Fokker block
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The **Fokker block** is one of the most notable inventions of the physicist and music theorist [[http://en.wikipedia.org/wiki/Adriaan_Fokker|Adriaan Fokker]]. While the idea generalizes easily to [[just intonation subgroups]], for ease of exposition we will suppose that we are in a [[Harmonic Limit|p-limit]] situation with n=pi(p) primes up to an including p. Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the [[Vals and Tuning Space|val]] v = <v2, v3, ... vp|. 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 v2<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 81/80, 128/125 and 64/63, the above procedure gives us the val v = <12 19 28 34|, and we will be looking at a 12-note scale in the 7-limit. Now choose a "chroma" for the Fokker block, which is 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 we choose doesn't actually matter, so if out commas are 81/80, 128/125 and 64/63, we could for instance choose 28/27, 25/24, 21/20, 16/15 or 15/14. Having selected a chroma, form the n by n matrix whose first row is the monzo for the chroma c, and whose other rows are monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be +-1. It is therefore a [[http://en.wikipedia.org/wiki/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 c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta_ij, where delta_ij is the [[http://en.wikipedia.org/wiki/Kronecker_delta|Kronecker delta]]. Stated anoth way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.
Original HTML content:
<html><head><title>Fokker blocks</title></head><body>The <strong>Fokker block</strong> is one of the most notable inventions of the physicist and music theorist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a>. While the idea generalizes easily to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>, for ease of exposition we will suppose that we are in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> situation with n=pi(p) primes up to an including p. <br /> <br /> Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> v = <v2, v3, ... vp|. 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 v2<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 81/80, 128/125 and 64/63, the above procedure gives us the val v = <12 19 28 34|, and we will be looking at a 12-note scale in the 7-limit.<br /> <br /> Now choose a "chroma" for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then <v|m> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x1. Precisely which interval we choose doesn't actually matter, so if out commas are 81/80, 128/125 and 64/63, we could for instance choose 28/27, 25/24, 21/20, 16/15 or 15/14. Having selected a chroma, form the n by n matrix whose first row is the monzo for the chroma c, and whose other rows are monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be +-1. It is therefore a 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 c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj)"></a><!-- ws:end:WikiTextHeadingRule:0 --> 1. Precisely which interval we choose doesn't actually matter, so if out commas are 81/80, 128/125 and 64/63, we could for instance choose 28/27, 25/24, 21/20, 16/15 or 15/14. Having selected a chroma, form the n by n matrix whose first row is the monzo for the chroma c, and whose other rows are monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be +-1. It is therefore a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>, 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 c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) </h1> delta_ij, where delta_ij is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated anoth way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.</body></html>