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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>2011-06-25 15:39:21 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-25 16:11:41 UTC</tt>.<br>

: The original revision id was <tt>~~238727423~~</tt>.<br>

: The original revision id was <tt>238729963</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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=Definition=

Given a reduced list of [[~~Prime~~ 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

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q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q)

In this way the transversal generators provide a [[~~Transverals~~|transversal]] of the p-limit, and hence the name.

In this way the transversal generators provide a [[Transversals|transversal]] of the p-limit, and hence the name.

=Examples=</pre></div>

=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].</pre></div>

<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><a href="#Definition">Definition</a> | <a href="#Examples">Examples</a>

<br />

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

Given a reduced list of <a class="wiki_link" href="/~~Prime~~%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 />

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 />

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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="/~~Transverals~~">transversal</a> of the p-limit, and hence the name.<br />

In this way the transversal generators provide a <a class="wiki_link" href="/Transversals">transversal</a> of the p-limit, and hence the name.<br />

<br />

<h1 id="toc1"><a name="Examples"></a>Examples</h1>

</body></html></pre></div>

Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [&lt;12 19 28 34|, &lt;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, [&lt;1 0 -4 -13|, &lt;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 />

[&lt;12 19 28 34|, &lt;19 30 44 53|, &lt;-4 -6 -9 -11|, &lt;-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 [&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>

@@ 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>2011-06-25 15:39:21 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-25 16:11:41 UTC</tt>.<br>
-: The original revision id was <tt>238727423</tt>.<br>
+: The original revision id was <tt>238729963</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 9: / Line 9: @@
 =Definition=
-Given a reduced list of [[Prime 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
@@ Line 15: / Line 15: @@
 q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q)
-In this way the transversal generators provide a [[Transverals|transversal]] of the p-limit, and hence the name.
+In this way the transversal generators provide a [[Transversals|transversal]] of the p-limit, and hence the name.
-=Examples=</pre></div>
+=Examples=
+Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [&lt;12 19 28 34|, &lt;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, [&lt;1 0 -4 -13|, &lt;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
+[&lt;12 19 28 34|, &lt;19 30 44 53|, &lt;-4 -6 -9 -11|, &lt;-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 [&lt;1 1 1 2|, &lt;0 2 1 1|, &lt;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>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Transversal generators&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:4:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:4 --&gt;&lt;!-- ws:start:WikiTextTocRule:5: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:5 --&gt;&lt;!-- ws:start:WikiTextTocRule:6: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;
-Given a reduced list of &lt;a class="wiki_link" href="/Prime%20limit"&gt;p-limit&lt;/a&gt; vals V, we may define a set of &lt;em&gt;transversal generators&lt;/em&gt; 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 &lt;em&gt;reduced&lt;/em&gt; 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.&lt;br /&gt;
+Given a reduced list of &lt;a class="wiki_link" href="/Harmonic%20limit"&gt;p-limit&lt;/a&gt; vals V, we may define a set of &lt;em&gt;transversal generators&lt;/em&gt; 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 &lt;em&gt;reduced&lt;/em&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
 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&lt;br /&gt;
@@ Line 28: / Line 37: @@
 q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q)&lt;br /&gt;
 &lt;br /&gt;
-In this way the transversal generators provide a &lt;a class="wiki_link" href="/Transverals"&gt;transversal&lt;/a&gt; of the p-limit, and hence the name.&lt;br /&gt;
+In this way the transversal generators provide a &lt;a class="wiki_link" href="/Transversals"&gt;transversal&lt;/a&gt; of the p-limit, and hence the name.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Examples&lt;/h1&gt;
-&lt;/body&gt;&lt;/html&gt;</pre></div>
+Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [&amp;lt;12 19 28 34|, &amp;lt;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, [&amp;lt;1 0 -4 -13|, &amp;lt;0 1 4 10|], which corresponds to the transversal generators [2, 3].&lt;br /&gt;
+&lt;br /&gt;
+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&lt;br /&gt;
+&lt;br /&gt;
+[&amp;lt;12 19 28 34|, &amp;lt;19 30 44 53|, &amp;lt;-4 -6 -9 -11|, &amp;lt;-5 -8 -12 -14|]&lt;br /&gt;
+&lt;br /&gt;
+This is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow"&gt;unimodular matrix&lt;/a&gt; defining a change of basis for the p-limit.&lt;br /&gt;
+&lt;br /&gt;
+For another example, consider [&amp;lt;1 1 1 2|, &amp;lt;0 2 1 1|, &amp;lt;0 0 2 1|] which is the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal val list&lt;/a&gt; for breed temperament, the temperament tempering out 2401/2400. A corresponding list of transversal generators is [2, 49/40, 10/7].&lt;/body&gt;&lt;/html&gt;</pre></div>