Generator preimage: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238729963 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 238731837 - 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>2011-06-25 16: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-25 16:39:27 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238731837</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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q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q) | q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q) | ||
In this way the transversal generators provide a [[ | In this way the transversal generators provide a [[transversal]] of the p-limit, and hence the name. | ||
=Examples= | =Examples= | ||
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This is a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]] defining a change of basis for the p-limit. | 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].</pre></div> | 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 pseudoinverse of V, call that U | |||
* Find a basis for the commas of V | |||
* For each row of U, clear denominators and append the monzos of the comma basis for V | |||
* Saturate the result to a list of monzos, call that S | |||
* Apply each val of V (dot product) to S | |||
* Insert V.S[i] in front of S[i] as the first element, obtaining 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 the corresponding transveral generator to the ith val of V; it may be reduced to an equivalent generator of minimal Tenney height by multiplying by the commas of V | |||
*</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: | <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: | <!-- 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: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 /> | ||
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q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q)<br /> | q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q)<br /> | ||
<br /> | <br /> | ||
In this way the transversal generators provide a <a class="wiki_link" href="/ | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> | ||
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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 /> | 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 /> | <br /> | ||
For another example, consider [&lt;1 1 1 2|, &lt;0 2 1 1|, &lt;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].</body></html></pre></div> | For another example, consider [&lt;1 1 1 2|, &lt;0 2 1 1|, &lt;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:&lt;h1&gt; --><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 pseudoinverse of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row of U, clear denominators and append the monzos of the comma basis for V</li><li>Saturate the result to a list of monzos, call that S</li><li>Apply each val of V (dot product) to S</li><li>Insert V.S[i] in front of S[i] as the first element, obtaining 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 the corresponding transveral generator to the ith val of V; it may be reduced to an equivalent generator of minimal Tenney height by multiplying by the commas of V</li></ul>*</body></html></pre></div> | |||