The Riemann zeta function and tuning
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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:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- 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:<h1> --><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]]<br/>
\sum_2^p (\frac{\|x \log_2 q}{\ln q})^2<br/>[[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]]<br/>
\frac{b}{N} - \log_2 q <br/>[[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>