Regular temperament: 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:genewardsmith|genewardsmith]] and made on <tt>2011-06-18 18:02:50 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-18 21:04:39 UTC</tt>.<br>

: The original revision id was <tt>~~237481103~~</tt>.<br>

: The original revision id was <tt>237496879</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>

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=Translation between methods of specifying temperaments=

The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.

The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.

==Wedgies==

To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational ~~arthimetic~~, not floating point numbers.

To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.

==Frobenius projection maps==

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<h1 id="toc0"><a name="Translation between methods of specifying temperaments"></a>Translation between methods of specifying temperaments</h1>

<br />

The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.<br />

The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.<br />

<br />

<h2 id="toc1"><a name="Translation between methods of specifying temperaments-Wedgies"></a>Wedgies</h2>

To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational ~~arthimetic~~, not floating point numbers.<br />

To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.<br />

<br />

<h2 id="toc2"><a name="Translation between methods of specifying temperaments-Frobenius projection maps"></a>Frobenius projection maps</h2>

@@ 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>2011-06-18 18:02:50 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-18 21:04:39 UTC</tt>.<br>
-: The original revision id was <tt>237481103</tt>.<br>
+: The original revision id was <tt>237496879</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>
@@ Line 36: / Line 36: @@
 =Translation between methods of specifying temperaments=
-The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.
+The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.
 ==Wedgies==
-To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arthimetic, not floating point numbers.
+To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.
 ==Frobenius projection maps==
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 &lt;br /&gt;
-The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.&lt;br /&gt;
+The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Translation between methods of specifying temperaments-Wedgies"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Wedgies&lt;/h2&gt;
-To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arthimetic, not floating point numbers.&lt;br /&gt;
+To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Translation between methods of specifying temperaments-Frobenius projection maps"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Frobenius projection maps&lt;/h2&gt;