The Riemann zeta function and tuning

From Xenharmonic Wiki
Revision as of 23:04, 6 April 2011 by Wikispaces>genewardsmith (**Imported revision 217958302 - Original comment: **)
Jump to navigation Jump to search

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2011-04-06 23:04:04 UTC.
The original revision id was 217958302.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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 - 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]] 

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:

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

If s is greater than one, this does converge.

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 - 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 />
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 />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
 \sum_2^p (\frac{\|x \log_2 q\|}{q^s})^2&lt;br/&gt;[[math]]
 --><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q\|}{q^s})^2</script><!-- ws:end:WikiTextMathRule:1 --> <br />
<br />
If s is greater than one, this does converge.</body></html>
Retrieved from "https://en.xen.wiki/index.php?title=The_Riemann_zeta_function_and_tuning&oldid=27733"