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 - 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\|}{\log_2 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^\infty \frac{\|x \log_2 q\|^2}{q^s}
[[math]]
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]], an [[http://en.wikipedia.org/wiki/Arithmetic_function|arithmetic function]] on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using 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.
Another consequence of the above definition which might be objected to is that it results in a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous function]], whereas a continuous and in fact a smooth function might be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s > 1
[[math]]
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}
[[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(0):
[[math]]
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}
[[math]]
This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the [[http://en.wikipedia.org/wiki/Riemann_zeta_function|Riemann zeta function]]:
[[math]]
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)
[[math]]
Original HTML content:
<html><head><title>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule:8:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:6 -->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]]<br/>
\sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2<br/>[[math]]
--><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 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]]<br/>
\sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}<br/>[[math]]
--><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>, an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_function" rel="nofollow">arithmetic function</a> on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using 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]]<br/>
\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}<br/>[[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 />
where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.<br />
<br />
Another consequence of the above definition which might be objected to is that it results in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous function</a>, whereas a continuous and in fact a smooth function might be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow">entire</a> function. Let us therefore now define for any s > 1<br />
<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]<br/>
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}<br/>[[math]]
--><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 />
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(0):<br />
<br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]<br/>
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}<br/>[[math]]
--><script type="math/tex">F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:4 --><br />
<br />
This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Riemann zeta function</a>:<br />
<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]<br/>
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)<br/>[[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>