Fokker Periodicity Blocks - Definition of Periodicity Blocks

Definition of Periodicity Blocks

Let an n-dimensional lattice (i.e. integer grid) embedded in n-dimensional space have a numerical value assigned to each of its nodes, such that moving within the lattice in one of the cardinal directions corresponds to a shift in pitch by a particular interval. Typically, n ranges from one to three. 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):

• In the one-dimensional case, the interval corresponding to a single step is generally taken to be a perfect fifth, with ratio 3/2, defining 3-limit just tuning. The lattice points correspond to the integers, with the point at position x being labeled with the pitch value 3x/2y for a number y chosen to make the resulting value lie in the range from 1 to 2. Thus, A(0) = 1, and surrounding it are the values

... 128/81, 32/27, 16/9, 4/3, 1, 3/2, 9/8, 27/16, 81/64, ...

• In the two-dimensional case, corresponding to 5-limit just tuning, the intervals defining the lattice are a perfect fifth and a major third, with ratio 5/4. This gives a square lattice in which the point at position (x,y) being labeled with the value 3x5y2z; again, z is chosen to be the unique integer that makes the resulting value lie in the interval [1,2).

• The three-dimensional case is similar, but adds the harmonic seventh to the set of defining intervals, leading to a cubic lattice in which the point at position (x,y,z) is labeled with a value 3x5y7z2w with w chosen to make this value lie in the interval [1,2).

Once the lattice and its labeling is fixed, one chooses n nodes of the lattice other than the origin whose values are close to either 1 or 2. The vectors from the origin to each one of these special nodes are called unison vectors. These vectors define a sublattice of the original lattice, which has a fundamental domain that in the two-dimensional case is a parallelogram bounded by unison vectors and their shifted copies, and in the three-dimensional case is a parallelepiped. These domains form the tiles in a tessellation of the original lattice.

The tile has an area or volume 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

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.