Fokker periodicity blocks

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Fokker periodicity blocks are devices used in a technique for constructing musical scales. They are named after Adriaan Daniël Fokker.

Note: this article uses mathematical symbols.

Let an n-dimensional lattice (i.e. grid) embedded in n-space have a numerical value assigned to each of its nodes. Let n be preferably equal either to 1, 2, or 3. In the two-dimensional case, the lattice is a square lattice. In the 3-D case, the lattice is cubic.

Examples of such lattices are the following (x, y, z and w are integers):

One-dimensional: 3-limit

A (0) = 1

$\forall x : 1 \le A(x) < 2$

$\forall x : \exists! z : A(x + 1) = 2^z \cdot {3\over 2} \cdot A(x)$

Two-dimensional: 5-limit

$\forall x : B(x, 0) = A(x)$

$\forall x : \forall y : 1 \le B(x, y) < 2$

$\forall x : \forall y : \exists! z : B(x, y + 1) = 2^z \cdot {5 \over 4} \cdot B(x, y)$

Three-dimensional: 7-limit

$\forall x : \forall y : C(x, y, 0) = B(x, y)$

$\forall x : \forall y : \forall z : 1 \le C(x, y, z) < 2$

$\forall x : \forall y : \forall z : \exists! w : C(x, y, z + 1) = 2^w \cdot {7 \over 4} \cdot C(x, y, z)$

Find n nodes on the lattice other than the origin such that their values are sufficiently close to either 1 or 2.

Vectors from the origin to each one of these special nodes are called unison vectors. A quantity n of unison vectors are enough to define an n-dimensional tiling pattern. Let the n unison vectors define the sides of a tile. In 1-D, a tile is a line segment. In 2-D, a tile is a parallelogram. In 3-D, a tile is a parallelepiped.

Each tile has an area given by the absolute value of the determinant of the matrix of unison vectors: i.e. in the 2-D case if the unison vectors are u and v, such that $\mathbf{u} = (u_x, u_y)$ and $\mathbf{v} = (v_x, v_y)$ then the area of a 2-D tile is

$\left| \begin{matrix} u_x & u_y \\ v_x & v_y \end{matrix} \right| = u_x v_y - u_y v_x.$

Each tile is called a Fokker periodicity block. The area of each block is always a natural number equal to the number of nodes falling within each block.

Choose the block containing the origin. Compile a list of the values of all the nodes contained by this block. Arrange the values in increasing numerical order. The result is a musical scale.

The periodicity blocks form a secondary, oblique lattice, superimposed on the first one. This lattice may be given by a function φ:

$\phi_B(x, y) := (x_0, y_0) + (x, y) \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}$

which is really a linear combination:

$\phi_B(x, y) := (x_0, y_0) + x\mathbf{u} + y\mathbf{v}$

where point (x₀, y₀) can be any point, preferably not a node of the primary lattice, and preferably so that points φ(0,1), φ(1,0) and φ(1,1) are not any nodes either.

Then membership of primary nodes within periodicity blocks may be tested analytically through the inverse φ function:

$\phi_B^{-1}(x, y) := \left( (x,y) - (x_0,y_0)\right) \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}^{-1}$

$= { \left( (x,y) - (x_0,y_0) \right) \over u_x v_y - u_y v_x} \begin{pmatrix} v_y & -u_y \\ -v_x & u_x \end{pmatrix}$

Let

$\nu_B (x,y) := ( \lfloor x\rfloor, \lfloor y\rfloor ),$

$\mu_B (x,y) := \nu_B (\phi_B^{-1}(x,y)),$

then let the pitch B(x,y) belong to the scale M_B iff $μ B (x, y) = μ B (0,0),$ i.e.

M B = {B (x, y):μ B (x, y) = μ B (0,0)}.

For the one-dimensional case:

φ A (x): = x 0 + L x

where L is the length of the unison vector,

$\phi_A^{-1}(x) = {x - x_0 \over L}$

$\mu_A (x) := \left\lfloor {x - x_0 \over L} \right\rfloor,$

M A = {A (x):μ A (x) = μ A (0)}.

For the three-dimensional case,

$\phi_C (x,y,z) := (x_0, y_0, z_0) + (x, y, z) \begin{pmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{pmatrix}$

$\phi_C^{-1}(x,y,z) = {((x,y,z) - (x_0,y_0,z_0)) \over \Delta} \begin{pmatrix} v_y w_z - v_z w_y & u_z w_y - u_y w_z & u_y v_z - u_z v_y \\ v_z w_x - v_x w_z & u_x w_z - u_z w_x & u_z v_x - u_x v_z \\ v_x w_y - v_y w_x & u_y w_x - u_x w_y & u_x v_y - u_y v_x \end{pmatrix}$