The Riemann zeta function and tuning: Difference between revisions
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Since θ is holomorphic on the strip with imaginary part between −{{frac|1|2}} and {{frac|1|2}}, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''. | Since θ is holomorphic on the strip with imaginary part between −{{frac|1|2}} and {{frac|1|2}}, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''. | ||
Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor | Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor <math>t = \frac{2\pi}{\ln(2)}x</math>, for ''x'' a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here]. | ||
If you have access to [[Wikipedia:Mathematica|Mathematica]], which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of Z({ | If you have access to [[Wikipedia:Mathematica|Mathematica]], which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of <math>Z\left(\frac{2\pi}{\ln(2)}x\right)</math> in the region around 12edo: | ||
[[File:plot12.png|alt=plot12.png|plot12.png]] | [[File:plot12.png|alt=plot12.png|plot12.png]] | ||
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== Zeta EDO lists == | == Zeta EDO lists == | ||
=== Record edos === | === Record edos === | ||
The prime-approximating strength of an edo can be determined by the magnitude of Z(''x''). Since a higher | The prime-approximating strength of an edo can be determined by the magnitude of Z(''x''). Since a higher peak correlates to a stronger tuning, we would like to find a sequence with succesively larger {{nowrap|{{!}}Z(''x''){{!}}}}-associated values satisfying some property. | ||
==== Zeta peak edos ==== | ==== Zeta peak edos ==== | ||
If we examine the increasingly larger peak values of {{ | If we examine the increasingly larger peak values of <math>Z\left(\frac{2\pi}{\ln(2)}x\right)</math>, we find they occur with values of ''x'' such that the local minima and maxima occur near integers, so that there is a sequence of [[edo]]s {{EDOs| 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} … of '''zeta peak edos'''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks are never exactly located at an integer but are merely close to integer values; this can be interpreted as the zeta function suggesting a [[stretched and compressed tuning|detuned (stretched or compressed) octave]] for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos." | ||
==== Zeta peak integer edos ==== | ==== Zeta peak integer edos ==== | ||
Alternatively (as [[groundfault]] has found), if we do not allow octave stretching or compression and only look at the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on the peak of 53. This definition may be better for measuring how accurate the edo itself is without stretched octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos." Similarly, we can look at pure-tritave EDTs, etc. | Alternatively (as [[groundfault]] has found), if we do not allow octave stretching or compression and only look at the record {{nowrap|{{!}}Z(''x''){{!}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on the peak of 53. This definition may be better for measuring how accurate the edo itself is without stretched octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos." | ||
Similarly, we can look at the peaks of <math>Z\left(\frac{2\pi}{\ln(3)}x\right)</math> for both pure-tritave and tempered-tritave EDTs, etc. | |||
==== Zeta integral edos ==== | ==== Zeta integral edos ==== | ||
Similarly, if we take the integral of {{ | Similarly, if we take the integral of <math>Z\left(\frac{2\pi}{\ln(2)}x\right)</math> between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers. This sequence, the '''zeta integral edos''', goes {{EDOs| 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973,}} … This is listed in the OEIS as {{OEIS|A117538}}. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes. | ||
==== Zeta gap edos ==== | ==== Zeta gap edos ==== | ||
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 {{EDOs| 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, 8539, 14348, 58973, 95524, }} … 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 {{OEIS|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. | Finally, taking the midpoints of the successively larger normalized gaps between the zeros of <math>Z\left(\frac{2\pi}{\ln(2)}x\right)</math> leads to a list of '''zeta gap edos'''. These are {{EDOs| 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, 8539, 14348, 58973, 95524, }} … 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 {{OEIS|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. | ||
==== Strict zeta edos ==== | ==== Strict zeta edos ==== | ||