Xenharmonic Wiki

The Riemann zeta function and tuning: Difference between revisions

← Older editNewer edit →
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 296849634 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 296850066 - Original comment: **
Line 1: Line 1:
<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>2012-01-30 21:36:31 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-30 21:38:13 UTC</tt>.<br>
: The original revision id was <tt>296849634</tt>.<br>
: The original revision id was <tt>296850066</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
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>
Line 114: Line 114:


=Zeta EDO lists=  
=Zeta EDO lists=  
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 [[1edo|1]], [[2edo|2]], [[3edo|3]], [[4edo|4]], [[5edo|5]], [[7edo|7]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[99edo|99]], [[118edo|118]], [[130edo|130]], [[152edo|152]], [[171edo|171]], [[217edo|217]], [[224edo|224]], [[270edo|270]], [[342edo|342]], [[422edo|422]], [[441edo|441]], [[494edo|494]], [[742edo|742]], [[764edo|764]], [[935edo|935]], [[954edo|954]], [[1012edo|1012]], [[1106edo|1106]], [[1178edo|1178]], [[1236edo|1236]], [[1395edo|1395]], [[1448edo|1448]], [[1578edo|1578]], [[246oedo|2460]], [[2684edo|2684]], [[3395edo|3395]], [[5585edo|5585]], [[6079edo|6079]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[11664edo|11664]] ... of //zeta peak edos//. This is listed in the On-Line Encyclopedia of Integer Sequences as [[http://oeis.org/A117536|sequence A117536]].
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 [[1edo|1]], [[2edo|2]], [[3edo|3]], [[4edo|4]], [[5edo|5]], [[7edo|7]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[99edo|99]], [[118edo|118]], [[130edo|130]], [[152edo|152]], [[171edo|171]], [[217edo|217]], [[224edo|224]], [[270edo|270]], [[342edo|342]], [[422edo|422]], [[441edo|441]], [[494edo|494]], [[742edo|742]], [[764edo|764]], [[935edo|935]], [[954edo|954]], [[1012edo|1012]], [[1106edo|1106]], [[1178edo|1178]], [[1236edo|1236]], [[1395edo|1395]], [[1448edo|1448]], [[1578edo|1578]], [[2460edo|2460]], [[2684edo|2684]], [[3395edo|3395]], [[5585edo|5585]], [[6079edo|6079]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[11664edo|11664]] ... 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 [[2edo|2]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[130edo|130]], [[171edo|171]], [[224edo|224]], [[270edo|270]], [[764edo|764]], [[954edo|954]], [[1178edo|1178]], [[1395edo|1395]], [[1578edo|1578]], [[2684edo|2684]], [[3395edo|3395]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[14348edo|14348]], [[16808edo|16808]], [[36269edo|36269]], [[58973edo|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.
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 [[2edo|2]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[130edo|130]], [[171edo|171]], [[224edo|224]], [[270edo|270]], [[764edo|764]], [[954edo|954]], [[1178edo|1178]], [[1395edo|1395]], [[1578edo|1578]], [[2684edo|2684]], [[3395edo|3395]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[14348edo|14348]], [[16808edo|16808]], [[36269edo|36269]], [[58973edo|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.
Line 276: Line 276:
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Zeta EDO lists"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Zeta EDO lists&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Zeta EDO lists"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Zeta EDO lists&lt;/h1&gt;
  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 &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;s &lt;a class="wiki_link" href="/1edo"&gt;1&lt;/a&gt;, &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/217edo"&gt;217&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/342edo"&gt;342&lt;/a&gt;, &lt;a class="wiki_link" href="/422edo"&gt;422&lt;/a&gt;, &lt;a class="wiki_link" href="/441edo"&gt;441&lt;/a&gt;, &lt;a class="wiki_link" href="/494edo"&gt;494&lt;/a&gt;, &lt;a class="wiki_link" href="/742edo"&gt;742&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764&lt;/a&gt;, &lt;a class="wiki_link" href="/935edo"&gt;935&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1012edo"&gt;1012&lt;/a&gt;, &lt;a class="wiki_link" href="/1106edo"&gt;1106&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1236edo"&gt;1236&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1448edo"&gt;1448&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/246oedo"&gt;2460&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/5585edo"&gt;5585&lt;/a&gt;, &lt;a class="wiki_link" href="/6079edo"&gt;6079&lt;/a&gt;, &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt;, &lt;a class="wiki_link" href="/11664edo"&gt;11664&lt;/a&gt; ... of &lt;em&gt;zeta peak edos&lt;/em&gt;. This is listed in the On-Line Encyclopedia of Integer Sequences as &lt;a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow"&gt;sequence A117536&lt;/a&gt;.&lt;br /&gt;
  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 &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;s &lt;a class="wiki_link" href="/1edo"&gt;1&lt;/a&gt;, &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/217edo"&gt;217&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/342edo"&gt;342&lt;/a&gt;, &lt;a class="wiki_link" href="/422edo"&gt;422&lt;/a&gt;, &lt;a class="wiki_link" href="/441edo"&gt;441&lt;/a&gt;, &lt;a class="wiki_link" href="/494edo"&gt;494&lt;/a&gt;, &lt;a class="wiki_link" href="/742edo"&gt;742&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764&lt;/a&gt;, &lt;a class="wiki_link" href="/935edo"&gt;935&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1012edo"&gt;1012&lt;/a&gt;, &lt;a class="wiki_link" href="/1106edo"&gt;1106&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1236edo"&gt;1236&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1448edo"&gt;1448&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/2460edo"&gt;2460&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/5585edo"&gt;5585&lt;/a&gt;, &lt;a class="wiki_link" href="/6079edo"&gt;6079&lt;/a&gt;, &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt;, &lt;a class="wiki_link" href="/11664edo"&gt;11664&lt;/a&gt; ... of &lt;em&gt;zeta peak edos&lt;/em&gt;. This is listed in the On-Line Encyclopedia of Integer Sequences as &lt;a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow"&gt;sequence A117536&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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 &lt;em&gt;zeta integral edos&lt;/em&gt;, goes &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt;, &lt;a class="wiki_link" href="/14348edo"&gt;14348&lt;/a&gt;, &lt;a class="wiki_link" href="/16808edo"&gt;16808&lt;/a&gt;, &lt;a class="wiki_link" href="/36269edo"&gt;36269&lt;/a&gt;, &lt;a class="wiki_link" href="/58973edo"&gt;58973&lt;/a&gt; ... This is listed in the OEIS as &lt;a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow"&gt;sequence A117538&lt;/a&gt;. 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.&lt;br /&gt;
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 &lt;em&gt;zeta integral edos&lt;/em&gt;, goes &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt;, &lt;a class="wiki_link" href="/14348edo"&gt;14348&lt;/a&gt;, &lt;a class="wiki_link" href="/16808edo"&gt;16808&lt;/a&gt;, &lt;a class="wiki_link" href="/36269edo"&gt;36269&lt;/a&gt;, &lt;a class="wiki_link" href="/58973edo"&gt;58973&lt;/a&gt; ... This is listed in the OEIS as &lt;a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow"&gt;sequence A117538&lt;/a&gt;. 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.&lt;br /&gt;
Retrieved from "https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning"