Mathematical theory of saturation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-10-29 11:59:07 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-10-29 12:00:11 UTC</tt>.

: The original revision id was <tt>~~377347028~~</tt>.

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<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 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 ~~scale~~, but disconnected. This is called a //contorsion// problem.

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 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 sytem of musical intervals, but disconnected. This has been called a //contorsion// problem.

For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group the [[monzo]]s |1 2 -3 1> and |-9 6 1 -1> 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 is not, and we have torsion. If we take the two vals <12 19 28 34| and <26 41 60 72| we similarly get contorsion. However, this 5- and 7-limit contorsion 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.

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<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 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 <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.

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 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 ~~scale~~, but disconnected. This is called a contorsion problem.

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 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 sytem of musical intervals, but disconnected. This has been called a contorsion problem.

For example, consider the &quot;temperament&quot; with commas generated by 126/125 and 3645/3584. The group the <a class="wiki_link" href="/monzo">monzo</a>s |1 2 -3 1&gt; and |-9 6 1 -1&gt; 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 is not, and we have torsion. If we take the two vals &lt;12 19 28 34| and &lt;26 41 60 72| we similarly get contorsion. However, this 5- and 7-limit contorsion 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.

@@ 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>2012-10-29 11:59:07 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-10-29 12:00:11 UTC</tt>.<br>
-: The original revision id was <tt>377347028</tt>.<br>
+: The original revision id was <tt>377347468</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>
@@ Line 8: / Line 8: @@
 <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 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 scale, but disconnected. This is called a //contorsion// problem.
+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 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 sytem of musical intervals, but disconnected. This has been called a //contorsion// problem.
 For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group the [[monzo]]s |1 2 -3 1&gt; and |-9 6 1 -1&gt; 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 is not, and we have torsion. If we take the two vals &lt;12 19 28 34| and &lt;26 41 60 72| we similarly get contorsion. However, this 5- and 7-limit contorsion 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.
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 <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 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 &lt;em&gt;are&lt;/em&gt; the same note. This is called a &lt;em&gt;torsion&lt;/em&gt; 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 scale, but disconnected. This is called a &lt;em&gt;contorsion&lt;/em&gt; problem.&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 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 &lt;em&gt;are&lt;/em&gt; the same note. This is called a &lt;em&gt;torsion&lt;/em&gt; 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 sytem of musical intervals, but disconnected. This has been called a &lt;em&gt;contorsion&lt;/em&gt; problem.&lt;br /&gt;
 &lt;br /&gt;
 For example, consider the &amp;quot;temperament&amp;quot; with commas generated by 126/125 and 3645/3584. The group the &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt;s |1 2 -3 1&amp;gt; and |-9 6 1 -1&amp;gt; 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 is not, and we have torsion. If we take the two vals &amp;lt;12 19 28 34| and &amp;lt;26 41 60 72| we similarly get contorsion. However, this 5- and 7-limit contorsion can be fixed in a way by extending to the 11-limit, and interpreting the &amp;quot;unobtainable&amp;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. &lt;br /&gt;