Radical interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 163752445 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 164161577 - 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>2010-09-19 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-21 01:19:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>164161577</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> | 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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In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning. | In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning. | ||
===Tunings in terms of fractional monzos=== | |||
If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r //eigenmonzos// e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvector]] for the eigenvalue 1. | If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r //eigenmonzos// e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvector]] for the eigenvalue 1. | ||
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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. | 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. | ||
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. | |||
===Algebraic considerations=== | ===Algebraic considerations=== | ||
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In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning.<br /> | In particular, this is true of matrices with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for a fifth in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--Tunings in terms of fractional monzos"></a><!-- ws:end:WikiTextHeadingRule:2 -->Tunings in terms of fractional monzos</h3> | |||
If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r <em>eigenmonzos</em> e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvector</a> for the eigenvalue 1.<br /> | If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r <em>eigenmonzos</em> e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvector</a> for the eigenvalue 1.<br /> | ||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | 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 /> | ||
<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 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 /> | |||
<br /> | |||
<!-- 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> | ||