Fokker periodicity blocks refer to a technique for constructing musical scales. It is named after Adriaan Daniël Fokker. In music, a scale is an unordered collection of notes or pitches, as opposed to a series of intervals, which is a musical mode. ... Adriaan Daniël Fokker (1887–1972) was a Dutch physicist and musician. ...

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. See lattice for other meanings of this term, both within and without mathematics. ... Upright square tiling. ...

Examples of such lattices are the following (x, y, z and w are integers): The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

• One-dimensional: 3-limit

A(0) = 1

• Two-dimensional: 5-limit

• Three-dimensional: 7-limit

Find n nodes on the lattice other than the origin such that their values are sufficiently close to either 1 or 2. Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ...

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. In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... A parallelogram. ... In geometry, a parallelepiped or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

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 and then the area of a 2-D tile is In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is...

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. In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

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

which is really a linear combination: In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

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: In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

Let

then let the pitch B(x,y) belong to the scale M_B iff μ_B(x,y) = μ_B(0,0), i.e. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

M_B = {B(x,y):μ_B(x,y) = μ_B(0,0)}.

For the one-dimensional case:

φ_A(x): = x₀ + Lx

where L is the length of the unison vector,

M_A = {A(x):μ_A(x) = μ_A(0)}.

For the three-dimensional case,

where Δ = u_xv_yw_z + u_yv_zw_x + u_zv_xw_y − u_xv_zw_y − u_yv_xw_z − u_zv_yw_x is the determinant of the matrix of unison vectors.

M_C = {C(x,y,z):μ_C(x,y,z) = μ_C(0,0,0)}.

External links

• A gentle introduction to Fokker periodicity blocks by Paul Erlich.