Temperament mapping 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:<br>

: This revision was by author [[User:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2012-07-31 11:16:19 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 13:07:10 UTC</tt>.<br>

: The original revision id was <tt>~~355698700~~</tt>.<br>

: The original revision id was <tt>355718280</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">=Basics=

The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a [[Regular Temperaments|regular temperament]], ~~which~~ is a group homomorphism **T**: J ~~->~~ K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism ~~between Z-modules~~. This homomorphism can also be represented by a integer matrix, called a **temperament mapping matrix**; when context is clear enough it's also sometimes just called a **mapping matrix** for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or **M-map** when context demands, as opposed to the [[Subgroup Mapping Matrices (V-maps)|V-map]] which is a mapping on vals.

The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a [[Regular Temperaments|regular temperament]], or more precisely an [[abstract regular temperament]], is a group homomorphism **T**: J → K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a **temperament mapping matrix**; when context is clear enough it's also sometimes just called a **mapping matrix** for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or **M-map** when context demands, as opposed to the [[Subgroup Mapping Matrices (V-maps)|V-map]] which is a mapping on vals.

Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the null module of M consists of the kernel of T, M is of full row rank, and the rows of M generate a submodule which is [[Saturation|saturated]]. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U*M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the tempered interval module.

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=Dual Transformation=

Any mapping matrix can be said to represent a linear map **M:** J ~~->~~ K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation **M*:** K* ~~->~~ J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of [[xenharmonic/tmonzos and tvals|tvals]] on K, so **M*** represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.

Any mapping matrix can be said to represent a linear map **M:** J → K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation **M*:** K* → J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of [[xenharmonic/tmonzos and tvals|tvals]] on K, so **M*** represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.

These two transformations correspond to different types of matrix multiplication: the ordinary transformation **M** corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation **M*** corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.

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

The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, ~~which~~ is a group homomorphism <strong>T</strong>: J ~~-&gt;~~ K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism ~~between Z-modules~~. This homomorphism can also be represented by a integer matrix, called a <strong>temperament mapping matrix</strong>; when context is clear enough it's also sometimes just called a <strong>mapping matrix</strong> for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &quot;monzo-map&quot; or <strong>M-map</strong> when context demands, as opposed to the <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> which is a mapping on vals.<br />

The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a>, or more precisely an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>, is a group homomorphism <strong>T</strong>: J → K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a <strong>temperament mapping matrix</strong>; when context is clear enough it's also sometimes just called a <strong>mapping matrix</strong> for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &quot;monzo-map&quot; or <strong>M-map</strong> when context demands, as opposed to the <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> which is a mapping on vals.<br />

<br />

Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the null module of M consists of the kernel of T, M is of full row rank, and the rows of M generate a submodule which is <a class="wiki_link" href="/Saturation">saturated</a>. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U*M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the tempered interval module.<br />

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<br />

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

Any mapping matrix can be said to represent a linear map <strong>M:</strong> J ~~-&gt;~~ K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation <strong>M*:</strong> K* ~~-&gt;~~ J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals">tvals</a> on K, so <strong>M</strong>* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.<br />

Any mapping matrix can be said to represent a linear map <strong>M:</strong> J → K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation <strong>M*:</strong> K* → J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals">tvals</a> on K, so <strong>M</strong>* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.<br />

<br />

These two transformations correspond to different types of matrix multiplication: the ordinary transformation <strong>M</strong> corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation <strong>M</strong>* corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.<br />

@@ 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 11:16:19 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 13:07:10 UTC</tt>.<br>
-: The original revision id was <tt>355698700</tt>.<br>
+: The original revision id was <tt>355718280</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">=Basics=
-The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a [[Regular Temperaments|regular temperament]], which is a group homomorphism **T**: J -&gt; K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism between Z-modules. This homomorphism can also be represented by a integer matrix, called a **temperament mapping matrix**; when context is clear enough it's also sometimes just called a **mapping matrix** for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or **M-map** when context demands, as opposed to the [[Subgroup Mapping Matrices (V-maps)|V-map]] which is a mapping on vals.
+The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a [[Regular Temperaments|regular temperament]], or more precisely an [[abstract regular temperament]], is a group homomorphism **T**: J → K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a **temperament mapping matrix**; when context is clear enough it's also sometimes just called a **mapping matrix** for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or **M-map** when context demands, as opposed to the [[Subgroup Mapping Matrices (V-maps)|V-map]] which is a mapping on vals.
 Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the null module of M consists of the kernel of T, M is of full row rank, and the rows of M generate a submodule which is [[Saturation|saturated]]. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U*M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the tempered interval module.
@@ Line 16: / Line 16: @@
 =Dual Transformation=
-Any mapping matrix can be said to represent a linear map **M:** J -&gt; K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation **M*:** K* -&gt; J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of [[xenharmonic/tmonzos and tvals|tvals]] on K, so **M*** represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.
+Any mapping matrix can be said to represent a linear map **M:** J → K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation **M*:** K* → J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of [[xenharmonic/tmonzos and tvals|tvals]] on K, so **M*** represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.
 These two transformations correspond to different types of matrix multiplication: the ordinary transformation **M** corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation **M*** corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.
@@ Line 83: / Line 83: @@
 <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;Temperament Mapping Matrices (M-maps)&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Basics&lt;/h1&gt;
-  The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt;, which is a group homomorphism &lt;strong&gt;T&lt;/strong&gt;: J -&amp;gt; K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a module homomorphism between Z-modules. This homomorphism can also be represented by a integer matrix, called a &lt;strong&gt;temperament mapping matrix&lt;/strong&gt;; when context is clear enough it's also sometimes just called a &lt;strong&gt;mapping matrix&lt;/strong&gt; for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &amp;quot;monzo-map&amp;quot; or &lt;strong&gt;M-map&lt;/strong&gt; when context demands, as opposed to the &lt;a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; which is a mapping on vals.&lt;br /&gt;
+  The multiplicative group of any set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt;, or more precisely an &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt;, is a group homomorphism &lt;strong&gt;T&lt;/strong&gt;: J → K from the group J of JI rationals to a group K of tempered intervals, also has the structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a &lt;strong&gt;temperament mapping matrix&lt;/strong&gt;; when context is clear enough it's also sometimes just called a &lt;strong&gt;mapping matrix&lt;/strong&gt; for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a &amp;quot;monzo-map&amp;quot; or &lt;strong&gt;M-map&lt;/strong&gt; when context demands, as opposed to the &lt;a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; which is a mapping on vals.&lt;br /&gt;
 &lt;br /&gt;
 Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the null module of M consists of the kernel of T, M is of full row rank, and the rows of M generate a submodule which is &lt;a class="wiki_link" href="/Saturation"&gt;saturated&lt;/a&gt;. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U*M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the tempered interval module.&lt;br /&gt;
@@ Line 92: / Line 92: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Dual Transformation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Dual Transformation&lt;/h1&gt;
-  Any mapping matrix can be said to represent a linear map &lt;strong&gt;M:&lt;/strong&gt; J -&amp;gt; K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation &lt;strong&gt;M*:&lt;/strong&gt; K* -&amp;gt; J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals"&gt;tvals&lt;/a&gt; on K, so &lt;strong&gt;M&lt;/strong&gt;* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.&lt;br /&gt;
+  Any mapping matrix can be said to represent a linear map &lt;strong&gt;M:&lt;/strong&gt; J → K, where J is a module of JI intervals and K is a module of tempered intervals. There is thus an associated dual transformation &lt;strong&gt;M*:&lt;/strong&gt; K* → J*, where J* and K* are the dual modules to J and K, respectively. J* is the module of vals on J, and K* is the module of &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals"&gt;tvals&lt;/a&gt; on K, so &lt;strong&gt;M&lt;/strong&gt;* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the submodule of vals supporting the associated temperament, and no two tvals map to the same val.&lt;br /&gt;
 &lt;br /&gt;
 These two transformations correspond to different types of matrix multiplication: the ordinary transformation &lt;strong&gt;M&lt;/strong&gt; corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation &lt;strong&gt;M&lt;/strong&gt;* corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.&lt;br /&gt;