The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:~~clumma~~|~~clumma~~]] and made on <tt>2012-07-16 16:25:52 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-16 21:11:13 UTC</tt>.<br>

: The original revision id was <tt>~~353391526~~</tt>.<br>

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For an example, consider x = 12, so that r = 12/ln(2) = 17.312. Then r ln(r) - r - 1/8 = 31.927, which rounded to the nearest integer is 32, so n = 32. Then (r + n + 1/8)/ln(r) = 17.338, corresponding to x = 12.0176, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents.

The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This ~~usually~~ works, but when x is extreme it may not; 49 is very sharp in tendency, but this method calls it as flat.

The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This works more often than not on the clearcut cases, but when x is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.

=Computing zeta=

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For an example, consider x = 12, so that r = 12/ln(2) = 17.312. Then r ln(r) - r - 1/8 = 31.927, which rounded to the nearest integer is 32, so n = 32. Then (r + n + 1/8)/ln(r) = 17.338, corresponding to x = 12.0176, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents. <br />

<br />

The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This ~~usually~~ works, but when x is extreme it may not; 49 is very sharp in tendency, but this method calls it as flat.<br />

The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This works more often than not on the clearcut cases, but when x is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.<br />

<br />

<h1 id="toc6"><a name="Computing zeta"></a>Computing zeta</h1>

@@ 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:clumma|clumma]] and made on <tt>2012-07-16 16:25:52 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-16 21:11:13 UTC</tt>.<br>
-: The original revision id was <tt>353391526</tt>.<br>
+: The original revision id was <tt>353427662</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 149: / Line 149: @@
 For an example, consider x = 12, so that r = 12/ln(2) = 17.312. Then r ln(r) - r - 1/8 = 31.927, which rounded to the nearest integer is 32, so n = 32. Then (r + n + 1/8)/ln(r) = 17.338, corresponding to x = 12.0176, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents.
-The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This usually works, but when x is extreme it may not; 49 is very sharp in tendency, but this method calls it as flat.
+The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This works more often than not on the clearcut cases, but when x is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.
 =Computing zeta=
@@ Line 326: / Line 326: @@
 For an example, consider x = 12, so that r = 12/ln(2) = 17.312. Then r ln(r) - r - 1/8 = 31.927, which rounded to the nearest integer is 32, so n = 32. Then (r + n + 1/8)/ln(r) = 17.338, corresponding to x = 12.0176, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents. &lt;br /&gt;
 &lt;br /&gt;
-The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This usually works, but when x is extreme it may not; 49 is very sharp in tendency, but this method calls it as flat.&lt;br /&gt;
+The fact that x is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for θ(2πr)/π, which was 31.927. Then 32 - 31.927 = 0.0726, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo x by computing floor(r ln(r) - r + 3/8) - r ln(r) + r + 1/8, where r = x/ln(2). This works more often than not on the clearcut cases, but when x is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Computing zeta"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&gt;Computing zeta&lt;/h1&gt;