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.
Read more about this topic: Fokker Periodicity Blocks
Famous quotes containing the words definition of, definition and/or blocks:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Good blocks of oak it was I split,
As large around as the chopping block;
And every piece I squarely hit
Fell splinterless as a cloven rock.”
—Robert Frost (1874–1963)