The Riemann zeta function and tuning: Difference between revisions

Revision as of 21:09, 6 April 2011

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This revision was by author genewardsmith and made on 2011-04-06 21:09:24 UTC.

The original revision id was 217929752.

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Original Wikitext content:

[[toc|flat]]
=Preliminaries= 
Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the x minus floor(x+1/2), then the [[Tenney-Euclidean metrics|Tenney-Euclidean error]] for the [[p-limit]] [[val]] obtained by rounding log2(q)*x to the nearest integer for each prime q up to p will be

[[math]]
 \sum_2^p (\frac{\|x \log_2 q}{\ln q})^2
[[math]] 

where E(q) is the error 

[[math]]
 \frac{b}{N} - \log_2 q 
[[math]]

of the [[patent val]] tuning, meaning the nearest to q, of the prime q, and the sum is over all primes up to p.

Original HTML content:

<html><head><title>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule:4:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: -->
<!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:2 -->Preliminaries</h1>
 Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the x minus floor(x+1/2), then the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for the <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> obtained by rounding log2(q)*x to the nearest integer for each prime q up to p will be<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
 \sum_2^p (\frac{\|x \log_2 q}{\ln q})^2&lt;br/&gt;[[math]]
 --><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q}{\ln q})^2</script><!-- ws:end:WikiTextMathRule:0 --> <br />
<br />
where E(q) is the error <br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
 \frac{b}{N} - \log_2 q &lt;br/&gt;[[math]]
 --><script type="math/tex"> \frac{b}{N} - \log_2 q </script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
of the <a class="wiki_link" href="/patent%20val">patent val</a> tuning, meaning the nearest to q, of the prime q, and the sum is over all primes up to p.</body></html>

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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:dguskin|dguskin]] and made on <tt>2011-04-06 20:33:29 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-06 21:09:24 UTC</tt>.<br>
-: The original revision id was <tt>217920262</tt>.<br>
+: The original revision id was <tt>217929752</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">[[toc|flat]]
 =Preliminaries=
-Consider to start out with [[Tenney-Euclidean metrics|Tenney-Euclidean error]]. For some [[equal]] division N in the [[p-limit]], this can be defined as the square root of the quantity
+Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the x minus floor(x+1/2), then the [[Tenney-Euclidean metrics|Tenney-Euclidean error]] for the [[p-limit]] [[val]] obtained by rounding log2(q)*x to the nearest integer for each prime q up to p will be
 [[math]]
-  \sum_2^p (\frac{E(q)}{\ln q})^2
+  \sum_2^p (\frac{\|x \log_2 q}{\ln q})^2
 [[math]]
@@ Line 24: / Line 24: @@
 <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;The Riemann Zeta Function and Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:4:&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:4 --&gt;&lt;!-- ws:start:WikiTextTocRule:5: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:5 --&gt;&lt;!-- ws:start:WikiTextTocRule:6: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Preliminaries"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Preliminaries&lt;/h1&gt;
-  Consider to start out with &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean error&lt;/a&gt;. For some &lt;a class="wiki_link" href="/equal"&gt;equal&lt;/a&gt; division N in the &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt;, this can be defined as the square root of the quantity&lt;br /&gt;
+  Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the x minus floor(x+1/2), then the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean error&lt;/a&gt; for the &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; obtained by rounding log2(q)*x to the nearest integer for each prime q up to p will be&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
-  \sum_2^p (\frac{E(q)}{\ln q})^2&amp;lt;br/&amp;gt;[[math]]
+  \sum_2^p (\frac{\|x \log_2 q}{\ln q})^2&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt; \sum_2^p (\frac{E(q)}{\ln q})^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt; &lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt; \sum_2^p (\frac{\|x \log_2 q}{\ln q})^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt; &lt;br /&gt;
 &lt;br /&gt;
 where E(q) is the error &lt;br /&gt;

The Riemann zeta function and tuning: Difference between revisions

Revision as of 21:09, 6 April 2011

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