The Riemann zeta function and tuning: Difference between revisions

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]

=Preliminaries=  

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.

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta gap edos//. These are [[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[46edo|46]], [[53edo|53]], [[72edo|72]], [[270edo|270]], [[311edo|311]], [[954edo|954]], [[1178edo|1178]], [[1308edo|1308]], [[1395edo|1395]], [[1578edo|1578]], [[3395edo|3395]], [[4190edo|4190]] ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.

=Removing primes=  

Removing 2 leads to increasing adjusted peak values corresponding to the division of 3 (the "tritave") into 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316 ... parts. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen-Pierce]] division of the tritave, but the multiples 26, 39 and 52 also.

When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[http://en.wikipedia.org/wiki/Asymptotic_expansion|asymptotic expansion]]

[[math]]

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.

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.

[[http://front.math.ucdavis.edu/0309.5433|X-Ray of Riemann zeta-function]] by Juan Arias-de-Reyna

[[http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/|Selberg's limit theorem]] by Terence Tao [[http://www.webcitation.org/5xrvgjW6T|Permalink]]

[[http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf|Computational estimation of the order of ζ(1/2 + it)]] by Tadej Kotnik</pre></div>

<h4>Original HTML content:</h4>

  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;

If you have access to &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow"&gt;Mathematica&lt;/a&gt;, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo:&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/x, and hence the size of the octave in the zeta peak value tuning for Nedo is N/x; if x is slightly larger than N as here with N=12, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with x less than N, we have stretched octaves.&lt;br /&gt;

For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.&lt;br /&gt;

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of &lt;em&gt;zeta gap edos&lt;/em&gt;. These are &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="/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="/46edo"&gt;46&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="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/311edo"&gt;311&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="/1308edo"&gt;1308&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="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/4190edo"&gt;4190&lt;/a&gt; ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as &lt;a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow"&gt;sequence A117537&lt;/a&gt;. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.&lt;br /&gt;

&lt;br /&gt;

  The &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/EulerProduct.html" rel="nofollow"&gt;Euler product&lt;/a&gt; for the Riemann zeta function is&lt;br /&gt;

&lt;br /&gt;

Removing 2 leads to increasing adjusted peak values corresponding to the division of 3 (the &amp;quot;tritave&amp;quot;) into 4, 7, 9, 13, 15, 17, 26, 32, 39, 45, 52, 56, 71, 75, 88, 131, 245, 316 ... parts. A striking feature of this list is the appearance not only of &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt;, the &lt;a class="wiki_link" href="/Bohlen-Pierce"&gt;Bohlen-Pierce&lt;/a&gt; division of the tritave, but the multiples 26, 39 and 52 also.&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:13:

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

&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;

  There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow"&gt;Dirichlet eta function&lt;/a&gt; which was introduced to mathematics by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow"&gt;Johann Peter Gustav Lejeune Dirichlet&lt;/a&gt;, who despite his name was a German and the brother-in-law of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow"&gt;Felix Mendelssohn&lt;/a&gt;.&lt;br /&gt;

&lt;br /&gt;

The Dirichlet series for the zeta function is absolutely convergent when s&amp;gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow"&gt;Euler summation&lt;/a&gt;.&lt;br /&gt;

&lt;br /&gt;

  &lt;a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow"&gt;X-Ray of Riemann zeta-function&lt;/a&gt; by Juan Arias-de-Reyna&lt;br /&gt;

&lt;a class="wiki_link_ext" href="http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/" rel="nofollow"&gt;Selberg's limit theorem&lt;/a&gt; by Terence Tao &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;br /&gt;

&lt;a class="wiki_link_ext" href="http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf" rel="nofollow"&gt;Computational estimation of the order of ζ(1/2 + it)&lt;/a&gt; by Tadej Kotnik&lt;/body&gt;&lt;/html&gt;</pre></div>