Fokker block: Difference between revisions

Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that V (''c'') = 1; that is, if m is the monzo for ''c'', then ⟨V|m⟩ = 1. Precisely which interval with this property we choose doesn't actually matter, so if our chromas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the ''n'' by ''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, the determinant of this matrix will be ±1. It is therefore a [[Wikipedia: Unimodular matrix|unimodular matrix]], that is, a square matrix with coefficients which are integers and with determinant ±1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call ''c'' "''c''_''n''", and label the chromas ''c''₁, ''c''₂, … , ''c''_{(''n'' - 1)}; and if we consider the columns of the inverse matrix to be vals and call them v₁, v₂, … , v_''n'', then by the definition of the inverse of a matrix, v_''i'' (c_''j'') = δ (''i'', ''j''), where δ (''i'', ''j'') is the [[Wikipedia: Kronecker delta|Kronecker delta]]. Stated another way, v_i (''c''_j) is 0 unless ''i'' equals ''j'', in which case v_''i'' (''c''_''i'') = 1.

<math>S[i+P] = S[i] + e_1 t_1 + e_2 t_2 + … + e_n t_n = S[i] + 1.</math>

The n-1 vals u1, u2, ..., u_(n-1) defined in the previous section gave us n-1 inequalities ak - P &lt; uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &lt; 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 &lt; 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.

Explicitly, S is a [http://en.wikipedia.org/wiki/Quasiperiodic_function quasiperiodic function] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = {{val|P ... }}, with P a positive integer; in other words, V is a P-edo val. For each of the n-1 wedgies Wk, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an [[Abstract_regular_temperament|abstract regular temperament]], by Tk[i] = Wk∨S[i]. The values Tk[i] are p-limit vals, and since Tk[P] = Wk∨S[i] = Wk∨2, Tk[P](2) = 0, and so Tk[i + P](2) = (Tk[i] + Tk[P])(2) = Tk[i](2). Hence Tk[i](2) takes on P or fewer values, with a ≤ Tk[i](2) ≤ b. The Graham complexity G(Wk) of S with respect to Wk is b-a, and if S is a Fokker block, for each Wk, G(Wk) &lt; P.

One way to generalize this is to allow the [[Just_intonation_subgroups|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 S[i + P] = E S[i] and replacing Tk[i](2) with Tk[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|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.