Radical interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 164161577 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 164163407 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-09-21 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-21 01:40:29 UTC</tt>.<br> | ||
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While minimax tuning for a list of target interval (such as an odd limit tonality diamond) can be found by setting it up as a linear programming problem, this is not necessary and in important respects not desirable. The problem is simple enough that it can be solved by brute force, which makes it easy to find exact answers in the form of fractional monzos. It can happen that the minimax solution is not unique, and the brute force approach allows ties to be broken using eg minimum squared error. | While minimax tuning for a list of target interval (such as an odd limit tonality diamond) can be found by setting it up as a linear programming problem, this is not necessary and in important respects not desirable. The problem is simple enough that it can be solved by brute force, which makes it easy to find exact answers in the form of fractional monzos. It can happen that the minimax solution is not unique, and the brute force approach allows ties to be broken using eg minimum squared error. | ||
If the target set is a q limit diamond, eigenmonzos in the range 1 < x < sqrt(2) may be found by simply finding all of the sets of intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error. | If the target set is a q limit diamond, eigenmonzos in the range 1 < x < sqrt(2) may be found by simply finding all of the sets of q-limit intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error. | ||
===Algebraic considerations=== | ===Algebraic considerations=== | ||
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While minimax tuning for a list of target interval (such as an odd limit tonality diamond) can be found by setting it up as a linear programming problem, this is not necessary and in important respects not desirable. The problem is simple enough that it can be solved by brute force, which makes it easy to find exact answers in the form of fractional monzos. It can happen that the minimax solution is not unique, and the brute force approach allows ties to be broken using eg minimum squared error.<br /> | While minimax tuning for a list of target interval (such as an odd limit tonality diamond) can be found by setting it up as a linear programming problem, this is not necessary and in important respects not desirable. The problem is simple enough that it can be solved by brute force, which makes it easy to find exact answers in the form of fractional monzos. It can happen that the minimax solution is not unique, and the brute force approach allows ties to be broken using eg minimum squared error.<br /> | ||
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If the target set is a q limit diamond, eigenmonzos in the range 1 &lt; x &lt; sqrt(2) may be found by simply finding all of the sets of intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error.<br /> | If the target set is a q limit diamond, eigenmonzos in the range 1 &lt; x &lt; sqrt(2) may be found by simply finding all of the sets of q-limit intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:4 -->Algebraic considerations</h3> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:4 -->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> | ||