Mathematical theory of saturation: Difference between revisions
Wikispaces>clumma **Imported revision 457893104 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509653666 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:46:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509653666</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">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, in which case Z^n is often called the [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]], or grid lattice. | <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"> | ||
[[image:mathhazard.jpg align="center"]] | |||
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, in which case Z^n is often called the [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]], or grid lattice. | |||
If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession //are// the same note. This is called a //torsion// problem. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals; this at least is an actual system of musical intervals, but disconnected. This has been called a //contorsion// problem. | If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession //are// the same note. This is called a //torsion// problem. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals; this at least is an actual system of musical intervals, but disconnected. This has been called a //contorsion// problem. | ||
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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 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, in which case Z^n 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 /> | <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><br /> | ||
<!-- ws:start:WikiTextLocalImageRule:0:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div><!-- ws:end:WikiTextLocalImageRule:0 -->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, in which case Z^n 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 /> | |||
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If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession <em>are</em> the same note. This is called a <em>torsion</em> problem. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals; this at least is an actual system of musical intervals, but disconnected. This has been called a <em>contorsion</em> problem.<br /> | If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession <em>are</em> the same note. This is called a <em>torsion</em> problem. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals; this at least is an actual system of musical intervals, but disconnected. This has been called a <em>contorsion</em> problem.<br /> | ||