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Wikispaces>genewardsmith **Imported revision 278342188 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509655488 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:59:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509655488</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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<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">[[toc|flat]] | <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">[[toc|flat]] | ||
[[image:mathhazard.jpg align="center"]] | |||
=Definition= | =Definition= | ||
Given a reduced list of [[Harmonic limit|p-limit]] vals V, we may define a set of //transversal generators// for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By //reduced// is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r. | Given a reduced list of [[Harmonic limit|p-limit]] vals V, we may define a set of //transversal generators// for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By //reduced// is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r. | ||
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* Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.) | * Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.) | ||
* Consider the rest to be a monzo, which may be converted to a rational number if you prefer | * Consider the rest to be a monzo, which may be converted to a rational number if you prefer | ||
* This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney height]] by multiplying by the commas of V | * This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney height]] by multiplying by the commas of V</pre></div> | ||
</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>Transversal generators</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Finding the transversal generators">Finding the transversal generators</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <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>Transversal generators</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Finding the transversal generators">Finding the transversal generators</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
<!-- ws:end:WikiTextTocRule:10 --><br /> | <!-- ws:end:WikiTextTocRule:10 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextLocalImageRule:31:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div><!-- ws:end:WikiTextLocalImageRule:31 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
Given a reduced list of <a class="wiki_link" href="/Harmonic%20limit">p-limit</a> vals V, we may define a set of <em>transversal generators</em> for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By <em>reduced</em> is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r.<br /> | Given a reduced list of <a class="wiki_link" href="/Harmonic%20limit">p-limit</a> vals V, we may define a set of <em>transversal generators</em> for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By <em>reduced</em> is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r.<br /> | ||
<br /> | <br /> | ||
Revision as of 13:59, 18 May 2014
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2014-05-18 13:59:41 UTC.
- The original revision id was 509655488.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[toc|flat]] [[image:mathhazard.jpg align="center"]] =Definition= Given a reduced list of [[Harmonic limit|p-limit]] vals V, we may define a set of //transversal generators// for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By //reduced// is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r. If v1, v2, ... vr are the vals of V and t1, t2, ... tr are the corresponding transversal generators, then for any p-limit q we have, modulo the regular temperament defined by V q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q) In this way the transversal generators provide a [[transversal]] of the p-limit, and hence the name. =Examples= Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [<12 19 28 34|, <19 30 44 53|]. Then a corresponding list of transversal generators is [49/48, 36/35]. 49/48 corresponds to one step of 12et, and zero steps of 19et, whereas 36/35 is zero steps of 12et, and one step of 19et. This gives us a septimal meantone transversal of the 7-limit where 3/2 is represented by (49/48)^7 * (36/35)^11, and 2 is represented by (49/48)^12 * (36/35)^19. A more familiar septimal meantone transversal starts from the normal val list, [<1 0 -4 -13|, <0 1 4 10|], which corresponds to the transversal generators [2, 3]. Given a list of transversal generators, we may append a comma basis for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain [<12 19 28 34|, <19 30 44 53|, <-4 -6 -9 -11|, <-5 -8 -12 -14|] This is a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]] defining a change of basis for the p-limit. For another example, consider [<1 1 1 2|, <0 2 1 1|, <0 0 2 1|] which is the [[Normal lists|normal val list]] for breed temperament, the temperament tempering out 2401/2400. A corresponding list of transversal generators is [2, 49/40, 10/7]. =Finding the transversal generators= We can find transveral generators for V by the following procedure: * Take the transpose of the [[Tenney-Euclidean Tuning#The pseudoinverse|pseudoinverse]] of V, call that U * Find a basis for the commas of V * For each row U[i] of U, clear denominators and append the monzos of the comma basis for V * [[http://xenharmonic.wikispaces.com/Saturation|Saturate]] the result to a list of monzos, call that S * Apply the ith val V[i] (dot product) to each element of S * Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T * Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.) * Consider the rest to be a monzo, which may be converted to a rational number if you prefer * This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney height]] by multiplying by the commas of V
Original HTML content:
<html><head><title>Transversal generators</title></head><body><!-- ws:start:WikiTextTocRule:6:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Finding the transversal generators">Finding the transversal generators</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> <!-- ws:end:WikiTextTocRule:10 --><br /> <!-- ws:start:WikiTextLocalImageRule:31:<div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="" title="" /></div> --><div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div><!-- ws:end:WikiTextLocalImageRule:31 --><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> Given a reduced list of <a class="wiki_link" href="/Harmonic%20limit">p-limit</a> vals V, we may define a set of <em>transversal generators</em> for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By <em>reduced</em> is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r.<br /> <br /> If v1, v2, ... vr are the vals of V and t1, t2, ... tr are the corresponding transversal generators, then for any p-limit q we have, modulo the regular temperament defined by V<br /> <br /> q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q)<br /> <br /> In this way the transversal generators provide a <a class="wiki_link" href="/transversal">transversal</a> of the p-limit, and hence the name.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [<12 19 28 34|, <19 30 44 53|]. Then a corresponding list of transversal generators is [49/48, 36/35]. 49/48 corresponds to one step of 12et, and zero steps of 19et, whereas 36/35 is zero steps of 12et, and one step of 19et. This gives us a septimal meantone transversal of the 7-limit where 3/2 is represented by (49/48)^7 * (36/35)^11, and 2 is represented by (49/48)^12 * (36/35)^19. A more familiar septimal meantone transversal starts from the normal val list, [<1 0 -4 -13|, <0 1 4 10|], which corresponds to the transversal generators [2, 3].<br /> <br /> Given a list of transversal generators, we may append a comma basis for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain<br /> <br /> [<12 19 28 34|, <19 30 44 53|, <-4 -6 -9 -11|, <-5 -8 -12 -14|]<br /> <br /> This is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a> defining a change of basis for the p-limit.<br /> <br /> For another example, consider [<1 1 1 2|, <0 2 1 1|, <0 0 2 1|] which is the <a class="wiki_link" href="/Normal%20lists">normal val list</a> for breed temperament, the temperament tempering out 2401/2400. A corresponding list of transversal generators is [2, 49/40, 10/7].<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Finding the transversal generators"></a><!-- ws:end:WikiTextHeadingRule:4 -->Finding the transversal generators</h1> We can find transveral generators for V by the following procedure:<br /> <br /> <ul><li>Take the transpose of the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The pseudoinverse">pseudoinverse</a> of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma basis for V</li><li><a href="http://xenharmonic.wikispaces.com/Saturation">Saturate</a> the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo, which may be converted to a rational number if you prefer</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal <a class="wiki_link" href="/Tenney%20height">Tenney height</a> by multiplying by the commas of V</li></ul></body></html>