=}} 0 }}, '''u'''<sub>''k''</sub> is a multiple of the generator step val of a [[normal lists|normal val list]], or mapping, for this rank-2 temperament; in fact it is ±''mG''<sub>''k''</sub>, where ''G''<sub>''k''</sub> is the generator step val and ''m'' is the number of periods to the octave. If we take the wedge product {{nowrap|'''v'''<sub>''n''</sub> ∧ ''G''<sub>''k''</sub>}} and reduce it to a [[wedgie]] ''W''<sub>''k''</sub>, then the [[interior product]]s {{nowrap|''W''<sub>''k''</sub> ∨ S[''i'']}} for ''i'' from 1 to ''P'' are ''P'' distinct vals '''w'''<sub>''i''</sub>, each of which have '''w'''<sub>''i''</sub>(2) in a range of ''P'' successive values. The ''W''<sub>''k''</sub> 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]] {{nowrap|''W''<sub>''k''</sub> ∨ ''S''}} represents a mos of the temperament defined by ''W''<sub>''k''</sub>. The Fokker block can be tempered in {{nowrap|''n'' − 1}} distinct rank-2 temperament ways to {{nowrap|''n'' − 1}} distinct mos scales (''not'' ignoring modal rotation), 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 {{nowrap|''n'' − 1}} ways to {{nowrap|''n'' − 1}} distinct mos scales. The arena of the Fokker block is defined equally well by the {{nowrap|''n'' − 1}} wedgies defining the {{nowrap|''n'' − 1}} distinct temperings as by the {{nowrap|''n'' − 1}} chromas introduced previously; these are dual points of view: if we take all but one of the {{nowrap|''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.

=== Third definition of a Fokker block ===
The {{nowrap|''n'' − 1}} vals '''u'''<sub>1</sub>, '''u'''<sub>2</sub>, …, '''u'''<sub style="white-space: nowrap;">(''n'' − 1)</sub> defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''<sub>''k''</sub> − ''P'' < '''u'''<sub>''k''</sub> (''q'') }} ≤&nbsp;''a''<sub>''k''</sub>, which apply to any ''q'' in the Fokker block. If we restrict ''q'' to {{nowrap| 1 ≤ ''q'' < 2 }}, and regard it as representing a pitch class, then it is associated to a lattice point in an {{nowrap|(''n'' − 1)}}-dimensional vector space, and in that space the {{nowrap|''n'' − 1}} inequalities define the boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepiped around (in '''R'''<sup>2</sup>) in all ways which retain the same orientation and have the unison inside them, we obtain an arena.

=== 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|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 ==
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.

== Expanding the definition ==
A Fokker block as we have so far defined it is an epimorphic periodic scale ''S'' with period ''P'' repeating at the octave, with values in ''p''-limit rational intonation, such that there exist {{nowrap|π(''p'') − 1 {{=}} ''n'' − 1}} different rank-2 wedgies {''W''<sub>''k''</sub>} such that ''S'' has Graham complexity less than ''P'' for each ''W''<sub>''k''</sub>. If we unpack that definition we can extend it in several distinct ways.

Explicitly, ''S'' is a {{w|quasiperiodic function}} from the integers to the ''p''-limit rational numbers, such that {{nowrap|''S''[0] {{=}} 1}} and {{nowrap|''S''[''i'' + ''P''] {{=}} 2''S''[''i'']}}, for which there is a val ''V'' such that {{nowrap|''V''(S[''i'']) {{=}} ''i''}}. This entails that {{nowrap|''V''(''S''[''P'']) {{=}} ''V''(2) {{=}} ''P''}}, so that {{nowrap|''V'' {{=}} {{val| ''P'' … }}}}, with ''P'' a positive integer; in other words, ''V'' is a ''P''-edo val. For each of the {{nowrap|''n'' − 1}} wedgies ''W''<sub>''k''</sub>, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an [[abstract regular temperament]], by {{nowrap|''T''<sub>''k''</sub>[''i''] {{=}} ''W''<sub>''k''</sub> ∨ ''S''[''i'']}}. The values ''T''<sub>''k''</sub>[''i''] are ''p''-limit vals, and since {{nowrap|''T''<sub>''k''</sub>[''P''] {{=}} ''W''<sub>''k''</sub> ∨ ''S''[''i'']}} {{nowrap|{{=}} ''W''<sub>''k''</sub> ∨ 2}}, {{nowrap|''T''<sub>''k''</sub>[''P''](2) {{=}} 0}}, and so {{nowrap|''T''<sub>''k''</sub>[''i'' + ''P''](2) {{=}} (''T''<sub>''k''</sub>[''i''] + ''T''<sub>''k''</sub>[''P''])(2)}} =&nbsp;''T''<sub>''k''</sub>[''i''](2). Hence ''T''<sub>''k''</sub>[''i''](2) takes on ''P'' or fewer values, with {{nowrap|''a'' ≤ ''T''<sub>''k''</sub>[''i''](2) ≤ ''b''}}. The Graham complexity {{nowrap|''G''(''W''<sub>''k''</sub>)}} of ''S'' with respect to ''W''<sub>''k''</sub> is {{nowrap|''b'' − ''a''}}, and if ''S'' is a Fokker block, for each {{nowrap|''W''<sub>''k''</sub>, ''G''(''W''<sub>''k''</sub>) < ''P''}}.

One way to generalize this is to allow the [[just intonation subgroup|group]] of the scale to be something other than the full ''p''-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number ''E'' which is not a power, so that {{nowrap|''S''[''i'' + ''P''] {{=}} ''ES''[''i'']}} and replacing ''T''<sub>''k''</sub>[''i''](2) with ''T''<sub>''k''</sub>[''i''](''E'').

Still another generalization is to consider regular temperament Fokker blocks. These are abstract temperament periodic scales ''S''[''i''] with values in an abstract regular temperament belonging to some ''r''-wedgie ''Y''. It can be determined if these scales are Fokker by a variant of the method used for JI Fokker blocks. First, a [[transversal]] for the abstract scale is obtained by truncating the ''p''-limit multivals to the the ''q''-limit which makes them [[codimension]]-one, and then taking the [[the dual|dual]] of each multival to obtain a monzo; the monzos then defining a scale in the ordinary way. If the transversal is a Fokker block, the original abstract scale is an abstract Fokker block. This procedure is illustrated below.

== Examples ==
=== Using a Fokker group basis ===
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 pajara–magic–orwell–porcupine arena.

If {{nowrap|Q(''a'', ''b'', ''c'', ''d'')}} is the {{nowrap|∑(''T''[''i''] − ''μ'')<sup>2</sup>}} quadratic form on {{nowrap|''a''&#x200A;·&#x200A;suprapyth + ''b''&#x200A;·&#x200A;pajara + ''c''&#x200A;·&#x200A;hedgehog + ''d''&#x200A;·&#x200A;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>

From this we can find {{nowrap|''Q''(pajara) {{=}} 880}}, {{nowrap|''Q''(magic) {{=}} 885.5}}, {{nowrap|''Q''(orwell) {{=}} 885.5}}, and {{nowrap|''Q''(porcupine) {{=}} 885.5}}, with the Graham complexity of ''S'' being 21 in magic, orwell, and porcupine, and 20 in pajara. If we look at the extrema of ''a'', ''b'', ''c'', and ''d'' separately after setting {{nowrap|''Q'' {{=}} 900}}, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog, and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.

=== Generator range and the first definition of a Fokker block ===
From the values for ''T''[''i''] for each of the four temperaments, we find that the generator range for pajara is −7 to 3, since we obtain the even numbers from −14 to 6. The others are magic from −9 to 12, orwell from −4 to 17, and porcupine from −8 to 13.

We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean tuning #Frobenius tuning and Frobenius projection matrix|Frobenius projection matrix]] ''P''<sub>''k''</sub> corresponding to each temperament wedgie ''W''<sub>''k''</sub>, and from that the dual projection matrix ''Q''<sub>''k''</sub>. ''Q''<sub>''k''</sub> has the property that each chroma except ''c''<sub>''k''</sub> is an eigenvector with eigenvalue 1. Hence, the matrix product of the ''Q''<sub>''i''</sub> with {{nowrap|''i'' ≠ ''k''}} has a single eigenvalue of 1, corresponding to ''c''<sub>''k''</sub>, which allows us to find ''c''<sub>''k''</sub>. From the Fokker group basis [pajara, magic, orwell, porcupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224, and 36/35, inverting and transposing, we obtain [ {{val| 12 19 28 34 42 }}, −{{val| 3 5 7 9 10 }}, {{val| 9 14 21 25 31 }}, −{{val| 7 11 16 20 24 }}, {{val| 22 35 51 62 76 }} ]. From this and the previously obtained generator ranges, we find that

<math>S[i] = (36/35)^i (385/384)^{\lfloor (12i + 14)/22 \rfloor}  (175/176)^{\lfloor (-3i + 9)/22 \rfloor}  (100/99)^{\lfloor (9i + 4)/22 \rfloor}  (224/225)^{\lfloor (-7i + 13)/22 \rfloor}</math>

is the periodic scale with which we began this analysis.

=== 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|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 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 ===
A Fokker block is not just a scale, but a little scale universe of tempered versions of that scale which identify various steps of the scale, as depicted below.

[[File:1000px-Pajmagorpor22_temperament_support_lattice.svg.png|750px]]

One has first the [[pajmagorpor22|original JI scale]]. Then there are codimension one temperings of the scale, in each of the commas associated to the Fokker block; in our example these are [[pajmagorpor22 225|225/224]], [[pajmagorpor22 100|100/99]], [[pajmagorpor22 176|176/175]], and [[pajmagorpor22 385|385/384]]. The next level gives [[pajmagorpor22apollo|apollo]], [[pajmagorpor22minerva|minerva]], [[pajmagorpor22marvel|marvel]], [[pajmagorpor22ares|ares]], [[pajmagorpor22supermagic|supermagic]], and [[pajmagorpor22zeus|zeus]]. Next come pajara, magic, orwell and porcupine, with the range of generators already given, and then finally 22 equal. Exploring the changes wrought by the various scales in such a Fokker universe, not to mention all of the modes and domes, would certainly give the interested composer plenty to work with.

=== Example of an abstract Fokker block ===
Let ''S'' be the abstract scale defined by, for scale steps from 1 to 22:

[
{{Multival|rank=3| 1 -2 0 1 1 -2 9 5 -10 -4 }}, {{multival|rank=3| 0 1 1 -1 -1 2 -4 -4 4 4 }}, {{multival|rank=3| 1 -1 1 0 0 0 5 1 -6 0 }},

{{Multival|rank=3| 0 0 0 2 2 -4 3 3 -6 -8 }}, {{multival|rank=3| 0 -2 -2 1 1 -2 6 6 -4 -4 }}, {{multival|rank=3| 0 1 1 1 1 -2 -1 -1 -2 -4 }},

{{Multival|rank=3| 0 -1 -1 0 0 0 2 2 0 0 }}, {{multival|rank=3| 1 -3 -1 1 1 -2 11 7 -10 -4 }}, {{multival|rank=3| 1 0 2 1 1 -2 4 0 -8 -4 }},

{{Multival|rank=3| 1 -2 0 0 0 0 7 3 -6 0 }}, {{multival|rank=3| 0 -1 -1 2 2 -4 5 5 -6 -8 }}, {{multival|rank=3| 1 -1 1 -1 -1 2 3 -1 -2 4 }},

{{Multival|rank=3| 0 0 0 1 1 -2 1 1 -2 -4 }}, {{multival|rank=3| 1 -2 0 2 2 -4 10 6 -12 -8 }}, {{multival|rank=3| 0 1 1 0 0 0 -3 -3 2 0 }},

{{Multival|rank=3| 1 -1 1 1 1 -2 6 2 -8 -4 }}, {{multival|rank=3| -1 0 -2 1 1 -2 -2 2 4 -4 }}, {{multival|rank=3| 1 0 2 0 0 0 2 -2 -4 0 }},

{{Multival|rank=3| 2 -2 2 1 1 -2 11 3 -14 -4 }}, {{multival|rank=3| 0 -1 -1 1 1 -2 3 3 -2 -4 }}, {{multival|rank=3| 2 -1 3 0 0 0 7 -1 -10 0 }},

{{Multival|rank=3| 0 0 0 0 0 0 -1 -1 2 0 }}
]

This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank-4 temperament. Working with it directly is more difficult than dealing with the [[transversal]] we may obtain by [[wedgies and multivals #Truncation of wedgies|truncation]]. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is [[the dual|dual]] to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way.

The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val {{val| 22 35 51 62 }}. Using a basis for the Fokker group, for instance the one listed in [[Minkowski reduced bases for Fokker groups of certain vals]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a mos, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-4 temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a mos.

=== Scale properties of Fokker blocks ===
By definition, a Fokker block is weakly epimorphic, which implies it is a weak constant structure (constant structure with negative steps allowed). 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''&#x200A;<sup>(''r'' − 1)</sup>}}.

== 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''{{!}}''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 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.

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])}}.

[[Category:Fokker block]]
[[Category:Math]]