The Riemann zeta function and tuning: Difference between revisions

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:genewardsmith|genewardsmith]] and made on <tt>2016-05-28 11:28:32 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-05-28 13:41:33 UTC</tt>.<br>

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

: The original revision id was <tt>584287197</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>

Line 19:

[[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.

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 ξ 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 194:

--><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 />

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 ξ 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:genewardsmith|genewardsmith]] and made on <tt>2016-05-28 11:28:32 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-05-28 13:41:33 UTC</tt>.<br>
-: The original revision id was <tt>584283017</tt>.<br>
+: The original revision id was <tt>584287197</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>
@@ Line 19: / Line 19: @@
 [[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.
+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 ξ 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 194: / Line 194: @@
   --&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;
+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 ξ 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;