The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 220881044 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-26 14:59:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238824469</tt>.<br> | ||
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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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Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +infinity. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3. | Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +infinity. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3. | ||
=Zeta integral EDOs= | |||
If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[edo]]s 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494 ... of //zeta peak edos//. This is listed in the | |||
On-Line Encyclopedia of Integer Sequences as [[http://oeis.org/A117536|sequence A117536]]. | |||
Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the //zeta integral edos//, goes 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973 ... This is listed in the OEIS as [[http://oeis.org/A117538|sequence A117538]]. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes. | |||
Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta zero gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line. | |||
=Computing zeta= | =Computing zeta= | ||
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[[http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/|Selberg's limit theorem]] by Terence Tao [[http://www.webcitation.org/5xrvgjW6T|Permalink]]</pre></div> | [[http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/|Selberg's limit theorem]] by Terence Tao [[http://www.webcitation.org/5xrvgjW6T|Permalink]]</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:24:&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:24 --><!-- ws:start:WikiTextTocRule:25: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Into the critical strip">Into the critical strip</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#The Z function">The Z function</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Zeta integral EDOs">Zeta integral EDOs</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Computing zeta">Computing zeta</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 function 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 function 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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If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2 pi x /ln(2)) in the region around 12edo:<br /> | If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2 pi x /ln(2)) in the region around 12edo:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:32:&lt;img src=&quot;/file/view/plot12.png/219376858/plot12.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot12.png/219376858/plot12.png" alt="plot12.png" title="plot12.png" /><!-- ws:end:WikiTextLocalImageRule:32 --><br /> | ||
<br /> | <br /> | ||
The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/x, and hence the size of the octave in the zeta peak value tuning for Nedo is N/x; if x is slightly larger than N as here with N=12, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with x less than N, we have stretched octaves.<br /> | The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/x, and hence the size of the octave in the zeta peak value tuning for Nedo is N/x; if x is slightly larger than N as here with N=12, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with x less than N, we have stretched octaves.<br /> | ||
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For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:<br /> | For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:33:&lt;img src=&quot;/file/view/plot270.png/219383970/plot270.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot270.png/219383970/plot270.png" alt="plot270.png" title="plot270.png" /><!-- ws:end:WikiTextLocalImageRule:33 --><br /> | ||
<br /> | <br /> | ||
Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +infinity. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.<br /> | Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +infinity. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc3"><a name="Computing zeta"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc3"><a name="Zeta integral EDOs"></a><!-- ws:end:WikiTextHeadingRule:18 -->Zeta integral EDOs</h1> | ||
If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of <a class="wiki_link" href="/edo">edo</a>s 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494 ... of <em>zeta peak edos</em>. This is listed in the <br /> | |||
On-Line Encyclopedia of Integer Sequences as <a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow">sequence A117536</a>. <br /> | |||
<br /> | |||
Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the <em>zeta integral edos</em>, goes 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973 ... This is listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow">sequence A117538</a>. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.<br /> | |||
<br /> | |||
Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of <em>zeta zero gap edos</em>. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow">sequence A117537</a>. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc4"><a name="Computing zeta"></a><!-- ws:end:WikiTextHeadingRule:20 -->Computing zeta</h1> | |||
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow">Dirichlet eta function</a> which was introduced to mathematics by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow">Johann Peter Gustav Lejeune Dirichlet</a>, who despite his name was a German and the brother-in-law of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow">Felix Mendelssohn</a>.<br /> | There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow">Dirichlet eta function</a> which was introduced to mathematics by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow">Johann Peter Gustav Lejeune Dirichlet</a>, who despite his name was a German and the brother-in-law of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow">Felix Mendelssohn</a>.<br /> | ||
<br /> | <br /> | ||
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The Dirichlet series for the zeta function is absolutely convergent when s&gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler summation</a>.<br /> | The Dirichlet series for the zeta function is absolutely convergent when s&gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler summation</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc5"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:22 -->Links</h1> | ||
<a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow">X-Ray of Riemann zeta-function</a> by Juan Arias-de-Reyna<br /> | <a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow">X-Ray of Riemann zeta-function</a> by Juan Arias-de-Reyna<br /> | ||
<a class="wiki_link_ext" href="http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/" rel="nofollow">Selberg's limit theorem</a> by Terence Tao <a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow">Permalink</a></body></html></pre></div> | <a class="wiki_link_ext" href="http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/" rel="nofollow">Selberg's limit theorem</a> by Terence Tao <a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow">Permalink</a></body></html></pre></div> | ||