The Riemann zeta function and tuning: Difference between revisions
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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> | ||
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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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;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]] | ||
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=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 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 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. | ||
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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"><html><head><title>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule:31:&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:31 --><!-- ws:start:WikiTextTocRule:32: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Into the critical strip">Into the critical strip</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#The Z function">The Z function</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Zeta EDO lists">Zeta EDO lists</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Removing primes">Removing primes</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#The Black Magic Formulas">The Black Magic Formulas</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Computing zeta">Computing zeta</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <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:31:&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:31 --><!-- ws:start:WikiTextTocRule:32: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Into the critical strip">Into the critical strip</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#The Z function">The Z function</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Zeta EDO lists">Zeta EDO lists</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Removing primes">Removing primes</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#The Black Magic Formulas">The Black Magic Formulas</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Computing zeta">Computing zeta</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | ||
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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 &quot;nonoctave&quot; divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the &quot;octave&quot; (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.<br /> | 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 &quot;nonoctave&quot; divisions as well. The Bohlen-Pierce scale, 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the &quot;octave&quot; (although the octave itself does not appear in this tuning), and would hence be represented by a value of x = 8.202.<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(2πx/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(2πx/ln(2)) in the region around 12edo:<br /> | ||
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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 /> | ||
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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.<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 /> | ||