Fokker periodicity block

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. Simultaneously 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 3^x/2^y 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) is labeled with the value 3^x5^y2^z. 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 3^x5^y7^z2^w 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 ${\displaystyle \mathbf {u} =(u_{x},u_{y})}$ and ${\displaystyle \mathbf {v} =(v_{x},v_{y})}$ then the area of a 2-D tile is

${\displaystyle \left|{\begin{matrix}u_{x}&u_{y}\\v_{x}&v_{y}\end{matrix}}\right|=u_{x}v_{y}-u_{y}v_{x}.}$

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.

Example 1: Take the 2-dimensional lattice of perfect fifths (ratio 3/2) and just major thirds (ratio 5/4). Choose the commas 128/125 (the diesis, the distance by which three just major thirds fall short of an octave, about 41 cents) and 81/80 (the syntonic comma, the difference between four perfect fifths and a just major third, about 21.5 cents). The result is a block of twelve, showing how twelve-tone equal temperament approximates the ratios of the 5-limit.

Example 2: However, if we were to reject the diesis as a unison vector and instead choose the difference between five major thirds (minus an octave) and a fourth, 3125/3072 (about 30 cents), the result is a block of 19, showing how 19-TET approximates ratios of the 5-limit.

Example 3: In the 3-dimensional lattice of perfect fifths, just major thirds, and just minor sevenths (ratio 7/4), the identification of the syntonic comma, the septimal kleisma (225/224, about 8 cents), and the ratio 1029/1024 (the difference between three septimal whole tones and a perfect fifth, about 8.4 cents) results in a block of 31, showing how 31-TET approximates ratios of the 7-limit.

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

${\displaystyle \phi _{B}(x,y):=(x_{0},y_{0})+(x,y){\begin{pmatrix}u_{x}&u_{y}\\v_{x}&v_{y}\end{pmatrix}}}$

which is really a linear combination:

${\displaystyle \phi _{B}(x,y):=(x_{0},y_{0})+x\mathbf {u} +y\mathbf {v} }$

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:

${\displaystyle \phi _{B}^{-1}(x,y):=\left((x,y)-(x_{0},y_{0})\right){\begin{pmatrix}u_{x}&u_{y}\\v_{x}&v_{y}\end{pmatrix}}^{-1}}$

${\displaystyle ={\left((x,y)-(x_{0},y_{0})\right) \over u_{x}v_{y}-u_{y}v_{x}}{\begin{pmatrix}v_{y}&-u_{y}\\-v_{x}&u_{x}\end{pmatrix}}}$

Let

${\displaystyle \nu _{B}(x,y):=(\lfloor x\rfloor ,\lfloor y\rfloor ),}$

${\displaystyle \mu _{B}(x,y):=\nu _{B}(\phi _{B}^{-1}(x,y)),}$

then let the pitch B(x,y) belong to the scale M_B iff ${\displaystyle \mu _{B}(x,y)=\mu _{B}(0,0),}$ i.e.

${\displaystyle M_{B}=\{B(x,y):\mu _{B}(x,y)=\mu _{B}(0,0)\}.}$

For the one-dimensional case:

${\displaystyle \phi _{A}(x):=x_{0}+Lx}$

where L is the length of the unison vector,

${\displaystyle \phi _{A}^{-1}(x)={x-x_{0} \over L}}$

${\displaystyle \mu _{A}(x):=\left\lfloor {x-x_{0} \over L}\right\rfloor ,}$

${\displaystyle M_{A}=\{A(x):\mu _{A}(x)=\mu _{A}(0)\}.}$

For the three-dimensional case,

${\displaystyle \phi _{C}(x,y,z):=(x_{0},y_{0},z_{0})+(x,y,z){\begin{pmatrix}u_{x}&u_{y}&u_{z}\\v_{x}&v_{y}&v_{z}\\w_{x}&w_{y}&w_{z}\end{pmatrix}}}$

${\displaystyle \phi _{C}^{-1}(x,y,z)={((x,y,z)-(x_{0},y_{0},z_{0})) \over \Delta }{\begin{pmatrix}v_{y}w_{z}-v_{z}w_{y}&u_{z}w_{y}-u_{y}w_{z}&u_{y}v_{z}-u_{z}v_{y}\\v_{z}w_{x}-v_{x}w_{z}&u_{x}w_{z}-u_{z}w_{x}&u_{z}v_{x}-u_{x}v_{z}\\v_{x}w_{y}-v_{y}w_{x}&u_{y}w_{x}-u_{x}w_{y}&u_{x}v_{y}-u_{y}v_{x}\end{pmatrix}}}$

where ${\displaystyle \Delta =u_{x}v_{y}w_{z}+u_{y}v_{z}w_{x}+u_{z}v_{x}w_{y}-u_{x}v_{z}w_{y}-u_{y}v_{x}w_{z}-u_{z}v_{y}w_{x}}$ is the determinant of the matrix of unison vectors.

${\displaystyle \nu _{C}(x,y,z):=(\lfloor x\rfloor ,\lfloor y\rfloor ,\lfloor z\rfloor )}$

${\displaystyle \mu _{C}(x,y,z):=\nu _{C}(\phi _{C}^{-1}(x,y,z))}$

${\displaystyle M_{C}=\{C(x,y,z):\mu _{C}(x,y,z)=\mu _{C}(0,0,0)\}.}$

• Fokker, A. D. (1969), "Unison vectors and periodicity blocks in the three-dimensional (3-5-7-)harmonic lattice of notes", Proc. Koninklijke Nederlandsche Akademie van Wetenschappen, B72 (3).
• Paul Erlich, (1999), A Gentle Introduction to Fokker Periodicity Blocks: Part 1; Part 2; etc.