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The Riemann zeta function and tuning: Difference between revisions

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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>
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-26 20:57:24 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-15 21:57:25 UTC</tt>.<br>
: The original revision id was <tt>314861008</tt>.<br>
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<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
=Preliminaries=  
=Preliminaries=  
Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves.
Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that x can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.
Suppose that x can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.


Now suppose that ||x|| denotes the difference between x and the integer nearest to x. For example, ||8.202|| would be .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function x - floor(x+1/2).
Now suppose that ||x|| denotes the difference between x and the integer nearest to x. For example, ||8.202|| would be .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function x - floor(x+1/2).
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<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Riemann Zeta Function and Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:28:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Into the critical strip"&gt;Into the critical strip&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#The Z function"&gt;The Z function&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Zeta EDO lists"&gt;Zeta EDO lists&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Removing primes"&gt;Removing primes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Computing zeta"&gt;Computing zeta&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Riemann Zeta Function and Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:28:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Into the critical strip"&gt;Into the critical strip&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#The Z function"&gt;The Z function&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Zeta EDO lists"&gt;Zeta EDO lists&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Removing primes"&gt;Removing primes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Computing zeta"&gt;Computing zeta&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;
&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Preliminaries"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Preliminaries&lt;/h1&gt;
&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Preliminaries"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Preliminaries&lt;/h1&gt;
  Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves.&lt;br /&gt;
  Suppose x is a variable representing some equal division of the octave. For example, if x = 80, x reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that x can also be continuous, so that it can also represent fractional or &amp;quot;nonoctave&amp;quot; divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the &amp;quot;octave&amp;quot; (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.&lt;br /&gt;
Suppose that x can also be continuous, so that it can also represent fractional or &amp;quot;nonoctave&amp;quot; divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the &amp;quot;octave&amp;quot; (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Now suppose that ||x|| denotes the difference between x and the integer nearest to x. For example, ||8.202|| would be .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function x - floor(x+1/2).&lt;br /&gt;
Now suppose that ||x|| denotes the difference between x and the integer nearest to x. For example, ||8.202|| would be .202, since it's the difference between 8.202 and the nearest integer, which is 8. ||7.95|| would be .05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, ||x|| denotes the function x - floor(x+1/2).&lt;br /&gt;
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