Mathematical theory of saturation: Difference between revisions
Wikispaces>genewardsmith **Imported revision 197296106 - Original comment: ** |
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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>2011-01-30 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-30 22:01:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>197297634</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> | ||
<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">The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise | <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">The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] of rank n. Its subgroups have the property of //saturation// if for any element a of Z^n, if an integer multiple m*a of a belongs to a sublattice V, then a already belongs to V. Another way to put it is that if some linear combination with rational coefficients q1*v1 + ... + qk*vk of elements of V belongs to Z^n, then it belongs to V. For the latter definition we consider Z^n to be contained in the n-dimensional [[http://en.wikipedia.org/wiki/Vector_space|real vector space]] R^n which is often called the [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]], or grid lattice. | ||
If | If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a somewhat temperament in which all of the notes cannot be reached by tempered intervals. | ||
For example, consider the temperament with commas generated by 126/125 and 3645/3584. The group the monzos |1 2 -3 1> and |-9 6 1 -1> for these generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the group. Hence (81/80)^2 is tempered out, but 81/80 isn't, and one gets two parallel meantones not connected by any 7-limit interval. Something similar happens with the two vals <12 19 28 34| and <26 41 60 72|; this however can be fixed in a way by extending to the 11-limit, and interpreting the "unobtainable" notes as notes reached in the 11-limit. Adding 245/242 to the commas (81/80 and 126/125) of septimal meantone is one way of reinterpreting the situation. In musical contexts, the first kind of problem has been called a "torsion problem" and the second kind "contorsion". | For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group the monzos |1 2 -3 1> and |-9 6 1 -1> for these generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the group. Hence (81/80)^2 is tempered out, but 81/80 isn't, and one gets two parallel meantones not connected by any 7-limit interval. Something similar happens with the two vals <12 19 28 34| and <26 41 60 72|; this however can be fixed in a way by extending to the 11-limit, and interpreting the "unobtainable" notes as notes reached in the 11-limit. Adding 245/242 to the commas (81/80 and 126/125) of septimal meantone is one way of reinterpreting the situation. In musical contexts, the first kind of problem has been called a "torsion problem" and the second kind "contorsion". | ||
Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to [[http://en.wikipedia.org/wiki/Smith_normal_form|Smith normal form]]. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such. | Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to [[http://en.wikipedia.org/wiki/Smith_normal_form|Smith normal form]]. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such. | ||
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To test for saturation, we may take the wedge product of the generators. Wedging <26 41 60 72| with <12 19 28 34| gives us <<2 8 20 8 26 24||; this is not zero, so the rank of the group these generate is two. However the coefficients have a gcd of two, and hence the group is not saturated; for saturation, the coefficients must be relatively prime, with a gcd of one.</pre></div> | To test for saturation, we may take the wedge product of the generators. Wedging <26 41 60 72| with <12 19 28 34| gives us <<2 8 20 8 26 24||; this is not zero, so the rank of the group these generate is two. However the coefficients have a gcd of two, and hence the group is not saturated; for saturation, the coefficients must be relatively prime, with a gcd of one.</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>Saturation</title></head><body>The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise | <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>Saturation</title></head><body>The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> of rank n. Its subgroups have the property of <em>saturation</em> if for any element a of Z^n, if an integer multiple m*a of a belongs to a sublattice V, then a already belongs to V. Another way to put it is that if some linear combination with rational coefficients q1*v1 + ... + qk*vk of elements of V belongs to Z^n, then it belongs to V. For the latter definition we consider Z^n to be contained in the n-dimensional <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">real vector space</a> R^n which is often called the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow">integer lattice</a>, or grid lattice.<br /> | ||
<br /> | <br /> | ||
If | If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a somewhat temperament in which all of the notes cannot be reached by tempered intervals.<br /> | ||
<br /> | <br /> | ||
For example, consider the temperament with commas generated by 126/125 and 3645/3584. The group the monzos |1 2 -3 1&gt; and |-9 6 1 -1&gt; for these generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the group. Hence (81/80)^2 is tempered out, but 81/80 isn't, and one gets two parallel meantones not connected by any 7-limit interval. Something similar happens with the two vals &lt;12 19 28 34| and &lt;26 41 60 72|; this however can be fixed in a way by extending to the 11-limit, and interpreting the &quot;unobtainable&quot; notes as notes reached in the 11-limit. Adding 245/242 to the commas (81/80 and 126/125) of septimal meantone is one way of reinterpreting the situation. In musical contexts, the first kind of problem has been called a &quot;torsion problem&quot; and the second kind &quot;contorsion&quot;.<br /> | For example, consider the &quot;temperament&quot; with commas generated by 126/125 and 3645/3584. The group the monzos |1 2 -3 1&gt; and |-9 6 1 -1&gt; for these generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the group. Hence (81/80)^2 is tempered out, but 81/80 isn't, and one gets two parallel meantones not connected by any 7-limit interval. Something similar happens with the two vals &lt;12 19 28 34| and &lt;26 41 60 72|; this however can be fixed in a way by extending to the 11-limit, and interpreting the &quot;unobtainable&quot; notes as notes reached in the 11-limit. Adding 245/242 to the commas (81/80 and 126/125) of septimal meantone is one way of reinterpreting the situation. In musical contexts, the first kind of problem has been called a &quot;torsion problem&quot; and the second kind &quot;contorsion&quot;.<br /> | ||
<br /> | <br /> | ||
Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smith_normal_form" rel="nofollow">Smith normal form</a>. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such.<br /> | Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smith_normal_form" rel="nofollow">Smith normal form</a>. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such.<br /> | ||