Temperament mapping matrix: Difference between revisions
Wikispaces>genewardsmith **Imported revision 355719170 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 355719418 - Original comment: ** |
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<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:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 13: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 13:15:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>355719418</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> | ||
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or, in shorthand, [|1 0 0 0 0>, |-1 1 0 0 0>], where it's understood in both cases that the kets represent columns. | or, in shorthand, [|1 0 0 0 0>, |-1 1 0 0 0>], where it's understood in both cases that the kets represent columns. | ||
The result of **P** | The result of **P**∙**M** is the matrix [|1 0>, |1 -3>], telling us that 2/1 maps to the tmonzo |1 0>, and that 3/2 maps to the tmonzo |1 -3>. This tells us that 2/1 maps to one of the generators for porcupine and that 3/2 is made up of one 2/1 minus three of the other generator. It so happens that the other generator is 10/9, which maps to |0 1>. | ||
We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of **P** by putting these intervals in monzo form as columns of a matrix **N**, which works out to be [|-1 -3 1 0 1>, |6 -2 0 -1 0>, |2 -2 2 0 -1>]. If we then evaluate the product **P∙N** we get the matrix [|0 0>, |0 0>, |0 0>], meaning that all of these intervals map to the unison tmonzo. This confirms that the kernel of porcupine does indeed lie in the null module of **P**. | We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of **P** by putting these intervals in monzo form as columns of a matrix **N**, which works out to be [|-1 -3 1 0 1>, |6 -2 0 -1 0>, |2 -2 2 0 -1>]. If we then evaluate the product **P∙N** we get the matrix [|0 0>, |0 0>, |0 0>], meaning that all of these intervals map to the unison tmonzo. This confirms that the kernel of porcupine does indeed lie in the null module of **P**. | ||
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or, in shorthand, [|1 0 0 0 0&gt;, |-1 1 0 0 0&gt;], where it's understood in both cases that the kets represent columns.<br /> | or, in shorthand, [|1 0 0 0 0&gt;, |-1 1 0 0 0&gt;], where it's understood in both cases that the kets represent columns.<br /> | ||
<br /> | <br /> | ||
The result of <strong>P</strong><strong> | The result of <strong>P</strong>∙<strong>M</strong> is the matrix [|1 0&gt;, |1 -3&gt;], telling us that 2/1 maps to the tmonzo |1 0&gt;, and that 3/2 maps to the tmonzo |1 -3&gt;. This tells us that 2/1 maps to one of the generators for porcupine and that 3/2 is made up of one 2/1 minus three of the other generator. It so happens that the other generator is 10/9, which maps to |0 1&gt;.<br /> | ||
<br /> | <br /> | ||
We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of <strong>P</strong> by putting these intervals in monzo form as columns of a matrix <strong>N</strong>, which works out to be [|-1 -3 1 0 1&gt;, |6 -2 0 -1 0&gt;, |2 -2 2 0 -1&gt;]. If we then evaluate the product <strong>P∙N</strong> we get the matrix [|0 0&gt;, |0 0&gt;, |0 0&gt;], meaning that all of these intervals map to the unison tmonzo. This confirms that the kernel of porcupine does indeed lie in the null module of <strong>P</strong>.<br /> | We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of <strong>P</strong> by putting these intervals in monzo form as columns of a matrix <strong>N</strong>, which works out to be [|-1 -3 1 0 1&gt;, |6 -2 0 -1 0&gt;, |2 -2 2 0 -1&gt;]. If we then evaluate the product <strong>P∙N</strong> we get the matrix [|0 0&gt;, |0 0&gt;, |0 0&gt;], meaning that all of these intervals map to the unison tmonzo. This confirms that the kernel of porcupine does indeed lie in the null module of <strong>P</strong>.<br /> | ||