Fokker block: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-30 03:05~~:12~~ UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-30 03:29:05 UTC</tt>.<br>

: The original revision id was <tt>~~151065243~~</tt>.<br>

: The original revision id was <tt>151066255</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>

Line 10:

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.

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 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 [[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)

Line 21:

Line 27:

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 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; ~~<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> 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>~~

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 />

~~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></pre></div>

<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 <br />

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)</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-30 03:05:12 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-30 03:29:05 UTC</tt>.<br>
-: The original revision id was <tt>151065243</tt>.<br>
+: The original revision id was <tt>151066255</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>
@@ Line 10: / Line 10: @@
 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 [[Vals and Tuning Space|val]] 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.
-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 &lt;v|m&gt; = 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.
+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 &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 [[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)
@@ Line 21: / Line 27: @@
 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&amp;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 &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; v = &amp;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&amp;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 = &amp;lt;12 19 28 34|, and we will be looking at a 12-note scale in the 7-limit.&lt;br /&gt;
 &lt;br /&gt;
-Now choose a &amp;quot;chroma&amp;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 &amp;lt;v|m&amp;gt;  &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;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)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow"&gt;unimodular matrix&lt;/a&gt;, 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) &lt;/h1&gt;
+Now choose a &amp;quot;chroma&amp;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 &amp;lt;v|m&amp;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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow"&gt;unimodular matrix&lt;/a&gt;, 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow"&gt;Kronecker delta&lt;/a&gt;. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.&lt;br /&gt;
- delta_ij, where delta_ij is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow"&gt;Kronecker delta&lt;/a&gt;. Stated anoth way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;br /&gt;
+These unimodular matricies define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow"&gt;change of basis&lt;/a&gt; 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 &lt;br /&gt;
+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&lt;br /&gt;
+&lt;br /&gt;
+q = c1^v1(q) * c2^v2(q) ... cn^vn(q)&lt;/body&gt;&lt;/html&gt;</pre></div>