Subgroup basis matrix: Difference between revisions
Wikispaces>genewardsmith **Imported revision 509654080 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 511015600 - 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 16:55:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>511015600</tt>.<br> | ||
: The revision comment was: <tt></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> | 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"]] | ||
=<span style="background-color: #ffffff;">Basics</span>= | =<span style="background-color: #ffffff;">Basics</span>= | ||
<span style="background-color: #ffffff;">A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a Z-module homomorphism (aka abelian group homomorphism) </span>**<span style="background-color: #ffffff;">T</span>**<span style="background-color: #ffffff;">: J → K from the free Z-module (abelian group) J of JI ratios to a new free Z-module K, where K then comes to represent tempered intervals, that is to say, intervals of an [[abstract regular temperament]]. We can also consider Z-module homomorphisms **S:** J* → L*, where J* is the Z-module of linear functionals (elements of Hom(J, Z)) on J, and where we map directly from J* to another Z-module of linear functionals L*; this Z-module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to [[xenharmonic/Smonzos and Svals|svals]] on a certain subgroup, and that the Z-module L which the elements of L* act on are [[xenharmonic/Smonzos and Svals|smonzos]]. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices **subgroup mapping matrices**, or "val-maps" or **V-maps** when context demands they be distinguished from their temperamental counterparts, the [[Temperament Mapping Matrices (M-maps)|M-maps]].</span> | <span style="background-color: #ffffff;">A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a Z-module homomorphism (aka abelian group homomorphism) </span>**<span style="background-color: #ffffff;">T</span>**<span style="background-color: #ffffff;">: J → K from the free Z-module (abelian group) J of JI ratios to a new free Z-module K, where K then comes to represent tempered intervals, that is to say, intervals of an [[abstract regular temperament]]. We can also consider Z-module homomorphisms **S:** J* → L*, where J* is the Z-module of linear functionals (elements of Hom(J, Z)) on J, and where we map directly from J* to another Z-module of linear functionals L*; this Z-module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to [[xenharmonic/Smonzos and Svals|svals]] on a certain subgroup, and that the Z-module L which the elements of L* act on are [[xenharmonic/Smonzos and Svals|smonzos]]. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices **subgroup mapping matrices**, or "val-maps" or **V-maps** when context demands they be distinguished from their temperamental counterparts, the [[Temperament Mapping Matrices (M-maps)|M-maps]].</span> | ||
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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>Subgroup Mapping Matrices (V-maps)</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>Subgroup Mapping Matrices (V-maps)</title></head><body><br /> | ||
<!-- ws:start:WikiTextLocalImageRule:13: | <!-- ws:start:WikiTextLocalImageRule:13:&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; align=&quot;left&quot; /&gt; --><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" align="left" /><!-- ws:end:WikiTextLocalImageRule:13 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:7 --><span style="background-color: #ffffff;">Basics</span></h1> | |||
<span style="background-color: #ffffff;">A <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">temperament mapping matrix</a>, or M-map, is a Z-module homomorphism (aka abelian group homomorphism) </span><strong><span style="background-color: #ffffff;">T</span></strong><span style="background-color: #ffffff;">: J → K from the free Z-module (abelian group) J of JI ratios to a new free Z-module K, where K then comes to represent tempered intervals, that is to say, intervals of an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>. We can also consider Z-module homomorphisms <strong>S:</strong> J* → L*, where J* is the Z-module of linear functionals (elements of Hom(J, Z)) on J, and where we map directly from J* to another Z-module of linear functionals L*; this Z-module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">svals</a> on a certain subgroup, and that the Z-module L which the elements of L* act on are <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">smonzos</a>. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices <strong>subgroup mapping matrices</strong>, or &quot;val-maps&quot; or <strong>V-maps</strong> when context demands they be distinguished from their temperamental counterparts, the <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">M-maps</a>.</span><br /> | <span style="background-color: #ffffff;">A <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">temperament mapping matrix</a>, or M-map, is a Z-module homomorphism (aka abelian group homomorphism) </span><strong><span style="background-color: #ffffff;">T</span></strong><span style="background-color: #ffffff;">: J → K from the free Z-module (abelian group) J of JI ratios to a new free Z-module K, where K then comes to represent tempered intervals, that is to say, intervals of an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>. We can also consider Z-module homomorphisms <strong>S:</strong> J* → L*, where J* is the Z-module of linear functionals (elements of Hom(J, Z)) on J, and where we map directly from J* to another Z-module of linear functionals L*; this Z-module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">svals</a> on a certain subgroup, and that the Z-module L which the elements of L* act on are <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">smonzos</a>. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices <strong>subgroup mapping matrices</strong>, or &quot;val-maps&quot; or <strong>V-maps</strong> when context demands they be distinguished from their temperamental counterparts, the <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">M-maps</a>.</span><br /> | ||
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