- Fokker periodicity blocks
**Fokker periodicity blocks**are a concept intuning theory used to mathematically relatemusical intervals injust 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 findvectors 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 arelinearly 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 inscale 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 residuesmodulo octave. This simplification is commonly known asoctave 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

integer s):*"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 aparallelogram . In 3-D, a tile is aparallelepiped .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:$left|\; egin\{matrix\}\; u\_x\; u\_y\; \backslash \; v\_x\; v\_y\; end\{matrix\}\; ight|\; =\; u\_x\; v\_y\; -\; u\_y\; v\_x.$Each tile is called a

**Fokker periodicity block**. The area of each block is always anatural 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 (thediesis , the distance by which three just major thirds fall short of an octave, about 41 cents) and 81/80 (thesyntonic 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-toneequal 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 how31-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::$phi\_B(x,\; y)\; :=\; (x\_0,\; y\_0)\; +\; (x,\; y)\; egin\{pmatrix\}\; u\_x\; u\_y\; \backslash \; v\_x\; v\_y\; end\{pmatrix\}$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::$phi\_B^\{-1\}(x,\; y)\; :=\; left(\; (x,y)\; -\; (x\_0,y\_0)\; ight)\; egin\{pmatrix\}\; u\_x\; u\_y\; \backslash \; v\_x\; v\_y\; end\{pmatrix\}^\{-1\}$::$=\; \{\; left(\; (x,y)\; -\; (x\_0,y\_0)\; ight)\; over\; u\_x\; v\_y\; -\; u\_y\; v\_x\}\; egin\{pmatrix\}\; v\_y\; -u\_y\; \backslash \; -v\_x\; u\_x\; end\{pmatrix\}$

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,:$phi\_C\; (x,y,z)\; :=\; (x\_0,\; y\_0,\; z\_0)\; +\; (x,\; y,\; z)\; egin\{pmatrix\}\; u\_x\; u\_y\; u\_z\; \backslash \; v\_x\; v\_y\; v\_z\; \backslash \; w\_x\; w\_y\; w\_z\; end\{pmatrix\}$

:$phi\_C^\{-1\}(x,y,z)\; =\; \{((x,y,z)\; -\; (x\_0,y\_0,z\_0))\; over\; Delta\}\; egin\{pmatrix\}\; v\_y\; w\_z\; -\; v\_z\; w\_y\; u\_z\; w\_y\; -\; u\_y\; w\_z\; u\_y\; v\_z\; -\; u\_z\; v\_y\; \backslash \; v\_z\; w\_x\; -\; v\_x\; w\_z\; u\_x\; w\_z\; -\; u\_z\; w\_x\; u\_z\; v\_x\; -\; u\_x\; v\_z\; \backslash \; v\_x\; w\_y\; -\; v\_y\; w\_x\; u\_y\; w\_x\; -\; u\_x\; w\_y\; u\_x\; v\_y\; -\; u\_y\; v\_x\; end\{pmatrix\}$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"*]

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