Temperament mapping matrix: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 355689440 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 355691458 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 10: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 10:36:14 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>355691458</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]] | ||
\begin{ | \[ \left[ \begin{array}{rrrrrrl} | ||
\langle 15 & 24 & 35 & 42 & 52|\\ | \langle & 15 & 24 & 35 & 42 & 52 & |\\ | ||
\langle 22 & 35 & 51 & 62 & 76| | \langle & 22 & 35 & 51 & 62 & 76 & | | ||
\end{ | \end{array} \right] \] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\begin{ | \[ \left[ \begin{array}{rrrrrrl} | ||
\langle 1 & 2 & 3 & 2 & 4|\\ | \langle & 1 & 2 & 3 & 2 & 4 & |\\ | ||
\langle 0 & -3 & -5 & 6 & -4| | \langle & 0 & -3 & -5 & 6 & -4 & | | ||
\end{ | \end{array} \right] \] | ||
[[math]] | [[math]] | ||
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We'll now right-multiply **P** by the following matrix **M** of two monzos, representing 2/1 and 3/2: | We'll now right-multiply **P** by the following matrix **M** of two monzos, representing 2/1 and 3/2: | ||
[[math] | [[math]] | ||
\begin{ | \[ \left[ \begin{array}{rr} | ||
1 & -1\\ | 1 & -1\\ | ||
0 & 1\\ | 0 & 1\\ | ||
| Line 51: | Line 51: | ||
0 & 0\\ | 0 & 0\\ | ||
0 & 0 | 0 & 0 | ||
\end{ | \end{array} \right] \] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\begin{ | \[ \left[ \begin{array}{rrrrrrl} | ||
|1 & 0 & 0 & 0 & 0\rangle\\ | | & 1 & 0 & 0 & 0 & 0 & \rangle\\ | ||
|-1 & 1 & 0 & 0 & 0\rangle | | & -1 & 1 & 0 & 0 & 0 & \rangle | ||
\end{ | \end{array} \right] \] | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\begin{ | \[ \left[ \begin{array}{rrrrrrl} | ||
\langle 7 & 11 & 16 & 20 & 24|\\ | \langle & 7 & 11 & 16 & 20 & 24 & |\\ | ||
\langle 15 & 24 & 35 & 42 & 52| | \langle & 15 & 24 & 35 & 42 & 52 & | | ||
\end{ | \end{array} \right] \] | ||
[[math]] | [[math]] | ||
for which the rows are the patent vals for [[7-EDO]] and [[15-EDO]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the <7 1| and <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 **V*P** is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [<1 2 3 2 4|, <0 -3 -5 6 -4|] as a result again.</pre></div> | for which the rows are the patent vals for [[7-EDO]] and [[15-EDO]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the <7 1| and <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 **V*P** is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [<1 2 3 2 4|, <0 -3 -5 6 -4|] as a result again.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Temperament Mapping Matrices (M-maps)</title></head><body><!-- ws:start:WikiTextHeadingRule: | <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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Temperament Mapping Matrices (M-maps)</title></head><body><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:5 -->Basics</h1> | ||
The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, which is a group homomorphism <strong>T</strong>: J -&gt; K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism between Z-modules. This homomorphism can also be represented by a integer matrix, called a <strong>mapping matrix</strong> or <strong>mapping</strong> for the temperament. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &quot;monzo-map&quot; or <strong>M-map</strong> when context demands, as opposed to the <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> which is a mapping on vals.<br /> | The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, which is a group homomorphism <strong>T</strong>: J -&gt; K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism between Z-modules. This homomorphism can also be represented by a integer matrix, called a <strong>mapping matrix</strong> or <strong>mapping</strong> for the temperament. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &quot;monzo-map&quot; or <strong>M-map</strong> when context demands, as opposed to the <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> which is a mapping on vals.<br /> | ||
<br /> | <br /> | ||
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Note also that since all mapping matrices for T will have the same row module, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same <a class="wiki_link" href="/Normal%20lists#x-Normal%20val%20lists">normal val list</a>, or more generally if they have the same Hermite normal form.<br /> | Note also that since all mapping matrices for T will have the same row module, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same <a class="wiki_link" href="/Normal%20lists#x-Normal%20val%20lists">normal val list</a>, or more generally if they have the same Hermite normal form.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc1"><a name="Dual Transformation"></a><!-- ws:end:WikiTextHeadingRule:7 -->Dual Transformation</h1> | ||
Any mapping matrix can be said to represent a linear map <strong>M:</strong> J -&gt; K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation <strong>M*:</strong> K* -&gt; J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals">tvals</a> on K, so <strong>M</strong>* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.<br /> | Any mapping matrix can be said to represent a linear map <strong>M:</strong> J -&gt; K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation <strong>M*:</strong> K* -&gt; J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals">tvals</a> on K, so <strong>M</strong>* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.<br /> | ||
<br /> | <br /> | ||
These two transformations correspond to different types of matrix multiplication: the ordinary transformation <strong>M</strong> corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation <strong>M</strong>* corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.<br /> | These two transformations correspond to different types of matrix multiplication: the ordinary transformation <strong>M</strong> corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation <strong>M</strong>* corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc2"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:9 -->Example</h1> | ||
11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for <a class="wiki_link" href="/15-EDO">15-EDO</a> and <a class="wiki_link" href="/22-EDO">22-EDO</a>. Since these two vals form a saturated submodule of the module of 11-limit vals, then the matrix formed by setting the rows to these vals is a valid mapping matrix for porcupine:<br /> | 11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for <a class="wiki_link" href="/15-EDO">15-EDO</a> and <a class="wiki_link" href="/22-EDO">22-EDO</a>. Since these two vals form a saturated submodule of the module of 11-limit vals, then the matrix formed by setting the rows to these vals is a valid mapping matrix for porcupine:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\begin{ | \[ \left[ \begin{array}{rrrrrrl}&lt;br /&gt; | ||
\langle 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52|\\&lt;br /&gt; | \langle &amp; 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52 &amp; |\\&lt;br /&gt; | ||
\langle 22 &amp; 35 &amp; 51 &amp; 62 &amp; 76|&lt;br /&gt; | \langle &amp; 22 &amp; 35 &amp; 51 &amp; 62 &amp; 76 &amp; |&lt;br /&gt; | ||
\end{ | \end{array} \right] \]&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\begin{ | --><script type="math/tex">\[ \left[ \begin{array}{rrrrrrl} | ||
\langle 15 & 24 & 35 & 42 & 52|\\ | \langle & 15 & 24 & 35 & 42 & 52 & |\\ | ||
\langle 22 & 35 & 51 & 62 & 76| | \langle & 22 & 35 & 51 & 62 & 76 & | | ||
\end{ | \end{array} \right] \]</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; | ||
\begin{ | \[ \left[ \begin{array}{rrrrrrl}&lt;br /&gt; | ||
\langle 1 &amp; 2 &amp; 3 &amp; 2 &amp; 4|\\&lt;br /&gt; | \langle &amp; 1 &amp; 2 &amp; 3 &amp; 2 &amp; 4 &amp; |\\&lt;br /&gt; | ||
\langle 0 &amp; -3 &amp; -5 &amp; 6 &amp; -4|&lt;br /&gt; | \langle &amp; 0 &amp; -3 &amp; -5 &amp; 6 &amp; -4 &amp; |&lt;br /&gt; | ||
\end{ | \end{array} \right] \]&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\begin{ | --><script type="math/tex">\[ \left[ \begin{array}{rrrrrrl} | ||
\langle 1 & 2 & 3 & 2 & 4|\\ | \langle & 1 & 2 & 3 & 2 & 4 & |\\ | ||
\langle 0 & -3 & -5 & 6 & -4| | \langle & 0 & -3 & -5 & 6 & -4 & | | ||
\end{ | \end{array} \right] \]</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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We'll now right-multiply <strong>P</strong> by the following matrix <strong>M</strong> of two monzos, representing 2/1 and 3/2:<br /> | We'll now right-multiply <strong>P</strong> by the following matrix <strong>M</strong> of two monzos, representing 2/1 and 3/2:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextMathRule:2: | |||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\[ \left[ \begin{array}{rr}&lt;br /&gt; | |||
&lt;br/&gt; | 1 &amp; -1\\&lt;br /&gt; | ||
0 &amp; 1\\&lt;br /&gt; | |||
& | 0 &amp; 0\\&lt;br /&gt; | ||
0 &amp; 0\\&lt;br /&gt; | |||
0 &amp; 0&lt;br /&gt; | |||
\end{array} \right] \]&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\[ \left[ \begin{array}{rr} | |||
1 & -1\\ | |||
0 & 1\\ | |||
0 & 0\\ | |||
0 & 0\\ | |||
0 & 0 | |||
\end{array} \right] \]</script><!-- ws:end:WikiTextMathRule:2 --><br /> | |||
<br /> | |||
we can also write this matrix as<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextMathRule:3: | |||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\[ \left[ \begin{array}{rrrrrrl}&lt;br /&gt; | |||
& | | &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle\\&lt;br /&gt; | ||
| &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle&lt;br /&gt; | |||
&lt;br /&gt; | \end{array} \right] \]&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\[ \left[ \begin{array}{rrrrrrl} | |||
&lt;br /&gt; | | & 1 & 0 & 0 & 0 & 0 & \rangle\\ | ||
& | | & -1 & 1 & 0 & 0 & 0 & \rangle | ||
\end{array} \right] \]</script><!-- ws:end:WikiTextMathRule:3 --><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 /> | ||
<br /> | |||
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 /> | |||
The result of | <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 | <br /> | ||
<br /> | |||
<strong>The Dual Transformation</strong><br /> | |||
To explore the dual transformation implied by <strong>P</strong>, we'll look at the tval matrix [&lt;7 1|, &lt;15 2|], where again the bras signify that the tvals represent matrix rows. The first tval maps the first generator (for which 2/1 is a <a class="wiki_link" href="/Transversal%20generators">transversal</a>) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval. If we call this matrix V, then the result of V*P is the matrix<br /> | |||
To explore the dual transformation implied by | <br /> | ||
</ | <!-- ws:start:WikiTextMathRule:4: | ||
[[math]]&lt;br/&gt; | |||
\[ \left[ \begin{array}{rrrrrrl}&lt;br /&gt; | |||
\langle &amp; 7 &amp; 11 &amp; 16 &amp; 20 &amp; 24 &amp; |\\&lt;br /&gt; | |||
\langle &amp; 15 &amp; 24 &amp; 35 &amp; 42 &amp; 52 &amp; |&lt;br /&gt; | |||
\end{array} \right] \]&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\[ \left[ \begin{array}{rrrrrrl} | |||
\langle & 7 & 11 & 16 & 20 & 24 & |\\ | |||
\langle & 15 & 24 & 35 & 42 & 52 & | | |||
\end{array} \right] \]</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> | ||