Subgroup basis matrix: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2012-07-31 12:08:11 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:48:57 UTC</tt>.

: The original revision id was <tt>~~355708344~~</tt>.

: The original revision id was <tt>355715168</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<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=

A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a module homomorphism **T**: J -> K from the module J of JI ratios to a new module K, where K then comes to represent tempered intervals. We can also consider module homomorphisms **S:** J* -> L*, where J* is module of linear functionals on J, and where we map directly from J* to another module of linear functionals L*; this 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 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]].

A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a Z-module homomorphism (aka abelian group homomorphism) **T**: 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. 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]].

If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup G of a JI module J if and only if the column module of V spans G and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that G must be saturated, so that we specifically allow for subgroups with prime powers and the like.

Line 100:

<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>Subgroup Mapping Matrices (V-maps)</title></head><body><h1 id="toc0"><a name="Basics"></a>Basics</h1>

A <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">temperament mapping matrix</a>, or M-map, is a module homomorphism T: J -&gt; K from the module J of JI ratios to a new module K, where K then comes to represent tempered intervals. We can also consider module homomorphisms S: J* -&gt; L*, where J* is module of linear functionals on J, and where we map directly from J* to another module of linear functionals L*; this 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 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 subgroup mapping matrices, or &quot;val-maps&quot; or V-maps 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>.

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) T: J -&gt; 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. We can also consider Z-module homomorphisms S: J* -&gt; 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 subgroup mapping matrices, or &quot;val-maps&quot; or V-maps 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>.

If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup G of a JI module J if and only if the column module of V spans G and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that G must be saturated, so that we specifically allow for subgroups with prime powers and the like.

@@ Line 1: / Line 1: @@
 <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:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 12:08:11 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:48:57 UTC</tt>.<br>
-: The original revision id was <tt>355708344</tt>.<br>
+: The original revision id was <tt>355715168</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">=&lt;span style="background-color: #ffffff;"&gt;Basics&lt;/span&gt;=
-&lt;span style="background-color: #ffffff;"&gt;A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a module homomorphism &lt;/span&gt;**&lt;span style="background-color: #ffffff;"&gt;T&lt;/span&gt;**&lt;span style="background-color: #ffffff;"&gt;: J -&gt; K from the module J of JI ratios to a new module K, where K then comes to represent tempered intervals. We can also consider module homomorphisms **S:** J* -&gt; L*, where J* is module of linear functionals on J, and where we map directly from J* to another module of linear functionals L*; this 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 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]].&lt;/span&gt;
+&lt;span style="background-color: #ffffff;"&gt;A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a Z-module homomorphism (aka abelian group homomorphism) &lt;/span&gt;**&lt;span style="background-color: #ffffff;"&gt;T&lt;/span&gt;**&lt;span style="background-color: #ffffff;"&gt;: J -&gt; 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. We can also consider Z-module homomorphisms **S:** J* -&gt; 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]].&lt;/span&gt;
 &lt;span style="background-color: #ffffff;"&gt;If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup G of a JI module J if and only if the column module of V spans G and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that G must be saturated, so that we specifically allow for subgroups with prime powers and the like.&lt;/span&gt;
@@ Line 100: / Line 100: @@
 <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;Subgroup Mapping Matrices (V-maps)&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;&lt;span style="background-color: #ffffff;"&gt;Basics&lt;/span&gt;&lt;/h1&gt;
-  &lt;span style="background-color: #ffffff;"&gt;A &lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;temperament mapping matrix&lt;/a&gt;, or M-map, is a module homomorphism &lt;/span&gt;&lt;strong&gt;&lt;span style="background-color: #ffffff;"&gt;T&lt;/span&gt;&lt;/strong&gt;&lt;span style="background-color: #ffffff;"&gt;: J -&amp;gt; K from the module J of JI ratios to a new module K, where K then comes to represent tempered intervals. We can also consider module homomorphisms &lt;strong&gt;S:&lt;/strong&gt; J* -&amp;gt; L*, where J* is module of linear functionals on J, and where we map directly from J* to another module of linear functionals L*; this module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals"&gt;svals&lt;/a&gt; on a certain subgroup, and that the module L which the elements of L* act on are &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals"&gt;smonzos&lt;/a&gt;. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices &lt;strong&gt;subgroup mapping matrices&lt;/strong&gt;, or &amp;quot;val-maps&amp;quot; or &lt;strong&gt;V-maps&lt;/strong&gt; when context demands they be distinguished from their temperamental counterparts, the &lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;M-maps&lt;/a&gt;.&lt;/span&gt;&lt;br /&gt;
+  &lt;span style="background-color: #ffffff;"&gt;A &lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;temperament mapping matrix&lt;/a&gt;, or M-map, is a Z-module homomorphism (aka abelian group homomorphism) &lt;/span&gt;&lt;strong&gt;&lt;span style="background-color: #ffffff;"&gt;T&lt;/span&gt;&lt;/strong&gt;&lt;span style="background-color: #ffffff;"&gt;: J -&amp;gt; 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. We can also consider Z-module homomorphisms &lt;strong&gt;S:&lt;/strong&gt; J* -&amp;gt; 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 &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals"&gt;svals&lt;/a&gt; on a certain subgroup, and that the Z-module L which the elements of L* act on are &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals"&gt;smonzos&lt;/a&gt;. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices &lt;strong&gt;subgroup mapping matrices&lt;/strong&gt;, or &amp;quot;val-maps&amp;quot; or &lt;strong&gt;V-maps&lt;/strong&gt; when context demands they be distinguished from their temperamental counterparts, the &lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;M-maps&lt;/a&gt;.&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup G of a JI module J if and only if the column module of V spans G and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that G must be saturated, so that we specifically allow for subgroups with prime powers and the like.&lt;/span&gt;&lt;br /&gt;