Fokker block: Difference between revisions

A '''Fokker block''' (or periodicity block) is a [[periodic scale|periodic]] [[scale]] that can be thought of as a tile on a lattice of [[pitch class]]es (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parallelogram, parallelepiped or higher-dimensional analog. It comprises those intervals in the lattice which fall inside the tile after moving the tile on the lattice to a place where no lattice point is on its boundary. (Different positions of the tile can create scales which are not rotations of each other.) The scale repeats at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes.  

The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]].  

Fokker blocks have a shape which {{w|tessellation|tiles}} the lattice; an interval between pitches which lie across an edge of two Fokker blocks within the tiling will be altered from its normal value by an interval corresponding to an edge of the parellelepiped. This edge turns out to be the difference between intervals that span the same number of steps in the scale, and so it is called a [[chroma]].

The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. In this way, Fokker blocks generalize mos scales.  

If the ratios of the cent values of two points on a Fokker block's lattice is always irrational, each scale formed from the block is [[constant structure|constant-structure]].  

Fokker blocks may be used to describe scales within JI subgroups or regular temperaments, or to describe rank-1 regular temperaments – that is, equal temperaments – themselves (by taking the chromas as commas to be tempered out).  

=== Arena ===

A Fokker arena contains all the periodic scales formable as Fokker blocks from the same list of commas.{{Todo|inline=1| add definition |text=Definitions in the "Terminology" section should not utilize mathematical jargon.}}

{{Todo|inline=1| reduce mathslang |comment=Try explaining without wedgies. }}

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, which we will assume are greater than 1 (i.e. ascending intervals). Call the n-1 commas ''c''₁, ''c''₂, … , ''c''_''n-1.'' There is a (nonunique) uniformizing step ''c''_''n'' which allows us to find ''n'' vals '''v'''₁, '''v'''₂, … , '''v'''_''n'' such that '''v'''_i tempers out all ''c''_''k'' except ''c''_''i'', 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>.

==== Proof of the existence of these vals ====

 

 

Let us set {{nowrap| ''e''_''i'' {{=}} '''v'''_''i''(2) }}, which is the number of steps 2/1 is mapped to under val ''i'', and also {{nowrap| ''P'' {{=}} ''e''_''n'' {{=}} '''v'''_''n''(2) }}; choose ''n'' non-negative integers ''a''₁, … , ''a''_''n'' with {{nowrap| 0 ≤ ''a''_''k'' < ''P'' }}. Here the choice of ''a''_''n'' does not matter and we can take it to be 0. Let {{nowrap| ''t''_''i'' {{=}} log₂(''c''_''i'')}}. Now {{nowrap| ''e''₁''t''₁ + ''e''₂''t''₂ + … + ''e''_''n''''t''_''n'' {{=}} 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''_''k'' &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''_''k'' satisfying {{nowrap|0 ≤ ''a''_''k'' < ''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''_''k'' 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'''_''k'' {{=}} ''P''&#x200A;'''v'''_''k'' − '''v'''_''k''(2) '''v'''_''n''}}. To apply these vals to S[''i''], note first that floor {{nowrap|((''e''_''n''''i'' + ''a''_''n'')/''P'')}} {{nowrap|{{=}} &lfloor;''i'' + ''a''_''n''/''P''&rfloor;}} =&nbsp;''i'', so that {{nowrap|'''v'''_''n''(S[''i'']) {{=}} ''i''}}. Hence {{nowrap|'''u'''_''n''(S[''i'']) {{=}} ''P''&#x200A;'''v'''_''n'' − '''v'''_''n''(2) '''v'''_''n''}} =&nbsp;0, while for {{nowrap|''k'' < ''n''}}, {{nowrap|'''u'''_''k''(S[''i'']) {{=}} ''P''&#x200A;'''v'''_''k''(S[''i'']) − '''v'''_''k''(2) ''i''}}. Since {{nowrap|''x'' − 1 < floor(''x'') ≤ ''x''}}, we have {{nowrap|(''e''_''k''''i'' + ''a''_''k'')/''P'' − 1 < &lfloor;(''e''_''k''''i'' + ''a''_''k'')/''P''&rfloor;}} {{nowrap|≤ (''e''_''k''''i'' + ''a''_''k'')/''P''}}, so that {{nowrap|''e''_''k''''i'' + ''a''_''k'' − ''P'' < ''P'''''v'''_''k'' (''S''[''i''])}} {{nowrap|≤ ''e''_''k''''i'' + ''a''_''k''}}. Since {{nowrap|''e''_''k'' {{=}} '''v'''_''k''(2)}}, this gives us {{nowrap|''a''_''k'' − ''P'' < '''u'''_''k''(''S''[''i'']) ≤ ''a''_''k''}}. This means that for each of the vals '''u'''_''k'', the scale is mapped to a set of ''P'' integers.

 

The val '''u'''_''k'' is a linear combination of '''v'''_''k'' and '''v'''_''n'', which are both vals of the rank-2 temperament defined by the set of chromas minus {''c''_''k''}. Since {{nowrap| '''u'''_''k''(2) {{=}} 0 }}, '''u'''_''k'' 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''_''k'', where ''G''_''k'' is the generator step val and ''m'' is the number of periods to the octave. If we take the wedge product {{nowrap|'''v'''_''n'' ∧ ''G''_''k''}} and reduce it to a [[wedgie]] ''W''_''k'', then the [[interior product]]s {{nowrap|''W''_''k'' ∨ S[''i'']}} for ''i'' from 1 to ''P'' are ''P'' distinct vals '''w'''_''i'', each of which have '''w'''_''i''(2) in a range of ''P'' successive values. The ''W''_''k'' 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''_''k'' ∨ ''S''}} represents a mos of the temperament defined by ''W''_''k''. 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.

 

The {{nowrap|''n'' − 1}} vals '''u'''₁, '''u'''₂, …, '''u'''<sub style="white-space: nowrap;">(''n'' − 1)</sub> defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''_''k'' − ''P'' < '''u'''_''k'' (''q'') }} ≤&nbsp;''a''_''k'', 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'''²) in all ways which retain the same orientation and have the unison inside them, we obtain an arena.

 

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 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.

 

 

 

 

 

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.

 

 

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

 

 

 

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''_''k'' corresponding to each temperament wedgie ''W''_''k'', and from that the dual projection matrix ''Q''_''k''. ''Q''_''k'' has the property that each chroma except ''c''_''k'' is an eigenvector with eigenvalue 1. Hence, the matrix product of the ''Q''_''i'' with {{nowrap|''i'' ≠ ''k''}} has a single eigenvalue of 1, corresponding to ''c''_''k'', which allows us to find ''c''_''k''. 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>

 

 

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.

 

 

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]]

 

 

 

 

 

 

 

 

 

 

 

 

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.

 

 

 

 

 

 

[[Category:Fokker block]]