Tempered monzos and vals: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

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_regular_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'''.

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

~~: This revision was by author [[User:clumma|clumma]] and made on <tt>2014-12-14 22:41:01 UTC</tt>.<br>~~

~~: The original revision id was <tt>535153194</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 [[~~abstract regular 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]].

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]].

=Example=

As an example, consider the mapping matrix

[<1 1 0|]

[<0 1 4|]

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>~~

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.

~~<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 → 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%20regular%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 />

~~<br />~~

~~<h1 id="toc0"><a name="Example"></a>Example</h1>~~

~~As an example, consider the mapping matrix<br />~~

~~[&lt;1 1 0|]<br />~~

~~[&lt;0 1 4|]<br />~~

~~<br />~~

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.</body></html></pre></div>

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+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_regular_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'''.
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:clumma|clumma]] and made on <tt>2014-12-14 22:41:01 UTC</tt>.<br>
-: The original revision id was <tt>535153194</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 [[abstract regular 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]].
+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]].
 =Example=
 As an example, consider the mapping matrix
 [&lt;1 1 0|]
 [&lt;0 1 4|]
-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>
+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.
-<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 → 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%20regular%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;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Example&lt;/h1&gt;
- As an example, consider the mapping matrix&lt;br /&gt;
-[&amp;lt;1 1 0|]&lt;br /&gt;
-[&amp;lt;0 1 4|]&lt;br /&gt;
-&lt;br /&gt;
-This matrix represents meantone temperament. If we right-multiply this matrix by the monzo |1 0 0&amp;gt;, representing 2/1, we get the tmonzo |1 0&amp;gt;. If we right-multiply it instead by |-1 1 0&amp;gt;, we get the tmonzo |0 1&amp;gt;. That 2/1 and 3/2 map to |1 0&amp;gt; and |0 1&amp;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&amp;gt;, representing 5/4, we get the tmonzo |-2 4&amp;gt;, telling us that the tempered 5/4 maps to four tempered 3/2's minus two tempered 2/1's.&lt;/body&gt;&lt;/html&gt;</pre></div>