Fokker block

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2012-02-28 17:21:01 UTC.

The original revision id was 306112932.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

[[toc|flat]]

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.

=Preliminaries=
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 an example step 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 with this property we choose doesn't actually matter, so if our 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 step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the 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(i,j), where delta(i,j) is the [[http://en.wikipedia.org/wiki/Kronecker_delta|Kronecker delta]]. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.

These unimodular matricies define a [[http://en.wikipedia.org/wiki/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 c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as

q = c1^v1(q) * c2^v2(q) ... cn^vn(q)

=First definition of a Fokker block=
Let us set ei = vi(2), and also P = e1 = v1(2), and choose n positive integers a1, ...., an with ak < P. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by

S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn

Here floor(x) is the [[http://en.wikipedia.org/wiki/Floor_and_ceiling_functions|floor function]], the [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] returning the largest integer less than or equal to x. When i=0, since ak < P each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have

S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1

Hence S satisfies the conditions for being a [[Periodic scale|periodic scale]], and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.

Original HTML content:

<html><head><title>Fokker blocks</title></head><body><!-- ws:start:WikiTextTocRule:4:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#First definition of a Fokker block">First definition of a Fokker block</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: -->
<!-- ws:end:WikiTextTocRule:7 --><br />
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 />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:0 -->Preliminaries</h1>
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&gt;, 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 = &lt;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&lt;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 = &lt;12 19 28 34|, and we will be looking at a 12-note scale in the 7-limit.<br />
<br />
Now choose an example step 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 &lt;v|m&gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our 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 step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the 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) = delta(i,j), where delta(i,j) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.<br />
<br />
These unimodular matricies define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow">change of basis</a> 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 c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as<br />
<br />
q = c1^v1(q) * c2^v2(q) ... cn^vn(q)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="First definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:2 -->First definition of a Fokker block</h1>
Let us set ei = vi(2), and also P = e1 = v1(2), and choose n positive integers a1, ...., an with ak &lt; P. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by<br />
<br />
S[i] = floor((e1*i + a1)/P)*t1 + ... + floor((en*i + an)/P)*tn<br />
<br />
Here floor(x) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">floor function</a>, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> returning the largest integer less than or equal to x. When i=0, since ak &lt; P each term is 0 and so S[0] = 0. Since for integer j, floor(x+j) = floor(x) + j, we have<br />
<br />
S[i + P] = S[i] + e1*t1 + e2*t2 + ... + en*tn = S[i] + 1<br />
<br />
Hence S satisfies the conditions for being a <a class="wiki_link" href="/Periodic%20scale">periodic scale</a>, and since our unit of measurement is the octave, ie we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block.</body></html>