Fokker periodicity block

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

These are included as the primary subset of what Erv Wilson refers to as constant structures, where "each interval occurs always subtended by the same number of steps".

[1] The basic idea of Fokker's periodicity blocks is to represent just ratios as points on a lattice, and 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; mathematically, this corresponds to finding the quotient group of the original lattice by the sublattice generated by the commas.

For an n-dimensional lattice, identifying n linearly independent commas reduces the dimension of the lattice to zero, meaning that the number of pitches in the lattice is finite; mathematically, its quotient is a finite abelian group.

This zero-dimensional set of pitches is a periodicity block.

Frequently, it forms a cyclic group, in which case 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.

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.

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

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 harmonic sevenths, 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.

Then membership of primary nodes within periodicity blocks may be tested analytically through the inverse φ function: Let then let the pitch B(x,y) belong to the scale MB iff