Tempered monzos and vals: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 509653968 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 511015568 - 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:genewardsmith|genewardsmith]] and made on <tt>2014-05- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-24 16:55:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>511015568</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> | ||
<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"> | <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"> | ||
[[image:mathhazard.jpg align=" | [[image:mathhazard.jpg align="left"]] | ||
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**. | 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**. | ||
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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>Tmonzos and Tvals</title></head><body><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><br /> | ||
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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 /> | |||
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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 /> | 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 /> | ||
Revision as of 16:55, 24 May 2014
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2014-05-24 16:55:05 UTC.
- The original revision id was 511015568.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[image:mathhazard.jpg align="left"]] 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]]. =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.
Original HTML content:
<html><head><title>Tmonzos and Tvals</title></head><body><br /> <!-- ws:start:WikiTextLocalImageRule:2:<img src="/file/view/mathhazard.jpg" alt="" title="" align="left" /> --><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" align="left" /><!-- ws:end:WikiTextLocalImageRule:2 --><br /> 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 /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1> As an example, consider the mapping matrix<br /> [<1 1 0|]<br /> [<0 1 4|]<br /> <br /> 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.</body></html>