Temperament mapping matrix: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 355676710 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 355688262 - 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 | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 10:19:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>355688262</tt>.<br> | ||
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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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11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for [[15-EDO]] and [[22-EDO]]. 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: | 11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for [[15-EDO]] and [[22-EDO]]. 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: | ||
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[<15 24 35 42 52|] | |||
[<22 35 51 62 76|] | |||
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where the row vectors are still placed in bras as a notational device to signify that these are vals. If we put this in [[xenharmonic/Normal lists#x-Normal%20val%20lists|normal val list]] form, we get | where the row vectors are still placed in bras as a notational device to signify that these are vals. If we put this in [[xenharmonic/Normal lists#x-Normal%20val%20lists|normal val list]] form, we get | ||
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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> | ||
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<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:1:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:1 -->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 /> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Dual Transformation"></a><!-- ws:end:WikiTextHeadingRule:3 -->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 /> | ||
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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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:5 -->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 /> | ||
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< | &lt;pre class=&quot;text&quot;&gt;[&amp;lt;15 24 35 42 52|]&lt;br/&gt;[&amp;lt;22 35 51 62 76|]&lt;/pre&gt; | ||
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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 /> | <style type="text/css"><!-- | ||
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</style><pre class="text">[&lt;15 24 35 42 52|] | |||
[&lt;22 35 51 62 76|]</pre> | |||
<!-- ws:end:WikiTextCodeRule:0 -->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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<strong><span style="font-family: 'Courier New',Courier,monospace;">[&lt;1 2 3 2 4|]</span></strong><br /> | <strong><span style="font-family: 'Courier New',Courier,monospace;">[&lt;1 2 3 2 4|]</span></strong><br /> | ||