Mathematical theory of saturation: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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 22:01:40 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-30 22:03:03 UTC</tt>.<br>

: The original revision id was <tt>~~197297634~~</tt>.<br>

: The original revision id was <tt>197297966</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>

<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 which 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">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, 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.

Line 18:

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>

<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 />

<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 />

<br />

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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 22:01:40 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-30 22:03:03 UTC</tt>.<br>
-: The original revision id was <tt>197297634</tt>.<br>
+: The original revision id was <tt>197297966</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>
 <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 which 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">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, 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.
@@ Line 18: / Line 18: @@
 To test for saturation, we may take the wedge product of the generators. Wedging &lt;26 41 60 72| with &lt;12 19 28 34| gives us &lt;&lt;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>
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Saturation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow"&gt;free abelian group&lt;/a&gt; of rank n. Its subgroups have the property of &lt;em&gt;saturation&lt;/em&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow"&gt;real vector space&lt;/a&gt; R^n which is often called the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow"&gt;integer lattice&lt;/a&gt;, or grid lattice.&lt;br /&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Saturation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow"&gt;free abelian group&lt;/a&gt; of rank n. Its subgroups have the property of &lt;em&gt;saturation&lt;/em&gt; 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow"&gt;real vector space&lt;/a&gt; R^n, in which case Z^n is often called the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow"&gt;integer lattice&lt;/a&gt;, or grid lattice.&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;br /&gt;