Interior product: Difference between revisions
Wikispaces>genewardsmith **Imported revision 278552322 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278559412 - 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>2011-11-23 14: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-23 14:31:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278559412</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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Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q. | Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q. | ||
The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, | The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit [[Marvel family#Marvel|marvel temperament]]. Then Marv∨45/44 = <<4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = <<-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = <<11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = <<-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = <<5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = <<6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = <<-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = <<6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨9801/9800 = <<-12 2 -20 6 31 2 51 -52 7 86||, wizard. | ||
The interior product is also useful in finding the temperament map given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the map. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [<0 0 -1 -2 3|, <0 1 0 2 -1|, <0 2 -2 0 4|, <0 -3 1 -4 0|, <-1 0 0 5 -12|, <-2 0 -5 0 -9|, <3 0 12 9 0|, <2 5 0 0 19|, <-1 -12 0 -19 0|, <4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|], the normal val list for 11-limit marvel.</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>Interior product</title></head><body>The <em>interior product</em> is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).<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>Interior product</title></head><body>The <em>interior product</em> is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).<br /> | ||
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Another application is the use of the interior product to define the intervals of the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q.<br /> | Another application is the use of the interior product to define the intervals of the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q.<br /> | ||
<br /> | <br /> | ||
The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, | The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit <a class="wiki_link" href="/Marvel%20family#Marvel">marvel temperament</a>. Then Marv∨45/44 = &lt;&lt;4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = &lt;&lt;-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = &lt;&lt;11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = &lt;&lt;-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = &lt;&lt;5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = &lt;&lt;6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = &lt;&lt;-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = &lt;&lt;6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨9801/9800 = &lt;&lt;-12 2 -20 6 31 2 51 -52 7 86||, wizard.<br /> | ||
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The interior product is also useful in finding the temperament map given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the map. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [&lt;0 0 -1 -2 3|, &lt;0 1 0 2 -1|, &lt;0 2 -2 0 4|, &lt;0 -3 1 -4 0|, &lt;-1 0 0 5 -12|, &lt;-2 0 -5 0 -9|, &lt;3 0 12 9 0|, &lt;2 5 0 0 19|, &lt;-1 -12 0 -19 0|, &lt;4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|], the normal val list for 11-limit marvel.</body></html></pre></div> | |||