Tp tuning: 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>2012-08-15 13:41:15 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-15 13:45:48 UTC</tt>.<br>

: The original revision id was <tt>~~357975036~~</tt>.<br>

: The original revision id was <tt>357975890</tt>.<br>

: The revision comment was: <tt></tt><br>

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=L2 tuning=

In the special case where p = 2, the Tp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is ~~L2 tuning, or TE~~ tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE%20error|TE error]].

In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the T2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE%20error|TE error]].

For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the L2 tuning map is <1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a ~~POL2~~ tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting <1200 1200 1200 1200 1200| gives <-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.</pre></div>

For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 tuning map is <1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting <1200 1200 1200 1200 1200| gives <-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.</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>Tp tuning</title></head><body><a href="#Definition">Definition</a> | <a href="#Dual norm">Dual norm</a> | <a href="#Applying the Hahn-Banach theorem">Applying the Hahn-Banach theorem</a> | <a href="#L2 tuning">L2 tuning</a>

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<br />

<h1 id="toc3"><a name="L2 tuning"></a>L2 tuning</h1>

In the special case where p = 2, the Tp norm becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is ~~L2 tuning, or TE~~ tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE%20error">TE error</a>.<br />

In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the T2 error, which is E2(S) for the temperament S, and which is approximately proportional to <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE%20error">TE error</a>.<br />

<br />

For an example, consider <a class="wiki_link" href="/Chromatic%20pairs#Indium">indium temperament</a>, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">usual methods</a>, in particular the pseudoinverse, we find that the L2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a ~~POL2~~ tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.</body></html></pre></div>

For an example, consider <a class="wiki_link" href="/Chromatic%20pairs#Indium">indium temperament</a>, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">usual methods</a>, in particular the pseudoinverse, we find that the T2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.</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>2012-08-15 13:41:15 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-15 13:45:48 UTC</tt>.<br>
-: The original revision id was <tt>357975036</tt>.<br>
+: The original revision id was <tt>357975890</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 31: / Line 31: @@
 =L2 tuning=
-In the special case where p = 2, the Tp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE%20error|TE error]].
+In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the T2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament measures#TE%20error|TE error]].
-For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the L2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POL2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.</pre></div>
+For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.</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;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:10:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt; | &lt;a href="#Applying the Hahn-Banach theorem"&gt;Applying the Hahn-Banach theorem&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#L2 tuning"&gt;L2 tuning&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;
@@ Line 62: / Line 62: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="L2 tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;L2 tuning&lt;/h1&gt;
-  In the special case where p = 2, the Tp norm becomes the L2 norm, which is the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean&lt;/a&gt; norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE%20error"&gt;TE error&lt;/a&gt;.&lt;br /&gt;
+  In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which is the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean&lt;/a&gt; norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the T2 error, which is E2(S) for the temperament S, and which is approximately proportional to &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE%20error"&gt;TE error&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-For an example, consider &lt;a class="wiki_link" href="/Chromatic%20pairs#Indium"&gt;indium temperament&lt;/a&gt;, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the &lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;usual methods&lt;/a&gt;, in particular the pseudoinverse, we find that the L2 tuning map is &amp;lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POL2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &amp;lt;1200 1200 1200 1200 1200| gives &amp;lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.&lt;/body&gt;&lt;/html&gt;</pre></div>
+For an example, consider &lt;a class="wiki_link" href="/Chromatic%20pairs#Indium"&gt;indium temperament&lt;/a&gt;, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the &lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;usual methods&lt;/a&gt;, in particular the pseudoinverse, we find that the T2 tuning map is &amp;lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &amp;lt;1200 1200 1200 1200 1200| gives &amp;lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives an error of 1.7414 cents.&lt;/body&gt;&lt;/html&gt;</pre></div>