Temperament mapping matrix: Difference between revisions
Wikispaces>guest **Imported revision 355780458 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509653838 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:47:39 UTC</tt>.<br> | ||
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<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">=Basics= | <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;white-space: pre-wrap ! important" class="old-revision-html"> | ||
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=Basics= | |||
The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an [[abstract regular temperament]], which is a group homomorphism **T**: J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a **temperament mapping matrix**; when context is clear enough it's also sometimes just called a **mapping matrix** for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or **M-map** when context demands, as opposed to the [[Subgroup Mapping Matrices (V-maps)|V-map]] which is a mapping on vals. | The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an [[abstract regular temperament]], which is a group homomorphism **T**: J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a **temperament mapping matrix**; when context is clear enough it's also sometimes just called a **mapping matrix** for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or **M-map** when context demands, as opposed to the [[Subgroup Mapping Matrices (V-maps)|V-map]] which is a mapping on vals. | ||
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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> | ||
<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:5:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:5 -->Basics</h1> | <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><br /> | ||
<!-- ws:start:WikiTextLocalImageRule:11:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div><!-- ws:end:WikiTextLocalImageRule:11 --><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:5 -->Basics</h1> | |||
The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>, which is a group homomorphism <strong>T</strong>: J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a <strong>temperament mapping matrix</strong>; when context is clear enough it's also sometimes just called a <strong>mapping matrix</strong> for the temperament in question. 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 generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>, which is a group homomorphism <strong>T</strong>: J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a <strong>temperament mapping matrix</strong>; when context is clear enough it's also sometimes just called a <strong>mapping matrix</strong> for the temperament in question. 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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