The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 217958302 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 217967382 - 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>2011-04-06 23: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-06 23:51:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>217967382</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]] | ||
\sum_2^p (\frac{\|x \log_2 q\|}{\ | \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2 | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\sum_2^ | \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s} | ||
[[math]] | [[math]] | ||
If s is greater than one, this does converge. </pre></div> | If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting used logarithms and error measures are consistent, then the logarithmic weighting canceled this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[http://en.wikipedia.org/wiki/Von_Mangoldt_function|Von Mangoldt function]], which is equal to ln p on prime powers p^n, and is zero elsewhere. This is writen with a capital lambda, and in terms of it we can include prime powers in our error function as | ||
[[math]] | |||
\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s} | |||
[[math]] | |||
where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.</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:5:&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:5 --><!-- ws:start:WikiTextTocRule:6: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:3 -->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 /> | ||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\sum_2^p (\frac{\|x \log_2 q\|}{\ | \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q\|}{\ | --><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2</script><!-- ws:end:WikiTextMathRule:0 --> <br /> | ||
<br /> | <br /> | ||
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:<br /> | Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:<br /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\sum_2^ | \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 | --><script type="math/tex"> \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}</script><!-- ws:end:WikiTextMathRule:1 --> <br /> | ||
<br /> | |||
If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting used logarithms and error measures are consistent, then the logarithmic weighting canceled this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow">Von Mangoldt function</a>, which is equal to ln p on prime powers p^n, and is zero elsewhere. This is writen with a capital lambda, and in terms of it we can include prime powers in our error function as<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextMathRule:2: | |||
[[math]]&lt;br/&gt; | |||
\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | |||
<br /> | <br /> | ||
where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.</body></html></pre></div> | |||