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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | =Definition= |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | 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. |
| : This revision was by author [[User:clumma|clumma]] and made on <tt>2014-12-14 22:42:06 UTC</tt>.<br>
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| : The original revision id was <tt>535153290</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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">=Definition=
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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. | |
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| 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 | | 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 |
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| q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q) | | q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q) |
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| 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|transversal]] of the p-limit, and hence the name. |
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| =Examples= | | =Examples= |
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| [<12 19 28 34|, <19 30 44 53|, <-4 -6 -9 -11|, <-5 -8 -12 -14|] | | [<12 19 28 34|, <19 30 44 53|, <-4 -6 -9 -11|, <-5 -8 -12 -14|] |
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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. |
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| 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]. | | 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]. |
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| =Finding the transversal generators= | | =Finding the transversal generators= |
| We can find transveral generators for V by the following procedure: | | We can find transveral generators for V by the following procedure: |
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| * Take the transpose of the [[Tenney-Euclidean Tuning#The pseudoinverse|pseudoinverse]] of V, call that U
| | <ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] 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>[[Saturation|Saturate]] 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 [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul> [[Category:generator]] |
| * Find a basis for the commas of V
| | [[Category:theory]] |
| * For each row U[i] of U, clear denominators and append the monzos of the comma basis for V
| | [[Category:todo:reduce_mathslang]] |
| * [[http://xenharmonic.wikispaces.com/Saturation|Saturate]] the result to a list of monzos, call that S
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| * Apply the ith val V[i] (dot product) to each element of S
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| * 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
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| * Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)
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| * Consider the rest to be a monzo, which may be converted to a rational number if you prefer
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| * 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>
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| <h4>Original HTML content:</h4>
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Transversal generators</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1>
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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 />
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| 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 />
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| q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q)<br />
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| 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 />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1>
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| 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 />
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| 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 />
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| [&lt;12 19 28 34|, &lt;19 30 44 53|, &lt;-4 -6 -9 -11|, &lt;-5 -8 -12 -14|]<br />
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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 />
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| 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 />
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| <!-- 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>
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| We can find transveral generators for V by the following procedure:<br />
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| <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></pre></div>
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Definition
Given a reduced list of 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 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 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 U[i] 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 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