Fokker block: Difference between revisions

The '''Fokker block''' is one of the most notable inventions of the physicist and music theorist [[Wikipedia: Adriaan Fokker|Adriaan Fokker]]. Informally, a Fokker block is a parallelogram-shaped tile of scale pitches (in a [[JI subgroup]] or a [[regular temperament]]) that can tessellate the entire lattice of pitch classes that it lives in ("Pitch class" means that the interval of equivalence is ignored). Fokker blocks in rank-''r'' temperaments live in (''r'' - 1)-dimensional pitch-class lattices. Fokker blocks are one way to generalize [[MOS]]es; MOSes are 1-dimensional Fokker blocks.

 

A Fokker block of rank ''r'' has [[maximum variety]] less than or equal to 2^{(''r'' - 1)}. For example, a rank-1 Fokker block has max variety ≤ 2 (hence is a MOS); a rank-2 Fokker block has max variety ≤ 4.  

While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[Harmonic limit|''p''-limit]] situation with ''n'' = π (''p'') primes up to an including ''p''.

Suppose we have ''n'' - 1 commas, which we will assume are greater than 1, and we form an ''n'' by ''n'' matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''₂ ''e''₃ ''e''₅ … ''e''_''p'' }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get ''w''₂''e''₂ + ''w''₃''e''₃ + … + ''w''_''p''''e''_''p'' where the ''w''₂, ''w''₃ … ''w''_''p'' are integers. We interpret this as the [[Vals and Tuning Space|val]] v = {{val| w₂ w₃ … w_''p'' }}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if ''w''₂ < 0 we reverse sign, and we have a val V which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us V = {{val| 22 35 51 62 76 }}, and we will be looking at a 22-note scale in the 11-limit. We may call the val V the epimorph val, and the ''n'' - 1 commas, which form a basis for the kernel of V, the chroma basis.

Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that V (''c'') = 1; that is, if m is the monzo for ''c'', then <V|m> = 1. Precisely which interval with this property we choose doesn't actually matter, so if our chromas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the ''n'' by ''n'' matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the ''n'' - 1 chromas. Because we have chosen ''c'' so that V (''c'') = 1, the determinant of this matrix will be ±1. It is therefore a [[Wikipedia: Unimodular matrix|unimodular matrix]], that is, a square matrix with coefficients which are integers and with determinant ±1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call ''c'' "''c''_''n''", and label the chromas ''c''₁, ''c''₂, … , ''c''_{(''n'' - 1)}; and if we consider the columns of the inverse matrix to be vals and call them v₁, v₂, … , v_''n'', then by the definition of the inverse of a matrix, v_''i'' (c_''j'') = δ (''i'', ''j''), where δ (''i'', ''j'') is the [[Wikipedia: Kronecker delta|Kronecker delta]]. Stated another way, v_i (''c''_j) is 0 unless ''i'' equals ''j'', in which case v_''i'' (''c''_''i'') = 1.

These unimodular matricies define a [[Wikipedia: Change of basis|change of basis]] for the ''p''-limit system of musical intervals: just as every ''p''-limit interval can be written as a product of primes up to ''p'' with integer exponents, every such interval is a product of ''c''₁, ''c''₂, … , ''c''_''n'' with integer exponents. To determine the exponents, we use v₁, v₂, … , v_''n'', so that if ''q'' is a ''p''-limit rational number, we may write it as

''q'' = ''c''₁^v₁(''q'') ''c''₂^v₂(''q'') … ''c''_''n''^{v_''n''(''q'')}