Fokker block
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- This revision was by author genewardsmith and made on 2010-06-29 06:27:40 UTC.
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Original Wikitext content:
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 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 a torsion problem. 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.
Original HTML content:
<html><head><title>Fokker blocks</title></head><body>The Fokker block 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. 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 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 a torsion problem. 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.</body></html>