The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2016-01-~~31 07~~:48:28 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-05-28 11:28:32 UTC</tt>.<br>

: The original revision id was <tt>~~573473057~~</tt>.<br>

: The original revision id was <tt>584283017</tt>.<br>

: The revision comment was: <tt>~~removed surplus "the", adjusted link text (I hope so)~~</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>

<h4>Original Wikitext content:</h4>

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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 "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)|.

For any value of x, we can construct a p-limit [[patent val]]. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. ~~The square of~~ the ~~[[Tenney-Euclidean metrics|Tenney-Euclidean error]] for this val will be~~

For any value of x, we can construct a p-limit [[patent val|generalized papent val]]. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. Now consider the function

[[math]]

\sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2

\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2

[[math]]

This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the [[Tenney-Euclidean tuning]]s of the octaves of the associated vals, while xi for these minima is the square of the [[Tenney-Euclidean metrics|Tenney-Euclidean relative error]] of the val.

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:

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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 />

<br />

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).<br />

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)|.<br />

<br />

For any value of x, we can construct a p-limit <a class="wiki_link" href="/patent%20val">~~patent~~ val</a>. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. ~~The square of~~ the ~~<a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for this val will be~~<br />

For any value of x, we can construct a p-limit <a class="wiki_link" href="/patent%20val">generalized papent val</a>. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. Now consider the function<br />

<br />

<!-- ws:start:WikiTextMathRule:0:

[[math]]&lt;br/&gt;

\sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2&lt;br/&gt;[[math]]

\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2&lt;br/&gt;[[math]]

--><script type="math/tex">\sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2</script><br />

--><script type="math/tex">\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2</script><br />

<br />

This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">Tenney-Euclidean tuning</a>s of the octaves of the associated vals, while xi for these minima is the square of the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean relative error</a> of the val.<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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-01-31 07:48:28 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-05-28 11:28:32 UTC</tt>.<br>
-: The original revision id was <tt>573473057</tt>.<br>
+: The original revision id was <tt>584283017</tt>.<br>
-: The revision comment was: <tt>removed surplus "the", adjusted link text (I hope so)</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>
 <h4>Original Wikitext content:</h4>
@@ Line 11: / Line 11: @@
 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.
-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)|.
-For any value of x, we can construct a p-limit [[patent val]]. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. The square of the [[Tenney-Euclidean metrics|Tenney-Euclidean error]] for this val will be
+For any value of x, we can construct a p-limit [[patent val|generalized papent val]]. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. Now consider the function
 [[math]]
-\sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2
+\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2
 [[math]]
+This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the [[Tenney-Euclidean tuning]]s of the octaves of the associated vals, while xi for these minima is the square of the [[Tenney-Euclidean metrics|Tenney-Euclidean relative error]] of the val.
 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:
@@ Line 183: / Line 185: @@
   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;
 &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;
 &lt;br /&gt;
-For any value of x, we can construct a p-limit &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt;. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. The square of the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean error&lt;/a&gt; for this val will be&lt;br /&gt;
+For any value of x, we can construct a p-limit &lt;a class="wiki_link" href="/patent%20val"&gt;generalized papent val&lt;/a&gt;. We do so by rounding log2(q)*x to the nearest integer for each prime q up to p. Now consider the function&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
-\sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2&amp;lt;br/&amp;gt;[[math]]
+\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the &lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;Tenney-Euclidean tuning&lt;/a&gt;s of the octaves of the associated vals, while xi for these minima is the square of the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean relative error&lt;/a&gt; of the val.&lt;br /&gt;
 &lt;br /&gt;
 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:&lt;br /&gt;