Fokker block
|
This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. The corresponding expert page for this topic is Mathematical theory of Fokker blocks . |
A Fokker block (or periodicity block) is a periodic scale that can be thought of as a tile on a lattice of pitch classes (of a JI subgroup or a regular temperament) shaped as a parallelogram, parallelepiped or higher-dimensional analog. It comprises those intervals in the lattice which fall inside the tile after moving the tile on the lattice to a place where no lattice point is on its boundary. (Different positions of the tile can create scales which are not rotations of each other.) The scale repeats at the interval of equivalence, which lies on the unison in the lattice of pitch classes.
The concept of the Fokker block was developed by the physicist and music theorist Adriaan Fokker.
Theory
|
Todo: add definition Either the "strong Fokker block" definition needs to be here, or the second and third paragraphs, which are not true for weak Fokker blocks, need to be removed. |
Fokker blocks have a shape which tiles the lattice; an interval between pitches which lie across an edge of two Fokker blocks within the tiling will be altered from its normal value by an interval corresponding to an edge of the parellelepiped. This edge turns out to be the difference between intervals that span the same number of steps in the scale, and so it is called a chroma.
The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-n Fokker block has n - 1 chromas: a consequence of this is that a Fokker block of rank n has maximum variety at most 2(n − 1) (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a mos), and a rank-3 Fokker block has max variety at most 4. In this way, Fokker blocks generalize mos scales.
If the ratios of the cent values of two points on a Fokker block's lattice is always irrational, each scale formed from the block is constant-structure.
Fokker blocks may be used to describe scales within JI subgroups or regular temperaments, or to describe rank-1 regular temperaments – that is, equal temperaments – themselves (by taking the chromas as commas to be tempered out).
Terminology
Arena
A Fokker arena contains all the periodic scales that can be constructed as Fokker blocks from the same list of commas.
Definition
Fokker blocks are defined on octave-equivalent lattices, which can be visualized as grids. These can be constructed by simply dropping the coordinate corresponding to the octave. For example, the 5-limit lattice is three dimensional: each interval with integer coordinates [math]\displaystyle{ (w, x, y) }[/math] corresponds to the frequency ratio [math]\displaystyle{ 2^w \cdot 3^x \cdot 5^y }[/math]. If we assume octave equivalence, this reduces to a grid of points with coordinates [math]\displaystyle{ (x, y) }[/math], each representing a class of equivalent intervals. In our example, the point [math]\displaystyle{ (0, 1) }[/math] now represents the class of just major thirds: 5/4, 5/2, 5/1, etc.
In general, if the original space is n-dimensional, this reduced space has n-1 dimensions. To construct a Fokker block, we take n-1 small intervals in this space, which we will call the chromas (called unison vectors in Fokker's original text). These interval vectors form a sublattice inside the original lattice. In the two-dimensional case, they form a parallellogram which tiles the plane, and in higher dimensions they form a parallelepiped. This tile is called the fundamental domain, and it's volume corresponds to the number of intervals contained inside of it. This number is always an integer, and it is the number of notes in the resulting scale. If we assemble the chromas in a matrix, the volume is given by the absolute value of the determinant of this matrix, as per the definition of the determinant as the signed area of the parallelepiped defined by the vectors of the matrix.
Examples
Ptolemy's intense diatonic
Let's take 5-limit just intation, and pick the just chromatic semitone (25/24) and the syntonic comma (81/80) as our chromas. The octave equivalent lattice is generated by fifths and just major thirds. Since [math]\displaystyle{ 25/24 = 2^{-3} \cdot 3^{-1} \cdot 5^2 }[/math], it has coordinates [math]\displaystyle{ (-1, 2) }[/math] in the octave-equivalent lattice. The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates [math]\displaystyle{ (4, -1) }[/math].
The corresponding Fokker block is Ptolemy's intense diatonic, also known as Zarlino, specifically the lydian mode.
Tempering out either of the two chromas gives a MOS scale related to the temperament.
- Tempering out the syntonic comma gives the diatonic scale LLLsLLs, in meantone.
- Tempering out the chromatic semitone gives the mosh LsLsLss (a 7-note neutral scale), in dicot.
If we temper out both 25/24 and 81/80, we get 7 equal temperament, which we can interpret as an equalized diatonic scale. This scale is a Fokker block in multiple ways: it is also possible to arrive at the same set of notes using 135/128 together with either 81/80 or 25/24 as the chromas.
Duodene and 12 equal temperament
Let's now use the diesis (128/125) and the syntonic comma (81/80) as our chromas, in 5-limit JI as above. The diesis is the difference between an octave and three major thirds, so it has coordinates [math]\displaystyle{ (0, -3) }[/math].
The number of notes in the tile will be 12, since the determinant is:
- [math]\displaystyle{ \begin{vmatrix} 0 & -3 \\ 4 & -1 \end{vmatrix} = (0 \cdot -1) - (-3 \cdot 4) = 12 }[/math]
This should not be surprising, as 12 equal temperament tempers out exactly these two commas, so each note in the resulting scale will correspond to an interval of 12et. The resulting just scale is known as duodene.