Fokker block: Difference between revisions
Wikispaces>guest **Imported revision 151747149 - Original comment: ** |
Wikispaces>guest **Imported revision 151747285 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:guest|guest]] and made on <tt>2010-07-06 20:17: | : This revision was by author [[User:guest|guest]] and made on <tt>2010-07-06 20:17:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>151747285</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 8: | Line 8: | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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=== | ===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. | 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. | ||
| Line 18: | Line 17: | ||
q = c1^v1(q) * c2^v2(q) ... cn^vn(q) | q = c1^v1(q) * c2^v2(q) ... cn^vn(q) | ||
===First definition of a Fokker block=== | ===First definition of a Fokker block=== | ||
There exist positive real numbers t1, ... tn, positive integers e1, ..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N, | There exist positive real numbers t1, ... tn, positive integers e1, ..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N, | ||
e1*t1 + ... + en*tn = O, and ak < 1 so that | e1*t1 + ... + en*tn = O, and ak < 1 so that | ||
| Line 27: | Line 26: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:0 -->Preliminaries</h3> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:0 -->Preliminaries</h3> | ||
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 /> | |||
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 /> | <br /> | ||
Now choose a &quot;chroma&quot; 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 chroma, form the n by n matrix whose first row is the monzo for the chroma 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 /> | Now choose a &quot;chroma&quot; 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 chroma, form the n by n matrix whose first row is the monzo for the chroma 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 /> | ||
| Line 37: | Line 35: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--First definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:2 -->First definition of a Fokker block</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--First definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:2 -->First definition of a Fokker block</h3> | ||
There exist positive real numbers t1, ... tn, positive integers e1, ..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N,<br /> | There exist positive real numbers t1, ... tn, positive integers e1, ..., en, and positive real numbers a1, ...., an with e1+e2+...+en = N,<br /> | ||
e1*t1 + ... + en*tn = O, and ak &lt; 1 so that<br /> | e1*t1 + ... + en*tn = O, and ak &lt; 1 so that<br /> | ||
<br /> | <br /> | ||
S[i] = floor((e1/N)*i + a1)*t1+...+floor((en/N)*i + an)*tn</body></html></pre></div> | S[i] = floor((e1/N)*i + a1)*t1+...+floor((en/N)*i + an)*tn</body></html></pre></div> | ||