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Wikispaces>genewardsmith **Imported revision 238733597 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 238745069 - 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 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-25 19:51:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238745069</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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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. | ||
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 | 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 | 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. | In this way the transversal generators provide a [[transversal]] of the p-limit, and hence the name. | ||
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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 /> | ||
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<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 /> | <br /> | ||
q | 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="/transversal">transversal</a> of the p-limit, and hence the name.<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 /> | ||