Radical interval: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 163026895 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 163059485 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-16 01:38:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>163059485</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/ | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents. | ||
Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. | |||
===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 = P. | |||
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=== | ||
For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (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 [[http://en.wikipedia.org/wiki/Divisible_group|divisible group]], 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 [[http://en.wikipedia.org/wiki/Vector_space|vector space]] (of dimension r) over the rational numbers. They are also torsion-free (equivalently, [[http://en.wikipedia.org/wiki/Flat_module|flat]]) abelian groups, and are the [[http://en.wikipedia.org/wiki/Injective_hull|injective hulls]] of the corresponding monzos.</pre></div> | For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (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 [[http://en.wikipedia.org/wiki/Divisible_group|divisible group]], 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 [[http://en.wikipedia.org/wiki/Vector_space|vector space]] (of dimension r) over the rational numbers. They are also torsion-free (equivalently, [[http://en.wikipedia.org/wiki/Flat_module|flat]]) abelian groups, and are the [[http://en.wikipedia.org/wiki/Injective_hull|injective hulls]] of the corresponding monzos.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26&gt; represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/ | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26&gt; represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with &lt;cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.<br /> | ||
<br /> | |||
Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. <br /> | |||
<br /> | |||
<!-- 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 = P. <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 /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- 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> | ||
Revision as of 01:38, 16 September 2010
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2010-09-16 01:38:26 UTC.
- The original revision id was 163059485.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents. Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. ===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 = P. 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=== For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (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 [[http://en.wikipedia.org/wiki/Divisible_group|divisible group]], 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 [[http://en.wikipedia.org/wiki/Vector_space|vector space]] (of dimension r) over the rational numbers. They are also torsion-free (equivalently, [[http://en.wikipedia.org/wiki/Flat_module|flat]]) abelian groups, and are the [[http://en.wikipedia.org/wiki/Injective_hull|injective hulls]] of the corresponding monzos.
Original HTML content:
<html><head><title>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.<br /> <br /> Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. <br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h3> --><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 = P. <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 /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><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>