Temperament mapping matrix: Difference between revisions
Wikispaces>genewardsmith **Imported revision 355718468 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 355719170 - 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>2012-07-31 13: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 13:13:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>355719170</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> | ||
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The multiplicative group generated by any finite 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. | The multiplicative group generated by any finite 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 | 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. | ||
The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of [[tmonzos and tvals|tmonzos]] for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T. | The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of [[tmonzos and tvals|tmonzos]] for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T. | ||
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The result of **P*****M** is the matrix [|1 0>, |1 -3>], telling us that 2/1 maps to the tmonzo |1 0>, and that 3/2 maps to the tmonzo |1 -3>. This tells us that 2/1 maps to one of the generators for porcupine and that 3/2 is made up of one 2/1 minus three of the other generator. It so happens that the other generator is 10/9, which maps to |0 1>. | The result of **P*****M** is the matrix [|1 0>, |1 -3>], telling us that 2/1 maps to the tmonzo |1 0>, and that 3/2 maps to the tmonzo |1 -3>. This tells us that 2/1 maps to one of the generators for porcupine and that 3/2 is made up of one 2/1 minus three of the other generator. It so happens that the other generator is 10/9, which maps to |0 1>. | ||
We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of **P** by putting these intervals in monzo form as columns of a matrix **N**, which works out to be [|-1 -3 1 0 1>, |6 -2 0 -1 0>, |2 -2 2 0 -1>]. If we then evaluate the product ** | We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of **P** by putting these intervals in monzo form as columns of a matrix **N**, which works out to be [|-1 -3 1 0 1>, |6 -2 0 -1 0>, |2 -2 2 0 -1>]. If we then evaluate the product **P∙N** we get the matrix [|0 0>, |0 0>, |0 0>], meaning that all of these intervals map to the unison tmonzo. This confirms that the kernel of porcupine does indeed lie in the null module of **P**. | ||
**The Dual Transformation** | **The Dual Transformation** | ||
To explore the dual transformation implied by **P**, we'll look at the tval matrix [<7 1|, <15 2|], where again the bras signify that the tvals represent matrix rows. The first tval maps the first generator (for which 2/1 is a [[Transversal generators|transversal]]) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval. If we call this matrix V, then the result of | To explore the dual transformation implied by **P**, we'll look at the tval matrix [<7 1|, <15 2|], where again the bras signify that the tvals represent matrix rows. The first tval maps the first generator (for which 2/1 is a [[Transversal generators|transversal]]) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval. If we call this matrix V, then the result of V∙P is the matrix | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
for which the rows are the patent vals for [[7-EDO]] and [[15-EDO]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the <7 1| and <15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix ** | for which the rows are the patent vals for [[7-EDO]] and [[15-EDO]], respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the <7 1| and <15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix **V∙P** is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [<1 2 3 2 4|, <0 -3 -5 6 -4|] as a result again.</pre></div> | ||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:5 -->Basics</h1> | <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><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:5 -->Basics</h1> | ||
The multiplicative group generated by any finite 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 /> | The multiplicative group generated by any finite 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 /> | <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 | 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 /> | ||
<br /> | <br /> | ||
The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of <a class="wiki_link" href="/tmonzos%20and%20tvals">tmonzos</a> for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T.<br /> | The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of <a class="wiki_link" href="/tmonzos%20and%20tvals">tmonzos</a> for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T.<br /> | ||
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The result of <strong>P</strong><strong>*M</strong> is the matrix [|1 0&gt;, |1 -3&gt;], telling us that 2/1 maps to the tmonzo |1 0&gt;, and that 3/2 maps to the tmonzo |1 -3&gt;. This tells us that 2/1 maps to one of the generators for porcupine and that 3/2 is made up of one 2/1 minus three of the other generator. It so happens that the other generator is 10/9, which maps to |0 1&gt;.<br /> | The result of <strong>P</strong><strong>*M</strong> is the matrix [|1 0&gt;, |1 -3&gt;], telling us that 2/1 maps to the tmonzo |1 0&gt;, and that 3/2 maps to the tmonzo |1 -3&gt;. This tells us that 2/1 maps to one of the generators for porcupine and that 3/2 is made up of one 2/1 minus three of the other generator. It so happens that the other generator is 10/9, which maps to |0 1&gt;.<br /> | ||
<br /> | <br /> | ||
We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of <strong>P</strong> by putting these intervals in monzo form as columns of a matrix <strong>N</strong>, which works out to be [|-1 -3 1 0 1&gt;, |6 -2 0 -1 0&gt;, |2 -2 2 0 -1&gt;]. If we then evaluate the product <strong> | We can use this technique to indeed confirm that 55/54, 64/63, and 100/99 form a basis for the null module of <strong>P</strong> by putting these intervals in monzo form as columns of a matrix <strong>N</strong>, which works out to be [|-1 -3 1 0 1&gt;, |6 -2 0 -1 0&gt;, |2 -2 2 0 -1&gt;]. If we then evaluate the product <strong>P∙N</strong> we get the matrix [|0 0&gt;, |0 0&gt;, |0 0&gt;], meaning that all of these intervals map to the unison tmonzo. This confirms that the kernel of porcupine does indeed lie in the null module of <strong>P</strong>.<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<strong>The Dual Transformation</strong><br /> | <strong>The Dual Transformation</strong><br /> | ||
To explore the dual transformation implied by <strong>P</strong>, we'll look at the tval matrix [&lt;7 1|, &lt;15 2|], where again the bras signify that the tvals represent matrix rows. The first tval maps the first generator (for which 2/1 is a <a class="wiki_link" href="/Transversal%20generators">transversal</a>) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval. If we call this matrix V, then the result of | To explore the dual transformation implied by <strong>P</strong>, we'll look at the tval matrix [&lt;7 1|, &lt;15 2|], where again the bras signify that the tvals represent matrix rows. The first tval maps the first generator (for which 2/1 is a <a class="wiki_link" href="/Transversal%20generators">transversal</a>) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval. If we call this matrix V, then the result of V∙P is the matrix<br /> | ||
<br /> | <br /> | ||
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for which the rows are the patent vals for <a class="wiki_link" href="/7-EDO">7-EDO</a> and <a class="wiki_link" href="/15-EDO">15-EDO</a>, respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the &lt;7 1| and &lt;15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix <strong> | for which the rows are the patent vals for <a class="wiki_link" href="/7-EDO">7-EDO</a> and <a class="wiki_link" href="/15-EDO">15-EDO</a>, respectively. Since the dual transformation is injective, these vals can be interpreted as the full-limit vals which are implied by the &lt;7 1| and &lt;15 2| tvals. Additionally, since the image of the dual transformation is the set of vals supporting porcupine, and since the above two vals are linearly independent, the resulting matrix <strong>V∙P</strong> is another valid mapping matrix for porcupine. We can confirm this by putting the matrix back into normal val list form and getting [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|] as a result again.</body></html></pre></div> | ||