The Riemann zeta function and tuning: Difference between revisions

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-16 01:~~59:10~~ UTC</tt>.<br>

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From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where r = t/2π = x/ln(2); hence while x is the number of equal steps to an octave, r is the number of equal steps to an "e-tave", meaning the interval of e, 1200/ln(2) = 1731.234 cents.

Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(~~rln~~(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.

Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(r ln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.

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.

=Computing zeta=

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From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where r = t/2π = x/ln(2); hence while x is the number of equal steps to an octave, r is the number of equal steps to an &quot;e-tave&quot;, meaning the interval of e, 1200/ln(2) = 1731.234 cents.<br />

<br />

Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(~~rln~~(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted &quot;Gram&quot; tuning from the orginal one.<br />

Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(r ln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted &quot;Gram&quot; tuning from the orginal one.<br />

<br />

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 />

<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:genewardsmith|genewardsmith]] and made on <tt>2012-07-16 01:59:10 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-16 16:01:30 UTC</tt>.<br>
-: The original revision id was <tt>353298870</tt>.<br>
+: The original revision id was <tt>353388144</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 145: / Line 145: @@
 From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where r = t/2π = x/ln(2); hence while x is the number of equal steps to an octave, r is the number of equal steps to an "e-tave", meaning the interval of e, 1200/ln(2) = 1731.234 cents.
-Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(rln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.
+Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(r ln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.
+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.
 =Computing zeta=
@@ Line 319: / Line 322: @@
 From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where r = t/2π = x/ln(2); hence while x is the number of equal steps to an octave, r is the number of equal steps to an &amp;quot;e-tave&amp;quot;, meaning the interval of e, 1200/ln(2) = 1731.234 cents.&lt;br /&gt;
 &lt;br /&gt;
-Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(rln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted &amp;quot;Gram&amp;quot; tuning from the orginal one.&lt;br /&gt;
+Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ} = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = x/ln(2), and if n = floor(r ln(r) - r + 3/8) is the nearest integer to θ(2πr)/π, then we may set r⁺ = (r + n + 1/8)/ln(r). This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted &amp;quot;Gram&amp;quot; tuning from the orginal one.&lt;br /&gt;
+&lt;br /&gt;
+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;
 &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;