Fokker block: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-25 23:18:40 UTC</tt>.<br>

: The original revision id was <tt>314501072</tt>.<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 [[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 225/224, 100/99, 176/175 and 385/384, the above procedure gives us the val v = &lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-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 &lt;v|m&gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. 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. By inverting the comma ci when vi(2) is negative, we can normalize the vals so that vi(2) is always positive, which we will assume.

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 225/224, 100/99, 176/175 and 385/384, the above procedure gives us the val v = &amp;lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit.&lt;br /&gt;

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 &amp;lt;v|m&amp;gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. 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 &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. By inverting the comma ci when vi(2) is negative, we can normalize the vals so that vi(2) is always positive, which we will assume.&lt;br /&gt;