Fokker periodicity blocks

Fokker periodicity blocks are a concept in tuning theory used to mathematically relate musical intervals in just intonation to those in equal tuning. They are named after Adriaan Daniël Fokker.

The basic idea of Fokker's periodicity blocks is to represent just ratios as points on a lattice, to find vectors in the lattice which represent very small intervals, known as commas. Treating pitches separated by a comma as equivalent "folds" the lattice, effectively reducing its dimension by one. For an "n"-dimensional lattice, identifying "n" commas (as long as they are linearly independent) reduces the dimension of the lattice to zero, meaning that the number of pitches in the lattice is finite. This zero-dimensional lattice is a periodicity block. Identifying the "m" pitches of the periodicity block with "m"-equal tuning gives equal tuning approximations of the just ratios that defined the original lattice.

Note that octaves are usually ignored in constructing periodicity blocks (as they are in scale theory generally) because it is assumed that for any pitch in the tuning system, all pitches differing from it by some number of octaves are also available in principle. In other words, all pitches and intervals can be considered as residues modulo octave. This simplification is commonly known as octave equivalence.

Definition of Periodicity Blocks

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 {3over 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| egin{matrix} u_x & u_y \ v_x & v_y end{matrix} ight| = 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.

Examples

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 1029/1028 (the difference between three septimal whole tones and a fourth, about 1.5 cents) results in a block of 31, showing how 31-TET approximates ratios of the 7-limit.

Mathematical Characteristics of Periodicity Blocks

The periodicity blocks form a secondary, oblique lattice, superimposed on the first one. This lattice may be given by a function φ::$$phi_B(x, y) := (x_0, y_0) + (x, y) egin{pmatrix} u_x & u_y \ v_x & v_y end{pmatrix} which is really a linear combination::$$phi_B(x, y) := (x_0, y_0) + xmathbf{u} + ymathbf{v} where point ("x"₀, "y"₀) 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::$$phi_B^{-1}(x, y) := left( (x,y) - (x_0,y_0) ight) egin{pmatrix} u_x & u_y \ v_x & v_y end{pmatrix}^{-1} ::$$ = { left( (x,y) - (x_0,y_0) ight) over u_x v_y - u_y v_x} egin{pmatrix} v_y & -u_y \ -v_x & u_x end{pmatrix}

Let :$$u_B (x,y) := ( lfloor x floor, lfloor y floor ),

:$$mu_B (x,y) := u_B (phi_B^{-1}(x,y)), then let the pitch "B"("x","y") belong to the scale "M"_"B" iff $$mu_B(x,y) = mu_B(0,0), i.e.:$$M_B = { B(x,y) : mu_B(x,y) = mu_B(0,0) }.

For the one-dimensional case::$$phi_A (x) := x_0 + L x where "L" is the length of the unison vector,:$$phi_A^{-1}(x) = {x - x_0 over L}

:$$mu_A (x) := leftlfloor {x - x_0 over L} ight floor,

:$$M_A = {A(x) : mu_A (x) = mu_A (0)}.

For the three-dimensional case,:$$phi_C (x,y,z) := (x_0, y_0, z_0) + (x, y, z) egin{pmatrix} u_x & u_y & u_z \ v_x & v_y & v_z \ w_x & w_y & w_z end{pmatrix}

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

:$$u_C (x,y,z) := (lfloor x floor, lfloor y floor, lfloor z floor)

:$$mu_C (x,y,z) := u_C (phi_C^{-1}(x,y,z))

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

External links

* [http://www.xs4all.nl/~huygensf/doc/fokkerpb.html Fokker, A. D. "Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Lattice of Notes"]
* [http://sonic-arts.org/td/erlich/intropblock1.htm Erlich, Paul "A Gentle Introduction to Fokker Periodicity Blocks"]