Generator preimage: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238732623 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 238733463 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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 17:00:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238733463</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> | ||
| Line 31: | Line 31: | ||
We can find transveral generators for V by the following procedure: | We can find transveral generators for V by the following procedure: | ||
* Take the transpose of the pseudoinverse of V, call that U | * Take the transpose of the [[http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning#The pseudoinverse|pseudoinverse]] of V, call that U | ||
* Find a basis for the commas of V | * 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 | * 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 | * [[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 | * 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 | * 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.) | * 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> | ||
| Line 67: | Line 67: | ||
We can find transveral generators for V by the following procedure:<br /> | We can find transveral generators for V by the following procedure:<br /> | ||
<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 U[i] 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 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 by multiplying by the commas of V</li></ul></body></html></pre></div> | <ul><li>Take the transpose of the [[<!-- ws:start:WikiTextUrlRule:58:http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning#The --><a href="http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning#The">http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning#The</a><!-- ws:end:WikiTextUrlRule:58 --> 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><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> | ||