The Riemann zeta function and tuning: Difference between revisions

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\def\ceil#1{\left\lceil{#1}\right\rceil}

\def\round#1{\left\lceil{#1}\right\rfloor}

\def\~~rfrac~~#1{\left\lfloor{#1}\right\rceil}

\def\rround#1{\left\lfloor{#1}\right\rceil}

</math>

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Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''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|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 {{nowrap|''x'' = 8.202}}.

Now suppose that ⌊x⌉ denotes the difference between ''x'' and the integer nearest to ''x''. For example, ⌊8.202⌉ would be 0.202, since it's the difference between 8.202 and the nearest integer, which is 8. ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, <math>\~~rfrac~~{x} = \abs{x - \floor{x + \frac{1}{2}}}</math>.

Now suppose that ⌊x⌉ denotes the difference between ''x'' and the integer nearest to ''x''. For example, ⌊8.202⌉ would be 0.202, since it's the difference between 8.202 and the nearest integer, which is 8. ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, <math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math>.

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

<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\~~rfrac~~{x \log_2 q}}{\log_2 q}\right)^2</math>

<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^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|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ''p'' for these minima is the square of the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

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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:

<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\~~rfrac~~{x \log_2 q}^2}{q^s}</math>

<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\rround{x \log_2 q}^2}{q^s}</math>

If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/''n'' for each prime power ''p''''n''. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[Wikipedia:Von Mangoldt function|Von Mangoldt function]], an [[Wikipedia:arithmetic function|arithmetic function]] on positive integers which is equal to ln ''p'' on prime powers ''p''''n'', and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as

<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\~~rfrac~~{x \log_2 n}^2}{n^s}</math>

<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\rround{x \log_2 n}^2}{n^s}</math>

where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.

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==== Valley edos ====

Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294 }}… These tunings tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.

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=== ''k''-ary-peak edos ===

{{Idiosyncratic terms|the term "k-ary-peak edos" itself, as well as the names for the different types of k-ary-peak edos. Proposed by [[User:Akselai]] and [[Budjarn Lambeth]].}}

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=== Zeta peak index ===

These octave-stretched edos are not the only tunings which can be produced from zeta peaks. They are only one type of tuning within a larger family of equal-step tunings called zeta peak indexes. They have their own article here, with a table of the first 500 or so: [[ZPI|zeta peak index (ZPI)]].

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}</math>

where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors (1-p~~^(-~~s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2~~^(-~~s))ζ(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors {{nowrap|(1 − p−s)}} for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 − 2−s)ζ(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

Along the critical line, |1 - p^(-1/2~~-i t~~)| may be written

Along the critical line, {{nowrap|1 − p^(−1/2 − it)|}} may be written

<math>\displaystyle{

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=== Black magic formulas ===

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 [[Wikipedia:asymptotic expansion|asymptotic expansion]]

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 {{nowrap|ζ(1/2 + ig)}} is real. This implies that exp(iθ(g)) is real, so that {{frac|θ(g)|π}} is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]

<math>\displaystyle{

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}</math>

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.

From this we may deduce that {{nowrap|θ(t)/π ≈ r ln(r) − r − 1/8}}, where {{nowrap|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, {{nowrap|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(~~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.

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 {{nowrap|r = x/ln(2)}}, and if {{nowrap|n = &lfloor;r ln(r) − r + 3/8&rfloor;}} is the nearest integer to {{nowrap|θ(2πr) / π}}, then we may set {{nowrap|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.

For an example, consider {{nowrap|x = 12}}, so that {{nowrap|r = 12/ln(2) = 17.312}}. Then {{nowrap|r ln(r) − r − 1/8 = 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|n = 32}}. Then {{nowrap|(r + n + 1/8) / ln(r) = 17.338}}, corresponding to {{nowrap|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 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.

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 {{nowrap|θ(2πr) / π}}, which was 31.927. Then {{nowrap|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 {{nowrap|&lfloor;r ln(r) − r + 3/8&rfloor; − r ln(r) + r + 1/8}}, where {{nowrap|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 ==

There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [[Wikipedia:Dirichlet eta function|Dirichlet eta function]] which was introduced to mathematics by [[Wikipedia:Johann Peter Gustav Lejeune Dirichlet|Johann Peter Gustav Lejeune Dirichlet]], who despite his name was a German and the brother-in-law of [[Wikipedia:Felix Mendelssohn|Felix Mendelssohn]].

The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at z=1 which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function:

The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|z = 1}} which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of {{nowrap|(z − 1)}}, but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function:

<math>\displaystyle{\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}

= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots}</math>

The Dirichlet series for the zeta function is absolutely convergent when s>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 [[Wikipedia:Euler summation|Euler summation]].

The Dirichlet series for the zeta function is absolutely convergent when {{nowrap|s > 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 {{nowrap|1 + 2πix/ln(2)}} corresponding to pure octave divisions along the line {{nowrap|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 [[Wikipedia:Euler summation|Euler summation]].

== Open problems ==

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* [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]

* [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch

* [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

* [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 {{nowrap|ζ(1/2 + it)}}] by Tadej Kotnik

* [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]

* [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]

@@ Line 9: / Line 9: @@
 \def\ceil#1{\left\lceil{#1}\right\rceil}
 \def\round#1{\left\lceil{#1}\right\rfloor}
-\def\rfrac#1{\left\lfloor{#1}\right\rceil}
+\def\rround#1{\left\lfloor{#1}\right\rceil}
 </math>
@@ Line 16: / Line 16: @@
 Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''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|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 {{nowrap|''x'' = 8.202}}.
-Now suppose that &#x230A;x&#x2309; denotes the difference between ''x'' and the integer nearest to ''x''. For example, &#x230A;8.202&#x2309; would be 0.202, since it's the difference between 8.202 and the nearest integer, which is 8. &#x230A;7.95&#x2309; would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, <math>\rfrac{x} = \abs{x - \floor{x + \frac{1}{2}}}</math>.
+Now suppose that &#x230A;x&#x2309; denotes the difference between ''x'' and the integer nearest to ''x''. For example, &#x230A;8.202&#x2309; would be 0.202, since it's the difference between 8.202 and the nearest integer, which is 8. &#x230A;7.95&#x2309; would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. Mathematically speaking, <math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math>.
 For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding ''x'' log<sub>2</sub>(''q'') to the nearest integer for each prime ''q'' up to ''p''. Now consider the function
-<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rfrac{x \log_2 q}}{\log_2 q}\right)^2</math>
+<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^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|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ<sub>''p''</sub> for these minima is the square of the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]] of the val&mdash;equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."
@@ Line 26: / Line 26: @@
 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:
-<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\rfrac{x \log_2 q}^2}{q^s}</math>
+<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\rround{x \log_2 q}^2}{q^s}</math>
 If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/''n'' for each prime power ''p''<sup>''n''</sup>. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[Wikipedia:Von Mangoldt function|Von Mangoldt function]], an [[Wikipedia:arithmetic function|arithmetic function]] on positive integers which is equal to ln ''p'' on prime powers ''p''<sup>''n''</sup>, and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as
-<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\rfrac{x \log_2 n}^2}{n^s}</math>
+<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\rround{x \log_2 n}^2}{n^s}</math>
 where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.
@@ Line 309: / Line 309: @@
 ==== Valley edos ====
 Instead of looking at |Z(x)| maxima, we can look at |Z(x)| ''minima'' for integer values of ''x''. These correspond to ''zeta valley edos'', and we get a list of edos {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294 }}… These tunings tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Keep in mind, however, that the ''most'' xenharmonic tunings would not contain octaves at all.
@@ Line 317: / Line 316: @@
 === ''k''-ary-peak edos ===
 {{Idiosyncratic terms|the term "k-ary-peak edos" itself, as well as the names for the different types of k-ary-peak edos. Proposed by [[User:Akselai]] and [[Budjarn Lambeth]].}}
@@ Line 436: / Line 434: @@
 === Zeta peak index ===
 These octave-stretched edos are not the only tunings which can be produced from zeta peaks. They are only one type of tuning within a larger family of equal-step tunings called zeta peak indexes. They have their own article here, with a table of the first 500 or so: [[ZPI|zeta peak index (ZPI)]].
@@ Line 446: / Line 443: @@
 }</math>
-where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying &zeta;(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))&zeta;(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.
+where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying &zeta;(s) by the corresponding factors {{nowrap|(1 &minus; p<sup>&minus;s</sup>)}} for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 &minus; 2<sup>&minus;s</sup>)&zeta;(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.
-Along the critical line, |1 - p^(-1/2-i t)| may be written
+Along the critical line, {{nowrap|1 &minus; p^(&minus;1/2 &minus; it)|}} may be written
 <math>\displaystyle{
@@ Line 459: / Line 456: @@
 === Black magic formulas ===
-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 &theta;(t). Recall that a Gram point is a point on the critical line where &zeta;(1/2 + ig) is real. This implies that exp(i&theta;(g)) is real, so that &theta;(g)/π is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]
+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 &theta;(t). Recall that a Gram point is a point on the critical line where {{nowrap|&zeta;(1/2 + ig)}} is real. This implies that exp(i&theta;(g)) is real, so that {{frac|&theta;(g)|π}} is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]
 <math>\displaystyle{
@@ Line 465: / Line 462: @@
 }</math>
-From this we may deduce that &theta;(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.
+From this we may deduce that {{nowrap|&theta;(t)/π ≈ r ln(r) &minus; r &minus; 1/8}}, where {{nowrap|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, {{nowrap|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 |&zeta;| = |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 &theta;(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 |&zeta;| = |Z|. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|r = x/ln(2)}}, and if {{nowrap|n = &lfloor;r ln(r) &minus; r + 3/8&rfloor;}} is the nearest integer to {{nowrap|&theta;(2πr) / π}}, then we may set {{nowrap|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.
+For an example, consider {{nowrap|x = 12}}, so that {{nowrap|r = 12/ln(2) = 17.312}}. Then {{nowrap|r ln(r) &minus; r &minus; 1/8 = 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|n = 32}}. Then {{nowrap|(r + n + 1/8) / ln(r) = 17.338}}, corresponding to {{nowrap|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 &theta;(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.
+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 {{nowrap|&theta;(2πr) / π}}, which was 31.927. Then {{nowrap|32 &minus; 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 {{nowrap|&lfloor;r ln(r) &minus; r + 3/8&rfloor; &minus; r ln(r) + r + 1/8}}, where {{nowrap|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 ==
 There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [[Wikipedia:Dirichlet eta function|Dirichlet eta function]] which was introduced to mathematics by [[Wikipedia:Johann Peter Gustav Lejeune Dirichlet|Johann Peter Gustav Lejeune Dirichlet]], who despite his name was a German and the brother-in-law of [[Wikipedia:Felix Mendelssohn|Felix Mendelssohn]].
-The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at z=1 which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function:
+The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|z = 1}} which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of {{nowrap|(z &minus; 1)}}, but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function:
 <math>\displaystyle{\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}
 = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots}</math>
-The Dirichlet series for the zeta function is absolutely convergent when s&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 [[Wikipedia:Euler summation|Euler summation]].
+The Dirichlet series for the zeta function is absolutely convergent when {{nowrap|s &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 {{nowrap|1 + 2πix/ln(2)}} corresponding to pure octave divisions along the line {{nowrap|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 [[Wikipedia:Euler summation|Euler summation]].
 == Open problems ==
@@ Line 490: / Line 487: @@
 * [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]
 * [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch
-* [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 &zeta;(1/2 + it)] by Tadej Kotnik
+* [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 {{nowrap|&zeta;(1/2 + it)}}] by Tadej Kotnik
 * [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]
 * [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]