Regular temperament: Difference between revisions
Wikispaces>genewardsmith **Imported revision 237496879 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 238238485 - Original comment: "here" is a poor link caption - from a psychological point of view** |
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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> | ||
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To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one. | To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one. | ||
Maple code for the parts of this which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found [[Basic abstract temperament translation code | Maple code for the parts of this which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]]. | ||
=The Geometry of Regular Temperaments= | =The Geometry of Regular Temperaments= | ||
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To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.<br /> | To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.<br /> | ||
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Maple code for the parts of this which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found <a class="wiki_link" href="/Basic%20abstract%20temperament%20translation%20code"> | Maple code for the parts of this which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article <a class="wiki_link" href="/Basic%20abstract%20temperament%20translation%20code">Basic abstract temperament translation code</a>. <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->The Geometry of Regular Temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:10 -->The Geometry of Regular Temperaments</h1> | ||