Interior product: Difference between revisions
Wikispaces>genewardsmith **Imported revision 278926644 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278928182 - 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>2011-11-25 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-25 01:59:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278928182</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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=Applications= | =Applications= | ||
One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero. | One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if and only if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero. | ||
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. It should be noted that V with this property is underdetermined, so that many possible vectors can be used to the same effect. In most cases, solving for V subject to the condition that all its coefficients except the first r are zero will work and this introduces some simplicifation; for example the dot product of Meantone∨q, where "Meantone" is the 7-limit wedgie, with [1200+300*log2(5), -1200, 0, 0] gives the value in cents of the [[quarter-comma meantone]] tuning of the interval denoted by Meantone∨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. It should be noted that V with this property is underdetermined, so that many possible vectors can be used to the same effect. In most cases, solving for V subject to the condition that all its coefficients except the first r are zero will work and this introduces some simplicifation; for example the dot product of Meantone∨q, where "Meantone" is the 7-limit wedgie, with [1200+300*log2(5), -1200, 0, 0] gives the value in cents of the [[quarter-comma meantone]] tuning of the interval denoted by Meantone∨q. | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:0 -->Applications</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:0 -->Applications</h1> | ||
One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.<br /> | One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if and only if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.<br /> | ||
<br /> | <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. It should be noted that V with this property is underdetermined, so that many possible vectors can be used to the same effect. In most cases, solving for V subject to the condition that all its coefficients except the first r are zero will work and this introduces some simplicifation; for example the dot product of Meantone∨q, where &quot;Meantone&quot; is the 7-limit wedgie, with [1200+300*log2(5), -1200, 0, 0] gives the value in cents of the <a class="wiki_link" href="/quarter-comma%20meantone">quarter-comma meantone</a> tuning of the interval denoted by Meantone∨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. It should be noted that V with this property is underdetermined, so that many possible vectors can be used to the same effect. In most cases, solving for V subject to the condition that all its coefficients except the first r are zero will work and this introduces some simplicifation; for example the dot product of Meantone∨q, where &quot;Meantone&quot; is the 7-limit wedgie, with [1200+300*log2(5), -1200, 0, 0] gives the value in cents of the <a class="wiki_link" href="/quarter-comma%20meantone">quarter-comma meantone</a> tuning of the interval denoted by Meantone∨q.<br /> | ||