Radical interval: Difference between revisions
Wikispaces>guest **Imported revision 163734097 - Original comment: ** |
Wikispaces>guest **Imported revision 163751933 - 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:guest|guest]] and made on <tt>2010-09-19 | : This revision was by author [[User:guest|guest]] and made on <tt>2010-09-19 15:52:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>163751933</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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For instance, if we take <19 30 44 53| and divide each coordinate twice by the corresponding coordinate of <31 49 72 87| we obtain a fractional eigenmonzo <19/961 30/2401 11/1296 53/7569|, and similarly from the 31et val we have <1/31 1/49 1/72 1/87|. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents. | For instance, if we take <19 30 44 53| and divide each coordinate twice by the corresponding coordinate of <31 49 72 87| we obtain a fractional eigenmonzo <19/961 30/2401 11/1296 53/7569|, and similarly from the 31et val we have <1/31 1/49 1/72 1/87|. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents. | ||
These methods suggest a wide range of tuning possibilities. For instance, Frobenius tuning uses unweighted eigenmonzos, whereas RMS-TOP doubly weights the vals to make left eigenvectors of them. Intermediate to the two is the tuning which val weights the vals and uses them as left eigenvectors; this gives a tuning which is less biased towards smaller primes than TOP-RMS. For rank two temperaments, using 2 and the product W of odd primes to p as eigenmonzos gives a tuning which has a simple expression in terms of roots of products of 2 and W, and where the errors in the tunings of the odd primes must cancel since W is tuned justly. | |||
===Algebraic considerations=== | ===Algebraic considerations=== | ||
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For instance, if we take &lt;19 30 44 53| and divide each coordinate twice by the corresponding coordinate of &lt;31 49 72 87| we obtain a fractional eigenmonzo &lt;19/961 30/2401 11/1296 53/7569|, and similarly from the 31et val we have &lt;1/31 1/49 1/72 1/87|. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents.<br /> | For instance, if we take &lt;19 30 44 53| and divide each coordinate twice by the corresponding coordinate of &lt;31 49 72 87| we obtain a fractional eigenmonzo &lt;19/961 30/2401 11/1296 53/7569|, and similarly from the 31et val we have &lt;1/31 1/49 1/72 1/87|. Using these as fractional eigenmonzos gives us a tuning which is already very close (less than 0.01 cents error for the primes) to TOP-RMS. Using 171et instead, the error is less than 0.0002 cents.<br /> | ||
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These methods suggest a wide range of tuning possibilities. For instance, Frobenius tuning uses unweighted eigenmonzos, whereas RMS-TOP doubly weights the vals to make left eigenvectors of them. Intermediate to the two is the tuning which val weights the vals and uses them as left eigenvectors; this gives a tuning which is less biased towards smaller primes than TOP-RMS. For rank two temperaments, using 2 and the product W of odd primes to p as eigenmonzos gives a tuning which has a simple expression in terms of roots of products of 2 and W, and where the errors in the tunings of the odd primes must cancel since W is tuned justly.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:2 -->Algebraic considerations</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:2 -->Algebraic considerations</h3> | ||
For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> (or equivalently, Z-module) of rank n equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow">divisible group</a>, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> (of dimension n) over the rational numbers. They are also torsion-free (equivalently, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow">flat</a>) abelian groups, and are the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hulls</a> of the corresponding monzos.</body></html></pre></div> | For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> (or equivalently, Z-module) of rank n equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow">divisible group</a>, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> (of dimension n) over the rational numbers. They are also torsion-free (equivalently, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow">flat</a>) abelian groups, and are the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hulls</a> of the corresponding monzos.</body></html></pre></div> | ||