Tempered monzos and vals: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:~~guest~~|~~guest~~]] and made on <tt>2012-07-~~31 18~~:33:32 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-04 12:16:38 UTC</tt>.<br>

: The original revision id was <tt>~~355780604~~</tt>.<br>

: The original revision id was <tt>356331752</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>

<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">A regular temperament T is an equivalence class of Z-module homomorphisms **T**: J ~~->~~ K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same temperament if they differ only by unimodular transformation. An element of K is called a **tmonzo**, and an element of the dual module K* is called a **tval**.

<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">A regular temperament T is an equivalence class of Z-module homomorphisms **T**: J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same [[abstract temperament|temperament]] if they differ only by unimodular transformation. An element of K is called a **tmonzo**, and an element of the dual module K* is called a **tval**.

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[Temperament Mapping Matrices (M-maps)|mapping matrix]] for the temperament which is in [[Normal lists|normal val list form]].

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This matrix represents meantone temperament. If we right-multiply this matrix by the monzo |1 0 0>, representing 2/1, we get the tmonzo |1 0>. If we right-multiply it instead by |-1 1 0>, we get the tmonzo |0 1>. That 2/1 and 3/2 map to |1 0> and |0 1> respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now right-multiply the matrix by the monzo |-2 0 1>, representing 5/4, we get the tmonzo |-2 4>, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.</pre></div>

<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>Tmonzos and Tvals</title></head><body>A regular temperament T is an equivalence class of Z-module homomorphisms <strong>T</strong>: J ~~-&gt;~~ K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same temperament if they differ only by unimodular transformation. An element of K is called a <strong>tmonzo</strong>, and an element of the dual module K* is called a <strong>tval</strong>.<br />

<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>Tmonzos and Tvals</title></head><body>A regular temperament T is an equivalence class of Z-module homomorphisms <strong>T</strong>: J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same <a class="wiki_link" href="/abstract%20temperament">temperament</a> if they differ only by unimodular transformation. An element of K is called a <strong>tmonzo</strong>, and an element of the dual module K* is called a <strong>tval</strong>.<br />

<br />

Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">mapping matrix</a> for the temperament which is in <a class="wiki_link" href="/Normal%20lists">normal val list form</a>.<br />

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2012-07-31 18:33:32 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-04 12:16:38 UTC</tt>.<br>
-: The original revision id was <tt>355780604</tt>.<br>
+: The original revision id was <tt>356331752</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>
 <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">A regular temperament T is an equivalence class of Z-module homomorphisms **T**: J -&gt; K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same temperament if they differ only by unimodular transformation. An element of K is called a **tmonzo**, and an element of the dual module K* is called a **tval**.
+<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">A regular temperament T is an equivalence class of Z-module homomorphisms **T**: J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same [[abstract temperament|temperament]] if they differ only by unimodular transformation. An element of K is called a **tmonzo**, and an element of the dual module K* is called a **tval**.
 Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the [[Temperament Mapping Matrices (M-maps)|mapping matrix]] for the temperament which is in [[Normal lists|normal val list form]].
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 This matrix represents meantone temperament. If we right-multiply this matrix by the monzo |1 0 0&gt;, representing 2/1, we get the tmonzo |1 0&gt;. If we right-multiply it instead by |-1 1 0&gt;, we get the tmonzo |0 1&gt;. That 2/1 and 3/2 map to |1 0&gt; and |0 1&gt; respectively tell us that the tempered versions of these intervals can serve as a basis for meantone. If we now right-multiply the matrix by the monzo |-2 0 1&gt;, representing 5/4, we get the tmonzo |-2 4&gt;, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tmonzos and Tvals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A regular temperament T is an equivalence class of Z-module homomorphisms &lt;strong&gt;T&lt;/strong&gt;: J -&amp;gt; K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same temperament if they differ only by unimodular transformation. An element of K is called a &lt;strong&gt;tmonzo&lt;/strong&gt;, and an element of the dual module K* is called a &lt;strong&gt;tval&lt;/strong&gt;.&lt;br /&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tmonzos and Tvals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A regular temperament T is an equivalence class of Z-module homomorphisms &lt;strong&gt;T&lt;/strong&gt;: J → K, where J is a Z-module of JI intervals, K is a Z-module of tempered intervals, and two homomorphisms are said to represent the same &lt;a class="wiki_link" href="/abstract%20temperament"&gt;temperament&lt;/a&gt; if they differ only by unimodular transformation. An element of K is called a &lt;strong&gt;tmonzo&lt;/strong&gt;, and an element of the dual module K* is called a &lt;strong&gt;tval&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
 Tmonzos are rather straightforward, and tvals act on tmonzos in the same way that vals act on monzos: they're linear functionals which map from tmonzos to a scalar representing a certain number of steps. Note that there is no restriction on which bases tmonzos can be written in, but one option is to use the basis corresponding to the &lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;mapping matrix&lt;/a&gt; for the temperament which is in &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal val list form&lt;/a&gt;.&lt;br /&gt;