The Riemann zeta function and tuning: Difference between revisions

Line 701:

Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. 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.

~~== 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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} is real. This implies that exp(''i''θ(''g'')) is real, so that {{frac|θ(''g'')|π}} is an integer. Theta has an {{w|asymptotic expansion}}

~~<math>\displaystyle{~~

~~\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots~~

~~}</math>~~

From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''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'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{cent}}}}.

Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{abs|ζ}} {{=}} {{abs|Z}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} ⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋}} is the nearest integer to {{sfrac|θ(2π''r'')|π}}, then we may set {{nowrap|''r''<sup>+</sup> {{=}} {{sfrac|''r'' + ''n'' + {{frac|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 {{nowrap|''x'' {{=}} 12}}, so that {{nowrap|''r'' {{=}} {{sfrac|12|ln(2)}} {{=}} 17.312}}. Then {{nowrap|''r'' ln(''r'') − ''r'' − {{sfrac|1|8}} {{=}} 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|''n'' {{=}} 32}}. Then {{nowrap|{{sfrac|''r'' + ''n'' + {{frac|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 {{nowrap|θ(2π''r'') / 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|⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋ − ''r'' ln(''r'') + ''r'' + {{frac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''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 {{w|Dirichlet eta function}} which was introduced to mathematics by {{w|Johann Peter Gustav Lejeune Dirichlet}}, who despite his name was a German and the brother-in-law of {{w|Felix Mendelssohn}}.

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 {{w|Euler product}} or {{w|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 {{nowrap|1 − 2<sup>1 − ''z''</sup>}}, leading to the eta function:

~~<math>\displaystyle{\eta(z) = \left(1-2^{1-z}\right)\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 {{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 + {{sfrac|2π''i''|ln(2)}}x}} 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 {{w|Euler summation}}.

== Open problems ==

# Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?

== Further information ==

* [[The Riemann zeta function and tuning/Appendix#4. Black magic formulas|How it can be shown what ETs have a sharp vs. flat tendency]]

* [[The Riemann zeta function and tuning/Appendix#5. Computing zeta|How do you actually compute zeta?]]

== Links ==

@@ Line 701: / Line 701: @@
 Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. 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.
-== 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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} is real. This implies that exp(''i''θ(''g'')) is real, so that {{frac|θ(''g'')|π}} is an integer. Theta has an {{w|asymptotic expansion}}
-<math>\displaystyle{
-\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots
-}</math>
-From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''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'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{cent}}}}.
-Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{abs|ζ}} {{=}} {{abs|Z}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} ⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋}} is the nearest integer to {{sfrac|θ(2π''r'')|π}}, then we may set {{nowrap|''r''<sup>+</sup> {{=}} {{sfrac|''r'' + ''n'' + {{frac|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 {{nowrap|''x'' {{=}} 12}}, so that {{nowrap|''r'' {{=}} {{sfrac|12|ln(2)}} {{=}} 17.312}}. Then {{nowrap|''r'' ln(''r'') − ''r'' − {{sfrac|1|8}} {{=}} 31.927}}, which rounded to the nearest integer is 32, so {{nowrap|''n'' {{=}} 32}}. Then {{nowrap|{{sfrac|''r'' + ''n'' + {{frac|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 {{nowrap|θ(2π''r'') / 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|⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋ − ''r'' ln(''r'') + ''r'' + {{frac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''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 {{w|Dirichlet eta function}} which was introduced to mathematics by {{w|Johann Peter Gustav Lejeune Dirichlet}}, who despite his name was a German and the brother-in-law of {{w|Felix Mendelssohn}}.
-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 {{w|Euler product}} or {{w|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 {{nowrap|1 − 2<sup>1&#x200A;−&#x200A;''z''</sup>}}, leading to the eta function:
-<math>\displaystyle{\eta(z) = \left(1-2^{1-z}\right)\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 {{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 + {{sfrac|2π''i''|ln(2)}}x}} 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 {{w|Euler summation}}.
 == Open problems ==
 # Are there metrics similar to zeta metrics, but for edos' performance at approximating arbitrary [[delta-rational]] chords?
+== Further information ==
+* [[The Riemann zeta function and tuning/Appendix#4. Black magic formulas|How it can be shown what ETs have a sharp vs. flat tendency]]
+* [[The Riemann zeta function and tuning/Appendix#5. Computing zeta|How do you actually compute zeta?]]
 == Links ==