Radical interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 163059485 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 163059973 - 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>2010-09-16 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-16 01:40:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>163059973</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> | ||
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===Fractional projection maps=== | ===Fractional projection maps=== | ||
A square matrix P is a [[http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29|projection]] if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and Interval Space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = | A square matrix P is a [[http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29|projection]] if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and Interval Space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. | ||
In particular, this is true of mstricies with rows consisting of fractional monzos. This is of interest since several of the most imortant 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 3 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 mstricies with rows consisting of fractional monzos. This is of interest since several of the most imortant 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 3 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. | ||
===Algebraic considerations=== | ===Algebraic considerations=== | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> | ||
A square matrix P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" rel="nofollow">projection</a> if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvalues</a> of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = | A square matrix P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" rel="nofollow">projection</a> if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvalues</a> of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. <br /> | ||
<br /> | <br /> | ||
In particular, this is true of mstricies with rows consisting of fractional monzos. This is of interest since several of the most imortant 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 3 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 mstricies with rows consisting of fractional monzos. This is of interest since several of the most imortant 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 3 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--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) I 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 r 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 r) 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) I 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 r 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 r) 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> | ||