Temperament mapping matrix: Difference between revisions
Wikispaces>clumma **Imported revision 535152924 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 630827909 - 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: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2018-07-06 16:18:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>630827909</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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[[math]] | [[math]] | ||
\left[ \begin{array}{rrrrrrl} | |||
\langle & 15 & 24 & 35 & 42 & 52 & |\\ | \langle & 15 & 24 & 35 & 42 & 52 & |\\ | ||
\langle & 22 & 35 & 51 & 62 & 76 & | | \langle & 22 & 35 & 51 & 62 & 76 & | | ||
\end{array} \right | \end{array} \right] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left[ \begin{array}{rrrrrrl} | |||
\langle & 1 & 2 & 3 & 2 & 4 & |\\ | \langle & 1 & 2 & 3 & 2 & 4 & |\\ | ||
\langle & 0 & -3 & -5 & 6 & -4 & | | \langle & 0 & -3 & -5 & 6 & -4 & | | ||
\end{array} \right | \end{array} \right] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left[ \begin{array}{rr} | |||
1 & -1\\ | 1 & -1\\ | ||
0 & 1\\ | 0 & 1\\ | ||
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0 & 0\\ | 0 & 0\\ | ||
0 & 0 | 0 & 0 | ||
\end{array} \right | \end{array} \right] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left[ \begin{array}{rrrrrrl} | |||
| & 1 & 0 & 0 & 0 & 0 & \rangle\\ | | & 1 & 0 & 0 & 0 & 0 & \rangle\\ | ||
| & -1 & 1 & 0 & 0 & 0 & \rangle | | & -1 & 1 & 0 & 0 & 0 & \rangle | ||
\end{array} \right | \end{array} \right] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\left[ \begin{array}{rrrrrrl} | |||
\langle & 7 & 11 & 16 & 20 & 24 & |\\ | \langle & 7 & 11 & 16 & 20 & 24 & |\\ | ||
\langle & 15 & 24 & 35 & 42 & 52 & | | \langle & 15 & 24 & 35 & 42 & 52 & | | ||
\end{array} \right | \end{array} \right] | ||
[[math]] | [[math]] | ||
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<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left[ \begin{array}{rrrrrrl}&lt;br /&gt; | |||
\langle &amp; 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52 &amp; |\\&lt;br /&gt; | \langle &amp; 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52 &amp; |\\&lt;br /&gt; | ||
\langle &amp; 22 &amp; 35 &amp; 51 &amp; 62 &amp; 76 &amp; |&lt;br /&gt; | \langle &amp; 22 &amp; 35 &amp; 51 &amp; 62 &amp; 76 &amp; |&lt;br /&gt; | ||
\end{array} \right | \end{array} \right]&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> | --><script type="math/tex">\left[ \begin{array}{rrrrrrl} | ||
\langle & 15 & 24 & 35 & 42 & 52 & |\\ | \langle & 15 & 24 & 35 & 42 & 52 & |\\ | ||
\langle & 22 & 35 & 51 & 62 & 76 & | | \langle & 22 & 35 & 51 & 62 & 76 & | | ||
\end{array} \right | \end{array} \right]</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
where the row vectors are still placed in bras as a notational device to signify that these are vals. If we put this in <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Normal%20lists#x-Normal%20val%20lists">normal val list</a> form, we get<br /> | where the row vectors are still placed in bras as a notational device to signify that these are vals. If we put this in <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Normal%20lists#x-Normal%20val%20lists">normal val list</a> form, we get<br /> | ||
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<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left[ \begin{array}{rrrrrrl}&lt;br /&gt; | |||
\langle &amp; 1 &amp; 2 &amp; 3 &amp; 2 &amp; 4 &amp; |\\&lt;br /&gt; | \langle &amp; 1 &amp; 2 &amp; 3 &amp; 2 &amp; 4 &amp; |\\&lt;br /&gt; | ||
\langle &amp; 0 &amp; -3 &amp; -5 &amp; 6 &amp; -4 &amp; |&lt;br /&gt; | \langle &amp; 0 &amp; -3 &amp; -5 &amp; 6 &amp; -4 &amp; |&lt;br /&gt; | ||
\end{array} \right | \end{array} \right]&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> | --><script type="math/tex">\left[ \begin{array}{rrrrrrl} | ||
\langle & 1 & 2 & 3 & 2 & 4 & |\\ | \langle & 1 & 2 & 3 & 2 & 4 & |\\ | ||
\langle & 0 & -3 & -5 & 6 & -4 & | | \langle & 0 & -3 & -5 & 6 & -4 & | | ||
\end{array} \right | \end{array} \right]</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
<br /> | <br /> | ||
or, in shorthand, [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]. We'll call this matrix <strong>P</strong>.<br /> | or, in shorthand, [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]. We'll call this matrix <strong>P</strong>.<br /> | ||
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<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left[ \begin{array}{rr}&lt;br /&gt; | |||
1 &amp; -1\\&lt;br /&gt; | 1 &amp; -1\\&lt;br /&gt; | ||
0 &amp; 1\\&lt;br /&gt; | 0 &amp; 1\\&lt;br /&gt; | ||
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0 &amp; 0\\&lt;br /&gt; | 0 &amp; 0\\&lt;br /&gt; | ||
0 &amp; 0&lt;br /&gt; | 0 &amp; 0&lt;br /&gt; | ||
\end{array} \right | \end{array} \right]&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> | --><script type="math/tex">\left[ \begin{array}{rr} | ||
1 & -1\\ | 1 & -1\\ | ||
0 & 1\\ | 0 & 1\\ | ||
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0 & 0\\ | 0 & 0\\ | ||
0 & 0 | 0 & 0 | ||
\end{array} \right | \end{array} \right]</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
<br /> | <br /> | ||
we can also write this matrix as<br /> | we can also write this matrix as<br /> | ||
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<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left[ \begin{array}{rrrrrrl}&lt;br /&gt; | |||
| &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle\\&lt;br /&gt; | | &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle\\&lt;br /&gt; | ||
| &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle&lt;br /&gt; | | &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle&lt;br /&gt; | ||
\end{array} \right | \end{array} \right]&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> | --><script type="math/tex">\left[ \begin{array}{rrrrrrl} | ||
| & 1 & 0 & 0 & 0 & 0 & \rangle\\ | | & 1 & 0 & 0 & 0 & 0 & \rangle\\ | ||
| & -1 & 1 & 0 & 0 & 0 & \rangle | | & -1 & 1 & 0 & 0 & 0 & \rangle | ||
\end{array} \right | \end{array} \right]</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <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 /> | 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 /> | ||
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<!-- ws:start:WikiTextMathRule:4: | <!-- ws:start:WikiTextMathRule:4: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left[ \begin{array}{rrrrrrl}&lt;br /&gt; | |||
\langle &amp; 7 &amp; 11 &amp; 16 &amp; 20 &amp; 24 &amp; |\\&lt;br /&gt; | \langle &amp; 7 &amp; 11 &amp; 16 &amp; 20 &amp; 24 &amp; |\\&lt;br /&gt; | ||
\langle &amp; 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52 &amp; |&lt;br /&gt; | \langle &amp; 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52 &amp; |&lt;br /&gt; | ||
\end{array} \right | \end{array} \right]&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> | --><script type="math/tex">\left[ \begin{array}{rrrrrrl} | ||
\langle & 7 & 11 & 16 & 20 & 24 & |\\ | \langle & 7 & 11 & 16 & 20 & 24 & |\\ | ||
\langle & 15 & 24 & 35 & 42 & 52 & | | \langle & 15 & 24 & 35 & 42 & 52 & | | ||
\end{array} \right | \end{array} \right]</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
<br /> | <br /> | ||
for which the rows are the patent vals for <a class="wiki_link" href="/7-EDO">7-EDO</a> and <a class="wiki_link" href="/15-EDO">15-EDO</a>, respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the &lt;7 1| and &lt;15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix <strong>V∙P</strong> is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|] as a result again.</body></html></pre></div> | for which the rows are the patent vals for <a class="wiki_link" href="/7-EDO">7-EDO</a> and <a class="wiki_link" href="/15-EDO">15-EDO</a>, respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the &lt;7 1| and &lt;15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix <strong>V∙P</strong> is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|] as a result again.</body></html></pre></div> | ||