The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 217986674 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 217990126 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2011-04-07 02: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 02:28:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>217990126</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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[[math]] | [[math]] | ||
This is a [[http://en.wikipedia.org/wiki/Uniform_convergence|uniformly convergent]] series of continuous functions, and hence is continuous. We can define essentially the same function by subtracting it from E_s( | This is a [[http://en.wikipedia.org/wiki/Uniform_convergence|uniformly convergent]] series of continuous functions, and hence is continuous. We can define essentially the same function by subtracting it from E_s(1/2)/2: | ||
[[math]] | [[math]] | ||
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F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2) | F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2) | ||
[[math]] | [[math]] | ||
If we take exponentials of both sides, then | |||
[[math]] | |||
\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)| | |||
[[math]] | |||
so that we see that the absolute value of the zeta function serves to measure error of an equal division. | |||
</pre></div> | </pre></div> | ||
<h4>Original HTML content:</h4> | <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>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule: | <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>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule:9:&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:9 --><!-- ws:start:WikiTextTocRule:10: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:7 -->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 - 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 /> | 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 - 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 /> | <br /> | ||
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--><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | --><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <br /> | ||
This is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Uniform_convergence" rel="nofollow">uniformly convergent</a> series of continuous functions, and hence is continuous. We can define essentially the same function by subtracting it from E_s( | This is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Uniform_convergence" rel="nofollow">uniformly convergent</a> series of continuous functions, and hence is continuous. We can define essentially the same function by subtracting it from E_s(1/2)/2:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextMathRule:4: | <!-- ws:start:WikiTextMathRule:4: | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)&lt;br/&gt;[[math]] | F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)&lt;br/&gt;[[math]] | ||
--><script type="math/tex">F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</script><!-- ws:end:WikiTextMathRule:5 --></body></html></pre></div> | --><script type="math/tex">F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</script><!-- ws:end:WikiTextMathRule:5 --><br /> | ||
<br /> | |||
If we take exponentials of both sides, then<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextMathRule:6: | |||
[[math]]&lt;br/&gt; | |||
\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|</script><!-- ws:end:WikiTextMathRule:6 --><br /> | |||
<br /> | |||
so that we see that the absolute value of the zeta function serves to measure error of an equal division.</body></html></pre></div> | |||