Normal forms: Difference between revisions

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<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>2010-05-~~14 21~~:11:52 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 01:58:58 UTC</tt>.<br>

: The original revision id was <tt>~~142155205~~</tt>.<br>

: The original revision id was <tt>142172347</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>

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The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because abstract regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can be also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal [[Comma sequences|comma sequence]] of septimal meantone.

==Normal val lists==

If L is a list of n vals, we may write it as an nxm mAtrix, where the rows of the matrix are the vals, and m = pi(p), where p is the prime limit. To get the normal val list, we do the following:

(1) Hermite reduce the matrix for L

(2) Throw away all rows which consist of nothing but zeros

(3) In the resulting kxm matrix, take the first k rows and columns, and test if the matrix is singular

(4) If it is, return the rows of the kxm matrix as the normalized list (in practice you would need to be doing something unusual for this to happen, and at that point you should probably work matters out for yourself.) If it is nonsingular, invert the matrix

and multiply the inverse matrix from the left by the row vector

[1 log2(3) log2(5) ... log2(p)]. If the ith entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.

The point of steps three and four is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively.) Just as a normal comma list can be used to classify an abstract regular temperament, so can a normal val list. The val list is what on [[Graham Breed]]'s [[http://x31eq.com/temper/|web site]] is called a "mapping", put into a canonical form.

</pre></div>

<h4>Original HTML content:</h4>

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(9) For any number q &lt; 1 on this list, replace q with 1/q<br />

<br />

The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because abstract regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can be also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal <a class="wiki_link" href="/Comma%20sequences">comma sequence</a> of septimal meantone.</body></html></pre></div>

The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because abstract regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can be also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal <a class="wiki_link" href="/Comma%20sequences">comma sequence</a> of septimal meantone.<br />

<br />

<h2 id="toc1"><a name="x-Normal val lists"></a>Normal val lists</h2>

<br />

If L is a list of n vals, we may write it as an nxm mAtrix, where the rows of the matrix are the vals, and m = pi(p), where p is the prime limit. To get the normal val list, we do the following:<br />

<br />

(1) Hermite reduce the matrix for L<br />

<br />

(2) Throw away all rows which consist of nothing but zeros<br />

<br />

(3) In the resulting kxm matrix, take the first k rows and columns, and test if the matrix is singular<br />

<br />

(4) If it is, return the rows of the kxm matrix as the normalized list (in practice you would need to be doing something unusual for this to happen, and at that point you should probably work matters out for yourself.) If it is nonsingular, invert the matrix<br />

and multiply the inverse matrix from the left by the row vector<br />

[1 log2(3) log2(5) ... log2(p)]. If the ith entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.<br />

<br />

The point of steps three and four is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively.) Just as a normal comma list can be used to classify an abstract regular temperament, so can a normal val list. The val list is what on <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">web site</a> is called a &quot;mapping&quot;, put into a canonical form.</body></html></pre></div>

@@ 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>2010-05-14 21:11:52 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-15 01:58:58 UTC</tt>.<br>
-: The original revision id was <tt>142155205</tt>.<br>
+: The original revision id was <tt>142172347</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 52: / Line 52: @@
 The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because abstract regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can be also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal [[Comma sequences|comma sequence]] of septimal meantone.
+==Normal val lists==
+If L is a list of n vals, we may write it as an nxm mAtrix, where the rows of the matrix are the vals, and m = pi(p), where p is the prime limit. To get the normal val list, we do the following:
+(1) Hermite reduce the matrix for L
+(2) Throw away all rows which consist of nothing but zeros
+(3) In the resulting kxm matrix, take the first k rows and columns, and test if the matrix is singular
+(4) If it is, return the rows of the kxm matrix as the normalized list (in practice you would need to be doing something unusual for this to happen, and at that point you should probably work matters out for yourself.) If it is nonsingular, invert the matrix
+and multiply the inverse matrix from the left by the row vector
+[1 log2(3) log2(5) ... log2(p)]. If the ith entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.
+The point of steps three and four is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively.) Just as a normal comma list can be used to classify an abstract regular temperament, so can a normal val list. The val list is what on [[Graham Breed]]'s [[http://x31eq.com/temper/|web site]]  is called a "mapping", put into a canonical form.
 </pre></div>
 <h4>Original HTML content:</h4>
@@ Line 100: / Line 113: @@
 (9) For any number q &amp;lt; 1 on this list, replace q with 1/q&lt;br /&gt;
 &lt;br /&gt;
-The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because abstract regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can be also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal &lt;a class="wiki_link" href="/Comma%20sequences"&gt;comma sequence&lt;/a&gt; of septimal meantone.&lt;/body&gt;&lt;/html&gt;</pre></div>
+The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because abstract regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can be also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal &lt;a class="wiki_link" href="/Comma%20sequences"&gt;comma sequence&lt;/a&gt; of septimal meantone.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Normal val lists"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Normal val lists&lt;/h2&gt;
+&lt;br /&gt;
+If L is a list of n vals, we may write it as an nxm mAtrix, where the rows of the matrix are the vals, and m = pi(p), where p is the prime limit. To get the normal val list, we do the following:&lt;br /&gt;
+&lt;br /&gt;
+(1) Hermite reduce the matrix for L&lt;br /&gt;
+&lt;br /&gt;
+(2) Throw away all rows which consist of nothing but zeros&lt;br /&gt;
+&lt;br /&gt;
+(3) In the resulting kxm matrix, take the first k rows and columns, and test if the matrix is singular&lt;br /&gt;
+&lt;br /&gt;
+(4) If it is, return the rows of the kxm matrix as the normalized list (in practice you would need to be doing something unusual for this to happen, and at that point you should probably work matters out for yourself.) If it is nonsingular, invert the matrix&lt;br /&gt;
+and multiply the inverse matrix from the left by the row vector&lt;br /&gt;
+[1 log2(3) log2(5) ... log2(p)]. If the ith entry in the result is negative, multiply the corresponding val by -1. Return the result as the normalized val list.&lt;br /&gt;
+&lt;br /&gt;
+The point of steps three and four is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively.) Just as a normal comma list can be used to classify an abstract regular temperament, so can a normal val list. The val list is what on &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow"&gt;web site&lt;/a&gt;  is called a &amp;quot;mapping&amp;quot;, put into a canonical form.&lt;/body&gt;&lt;/html&gt;</pre></div>