Fokker block: Difference between revisions

The '''Fokker block''' is one of the most notable inventions of the physicist and music theorist [http://en.wikipedia.org/wiki/Adriaan_Fokker Adriaan Fokker]. While the idea generalizes easily to [[Just_intonation_subgroups|just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[Harmonic_Limit|p-limit]] situation with n=pi(p) primes up to an including p.

Suppose we have n-1 commas, which we will assume are greater than 1, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get w2*e2+w3*e3+...+wp*ep where the w2, w3 ... wp are integers. We interpret this as the [[Vals_and_Tuning_Space|val]] v = <w2 w3 ... wp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if w2<0 we reverse sign, and we have a val V which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us V = <22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit. We may call the val V the epimorph val, and the n-1 commas, which form a basis for the kernel of V, the chroma basis.

q = c1^v1(q) * c2^v2(q) ... cn^vn(q)

Let us set ei = vi(2), and also P = en = vn(2), and choose n non-negative integers a1, ...., an with 0 ≤ ak < P. Here the choice of an doesn't matter and we can take it to be 0. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by

By choosing various ak satisfying 0 ≤ ak < P, for any Fokker block we may find the various [[Periodic_scale#Definition-Rotations|rotations]], of the block. However, other choices lead to blocks which are not rotations of our initial block, but instead belong to [[Dome|domes]] which are disjoint from the dome of the initial block. The collection of all Fokker blocks for any of the allowed values of the ak offsets is an ''arena''; a Fokker arena is defined entirely by its chromas.

Let us define a new set of vals by uk = P*vk - vk(2)*vn. To apply these vals to S[i], note first that floor((en*i+an)/P) = floor(i+an/P) = i, so that vn(S[i]) = i. Hence un(S[i]) = P*vn - vn(2)*vn = 0, while for k<n, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 < floor(x) ≤ x, we have (ek*i + ak)/P-1 < floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that ek*i + ak - P < P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P < uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.

The val uk is a linear combination of vk and vn, which are both vals of the rank two temperament defined by the set of chromas minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a [[Normal_lists|normal val list]], or map, for this rank two temperament; in fact it is ±mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product vn∧Gk and reduce it to a [[The_wedgie|wedgie]] Wk, then the [[Interior_product|interior products]] Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk 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|periodic scale]] Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, 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 n-1 ways to n-1 distinct MOS. The arena of the Fokker block is defined equally well by the n-1 wedgies defining the n-1 distinct temperings as by the n-1 chromas introduced previously; these are dual points of view: if we take all but one of the 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 n-1 vals u1, u2, ..., u_(n-1) defined in the previous section gave us n-1 inequalities ak - P < uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q < 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the 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 parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.

The n-1 abstract MOS scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[Product_word|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 n-1 abstract MOS scales lead to Fokker blocks. Given the 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 nxn 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 V(S[i]) = i have a solution. [[Scala|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 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 ±Wk 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 r-1 wedgies each of which gives a [[Graham_complexity|Graham complexity]] to the scale reduced to the octave; that is, to 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 (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to 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 (W∨S[i])(2) in the first period of the octave, (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 P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.

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 pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways.

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|transversal]] for the abstract scale is obtained by truncating the p-limit multivals to the the q-limit which makes them [[Rank_and_codimension|codimension]] one, and then taking [[The_dual|the 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.

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 = <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 {<<1 9 -2 -6 12 -6 -13 -30 -45 -10||, <<2 -4 -4 -12 -11 -12 -26 2 -14 -20||, <<6 10 10 8 2 -1 -8 -5 -16 -12||, <<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 Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. 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.

is the periodic scale with which we began this analysis.

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 MOS, each of which has two kinds of steps, expressed as vals. If a = -<10 16 23 28 34| and b = <12 19 28 34 42|, then pajara applied to the steps gives abababaabababababaabab. If c = -<3 5 7 9 10| and d = <19 30 44 53 66|, then magic gives cccdccccccdccccccdcccc. If e = <9 14 21 25 31| and f = -<13 21 30 37 45|, then orwell gives efeefefeefefeefefeefef. Finally, if g = <7 11 16 20 24| and h = -<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 <<<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 = <<1 -3 5 -1 -7 5 -5 20 8 -20||, x = <<-3 5 -9 1 15 -6 12 -35 -15 34||, y = <<4 2 -1 3 -6 -13 -9 -8 0 12||, and z = <<-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.

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.

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.

Let S be the abstract scale defined by, for scale steps from 1 to 22:

The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val <22 35 51 62|. Using a basis for the Fokker group, for instance the one listed [[Minkowski_reduced_bases_for_Fokker_groups_of_certain_vals|here]], 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-four 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 MOS, 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))º/i^(r-1).

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 an 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 offet values A = [a1, a2, ..., 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 Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokblock(C, A) is a Fokker block, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of [[Modal_UDP_Notation|modal UDP notation]], the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values.

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|nofives]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]).

[[Category:Fokker block]]

[[Category:Math]]

[[Category:Theory]]