Subgroup basis matrix: 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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:51:43 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:56:13 UTC</tt>.

: The original revision id was <tt>~~355715616~~</tt>.

: The original revision id was <tt>355716414</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 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, 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]].

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, 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]].

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.

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=Dual Transformation=

Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation **S:** J* ~~->~~ L*, the associated dual transformation is **S*:** L ~~->~~ J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, **S*** maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals ~~->~~ vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.

Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation **S:** J* → L*, the associated dual transformation is **S*:** L → J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, **S*** maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals → vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.

The main transformation of any V-map V can be applied by matrix multiplication with the V-map on the right and a matrix with vals as rows on the left. Conversely, the dual transformation of V can be applied by matrix multiplication with the V-map on the left and a matrix with smonzos as columns on the right.

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<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 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, 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 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>.

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 → 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 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 <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.

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<h1 id="toc1"><a name="Dual Transformation"></a>Dual Transformation</h1>

Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation S: J* ~~-&gt;~~ L*, the associated dual transformation is S*: L ~~-&gt;~~ J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, S* maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals ~~-&gt;~~ vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.

Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation S: J* → L*, the associated dual transformation is S*: L → J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, S* maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals → vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.

The main transformation of any V-map V can be applied by matrix multiplication with the V-map on the right and a matrix with vals as rows on the left. Conversely, the dual transformation of V can be applied by matrix multiplication with the V-map on the left and a matrix with smonzos as columns on the right.

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:51:43 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:56:13 UTC</tt>.<br>
-: The original revision id was <tt>355715616</tt>.<br>
+: The original revision id was <tt>355716414</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 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, that is to say, intervals of an [[abstract regular temperament]]. 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;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 → 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]].&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 18: / Line 18: @@
 =Dual Transformation=
-Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation **S:** J* -&gt; L*, the associated dual transformation is **S*:** L -&gt; J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, **S*** maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals -&gt; vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.
+Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation **S:** J* → L*, the associated dual transformation is **S*:** L → J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, **S*** maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals → vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.
 The main transformation of any V-map V can be applied by matrix multiplication with the V-map on the right and a matrix with vals as rows on the left. Conversely, the dual transformation of V can be applied by matrix multiplication with the V-map on the left and a matrix with smonzos as columns on the right.
@@ 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 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, that is to say, intervals of an &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt;. 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;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 → 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 &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt;. We can also consider Z-module homomorphisms &lt;strong&gt;S:&lt;/strong&gt; 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 &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;
@@ Line 111: / Line 111: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Dual Transformation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Dual Transformation&lt;/h1&gt;
-  Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation &lt;strong&gt;S:&lt;/strong&gt; J* -&amp;gt; L*, the associated dual transformation is &lt;strong&gt;S*:&lt;/strong&gt; L -&amp;gt; J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, &lt;strong&gt;S&lt;/strong&gt;* maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals -&amp;gt; vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.&lt;br /&gt;
+  Much like with temperament mapping matrices, subgroup mapping matrices also have an associated dual transformation. Since the V-map represents a linear transformation &lt;strong&gt;S:&lt;/strong&gt; J* → L*, the associated dual transformation is &lt;strong&gt;S*:&lt;/strong&gt; L → J. Since L is the module that the module L* of svals acts on, we can identify L with smonzos, and since J is the module of JI monzos, &lt;strong&gt;S&lt;/strong&gt;* maps from smonzos back to monzos. As with the dual transformation on a mapping matrix sending tvals → vals, this mapping is generally injective but not surjective. No two smonzos will map to the same monzo, and the only monzos in the image of this transformation are those lying in the submodule of J denoted by G.&lt;br /&gt;
 &lt;br /&gt;
 The main transformation of any V-map V can be applied by matrix multiplication with the V-map on the right and a matrix with vals as rows on the left. Conversely, the dual transformation of V can be applied by matrix multiplication with the V-map on the left and a matrix with smonzos as columns on the right.&lt;br /&gt;