Fokker block: Difference between revisions

The '''Fokker block''' is one of the most notable inventions of the physicist and music theorist [[Wikipedia: Adriaan Fokker|Adriaan Fokker]]. A Fokker block can be thought of as a parallelogram-shaped tile of scale pitches (in a [[JI subgroup]] or a [[regular temperament]]) that can tessellate the entire lattice of pitch classes that it lives in ("Pitch class" means that the interval of equivalence is ignored). Fokker blocks in [[rank]]-''r'' temperaments live in (''r'' - 1)-dimensional pitch-class lattices. Fokker blocks are one way to generalize [[mos]]ses; mosses are "1-dimensional Fokker blocks".  

A Fokker block of rank ''r'' has [[maximum variety]] at most 2^{(''r'' - 1)}. For example, a rank-2 Fokker block has max variety at most 2 (hence is a mos); a rank-3 Fokker block has max variety at most 4.

Consider the periodic scale S[''i''] with quasiperiod ''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 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, magic = pajara + hedgehog - suprapyth - pajarous, orwell = pajara + hedgehog - suprapyth, and porcupine = suprapyth + pajarous; hence, S is a Fokker block, in the pajara-magic-orwell-porcupine arena.

If Q (''a'', ''b'', ''c'', ''d'') is the ∑(T[''i''] - ''μ'')² quadratic form on ''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>

From this we can find Q (pajara) = 880, Q (magic) = 885.5, Q (orwell) = 885.5, and 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 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.

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 ''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: 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 a = -{{val| 10 16 23 28 34 }} and b = {{val| 12 19 28 34 42 }}, then pajara applied to the steps gives abababaabababababaabab. If c = -{{val| 3 5 7 9 10 }} and d = {{val| 19 30 44 53 66 }}, then magic gives cccdccccccdccccccdcccc. If e = {{val| 9 14 21 25 31 }} and f = -{{val| 13 21 30 37 45 }}, then orwell gives efeefefeefefeefefeefef. Finally, if g = {{val| 7 11 16 20 24 }} and h = -{{val| 15 24 35 42 52 }}, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, 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.

As noted above, pajara, magic, orwell and porcupine correspond to the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 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 w = {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }}, x = {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }}, y = {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }}, and z = {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }}. If we set Orw[''i''] = orwell∨steps[''i''] and Por[''i''] = porcupine∨steps[''i''], then 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.

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.

{{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.

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 ''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 ''r'' - 1 abstract mosses, that means each interval class in the scale has at most 2^{(''r'' - 1)} possible values; in other words, it has maximum variety less than or equal to 2^{(''r'' - 1)}.

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 ''r'' - 1 abstract mos vals for ''i'', taking the dual, and dividing by ''i''^{(''r'' - 1)}, 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 (v₁∧v₂∧…∧v_{(''r'' - 1)})°/''i''<sup>(''r'' - 1)</sub>.

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, C = [c₁, c₂, …, c_{(''n'' - 1)}] together with the offset values A = [''a''₁, ''a''₂, …, a_{(''n'' - 1)}]. Hence we may define a function Fokblock (C, A) from ''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 [w₁, w₂, …, 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 ''n'' - 1 elements of w_''i''. If S = Fokblock (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''/''P''_''k'', where ''a''_''i'' is the smallest value giving S, and the value for ''D'' is V(2)/''P''_''k'' - ''U'' - 1. Hence, the UDP notation for the mos is ''U''|''D''(''P''_''k''), with these values.

Returning to our pajmagorpor22 example, we have that pajmagorpor22 = Fokblock ([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13]). It is also equal to Fokblock ([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 ''P''₁ = 2. Hence the pajara mos mode is 7|3(2) in UDP notation. Finding the others by the fact that for them ''P''_''k'' = 1 and ''a''_''k'' = ''U'', we have that the block, in product word form, is (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''·''U'', and so display S in terms of Fokblock.

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 Fokblock ([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 pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = {{val| 2 0 11 12 7 }}. Note that 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 Fokblock ([64/63, 729/686, 5], [3, 4, 0]).