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Consequently, the list below is inherently '''biased toward a handful of functions''' and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon. | Consequently, the list below is inherently '''biased toward a handful of functions''' and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon. | ||
=== Notable Local Maxima of the Riemann Zeta Function === | === Notable Local Maxima of the Riemann Zeta Function === | ||
{|class="wikitable sortable" | {|class="wikitable sortable" | ||
|+ style="font-size: 105%;" | | |+ style="font-size: 105%;" | | ||
|- | |- | ||
!colspan="3"|Tuning | !colspan="3"|Tuning | ||
| Line 177: | Line 175: | ||
|16 | |16 | ||
|10 | |10 | ||
|- | |||
|[[196zpi (σ = 1)]] | |||
|43.0234004818 | |||
|27.8917981043 | |||
|2.78019 | |||
|[[43edo]] | |||
|1199.34731849 | |||
|8 | |||
|8 | |||
|- | |- | ||
|[[214zpi (σ = 1)]] | |[[214zpi (σ = 1)]] | ||
| Line 312: | Line 319: | ||
|24 | |24 | ||
|15 | |15 | ||
|- | |- | ||
|[[568zpi (σ = 1)]] | |[[568zpi (σ = 1)]] | ||
| Line 330: | Line 328: | ||
|12 | |12 | ||
|12 | |12 | ||
|- | |||
|[[596zpi (σ = 1)]] | |||
|102.936325452 | |||
|11.6576922163 | |||
|3.25007 | |||
|[[103edo]] | |||
|1200.74229828 | |||
|15 | |||
|15 | |||
|- | |||
|[[655zpi (σ = 1)]] | |||
|111.058159333 | |||
|10.8051493669 | |||
|3.39509 | |||
|[[111edo]] | |||
|1199.37157972 | |||
|22 | |||
|16 | |||
|- | |||
|[[706zpi (σ = 1)]] | |||
|117.971388652 | |||
|10.1719579104 | |||
|3.62695 | |||
|[[118edo]] | |||
|1200.29103343 | |||
|12 | |||
|12 | |||
|- | |||
|[[796zpi (σ = 1)]] | |||
|130.004267285 | |||
|9.23046623824 | |||
|3.72487 | |||
|[[130edo]] | |||
|1199.96061097 | |||
|16 | |||
|16 | |||
|- | |||
|[[872zpi (σ = 1)]] | |||
|139.992781938 | |||
|8.57187051639 | |||
|3.60746 | |||
|[[140edo]] | |||
|1200.06187229 | |||
|10 | |||
|10 | |||
|- | |||
|[[965zpi (σ = 1)]] | |||
|152.050659206 | |||
|7.89210652729 | |||
|3.68901 | |||
|[[152edo]] | |||
|1199.60019215 | |||
|15 | |||
|15 | |||
|- | |||
|[[1114zpi (σ = 1)]] | |||
|170.995049914 | |||
|7.01774700849 | |||
|3.82285 | |||
|[[171edo]] | |||
|1200.03473845 | |||
|14 | |||
|14 | |||
|- | |||
|[[1210zpi (σ = 1)]] | |||
|183.000273182 | |||
|6.55736726036 | |||
|3.76064 | |||
|[[183edo]] | |||
|1199.99820865 | |||
|18 | |||
|18 | |||
|} | |||
=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product === | |||
{|class="wikitable sortable" | |||
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.6 and cents ≥ 15.0) or (height ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.1 and cents ≥ 6.0) | |||
!colspan="3"|Tuning | |||
!colspan="1"|Strength | |||
!colspan="2"|Closest EDO | |||
!colspan="2"|No-3 Integer limit | |||
|- | |||
!No-3 ZPI analog | |||
!Steps per octave | |||
!Cents | |||
!Height | |||
!EDO | |||
!Octave | |||
!Consistent | |||
!Distinct | |||
|- | |||
|[[no-3 51zpi (σ = 1)]] | |||
|15.9687074547 | |||
|75.1469712502 | |||
|2.56677 | |||
|[[16edo]] | |||
|1202.35154000 | |||
|26 | |||
|8 | |||
|- | |||
|[[no-3 75zpi (σ = 1)]] | |||
|21.0417134383 | |||
|57.0295762045 | |||
|2.60042 | |||
|[[21edo]] | |||
|1197.62110029 | |||
|17 | |||
|10 | |||
|- | |||
|[[no-3 95zpi (σ = 1)]] | |||
|24.9617781085 | |||
|48.0734984016 | |||
|2.64675 | |||
|[[25edo]] | |||
|1201.83746004 | |||
|14 | |||
|11 | |||
|- | |||
|[[no-3 127zpi (σ = 1)]] | |||
|31.0146799866 | |||
|38.6913552073 | |||
|2.60405 | |||
|[[31edo]] | |||
|1199.43201143 | |||
|11 | |||
|11 | |||
|- | |||
|[[no-3 161zpi (σ = 1)]] | |||
|37.0135086000 | |||
|32.4205957606 | |||
|2.92705 | |||
|[[37edo]] | |||
|1199.56204314 | |||
|22 | |||
|16 | |||
|- | |||
|[[no-3 196zpi (σ = 1)]] | |||
|43.0494972034 | |||
|27.8748900209 | |||
|2.71380 | |||
|[[43edo]] | |||
|1198.62027090 | |||
|22 | |||
|19 | |||
|- | |||
|[[no-3 220zpi (σ = 1)]] | |||
|47.0043385196 | |||
|25.5295582875 | |||
|2.69328 | |||
|[[47edo]] | |||
|1199.88923951 | |||
|10 | |||
|10 | |||
|- | |||
|[[no-3 276zpi (σ = 1)]] | |||
|55.9891415481 | |||
|21.4327272543 | |||
|2.76321 | |||
|[[56edo]] | |||
|1200.23272624 | |||
|20 | |||
|19 | |||
|- | |||
|[[no-3 340zpi (σ = 1)]] | |||
|65.9204029312 | |||
|18.2037722259 | |||
|2.65263 | |||
|[[66edo]] | |||
|1201.44896691 | |||
|16 | |||
|16 | |||
|- | |||
|[[no-3 354zpi (σ = 1)]] | |||
|68.0229453080 | |||
|17.6411061674 | |||
|2.76285 | |||
|[[68edo]] | |||
|1199.59521939 | |||
|11 | |||
|11 | |||
|- | |||
|[[no-3 394zpi (σ = 1)]] | |||
|74.0566473758 | |||
|16.2038121158 | |||
|2.76672 | |||
|[[74edo]] | |||
|1199.08209657 | |||
|16 | |||
|16 | |||
|- | |||
|[[no-3 421zpi (σ = 1)]] | |||
|78.0097604150 | |||
|15.3826904943 | |||
|2.81219 | |||
|[[78edo]] | |||
|1199.84985856 | |||
|17 | |||
|16 | |||
|- | |||
|[[no-3 525zpi (σ = 1)]] | |||
|93.0066513531 | |||
|12.9023030347 | |||
|2.97919 | |||
|[[93edo]] | |||
|1199.91418223 | |||
|35 | |||
|19 | |||
|- | |||
|[[no-3 751zpi (σ = 1)]] | |||
|124.013627761 | |||
|9.67635591079 | |||
|3.13747 | |||
|[[124edo]] | |||
|1199.86813294 | |||
|28 | |||
|26 | |||
|} | |} | ||
| Line 335: | Line 550: | ||
{|class="wikitable sortable" | {|class="wikitable sortable" | ||
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.075 and cents ≥ 6.0) | |||
!colspan="3"|Tuning | !colspan="3"|Tuning | ||
!colspan=" | !colspan="1"|Strength | ||
!colspan="2"|Closest EDT | !colspan="2"|Closest EDT | ||
!colspan="2"|No-2 Integer limit | !colspan="2"|No-2 Integer limit | ||
|- | |- | ||
!No-2 ZPI | !No-2 ZPI (σ = 1) | ||
!Steps per octave | !Steps per octave | ||
!Cents | !Cents | ||
!Height | !Height | ||
!EDT | !EDT | ||
!Tritave | !Tritave | ||
| Line 351: | Line 565: | ||
!Distinct | !Distinct | ||
|- | |- | ||
|[[no-2 93zpi | |[[no-2 93zpi (σ = 1)]] | ||
|24. | |24.5747239922 | ||
|48. | |48.8306603314 | ||
| | |2.12985 | ||
|[[39edt]] | |[[39edt]] | ||
|1904. | |1904.39575293 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 151zpi | |[[no-2 151zpi (σ = 1)]] | ||
|35. | |35.3061077059 | ||
|33. | |33.9884534992 | ||
| | |2.08576 | ||
|[[56edt]] | |[[56edt]] | ||
|1903. | |1903.35339595 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 207zpi (σ = 1)]] | |||
|44.8164999984 | |||
|26.7758526445 | |||
|2.10342 | |||
|[[no-2 207zpi | |||
|44. | |||
|26. | |||
| | |||
|[[71edt]] | |[[71edt]] | ||
|1901. | |1901.08553776 | ||
|17 | |17 | ||
|17 | |17 | ||
|- | |- | ||
|[[no-2 222zpi | |[[no-2 222zpi (σ = 1)]] | ||
|47. | |47.3516876312 | ||
|25. | |25.3422857776 | ||
| | |2.11876 | ||
|[[75edt]] | |[[75edt]] | ||
|1900. | |1900.67143332 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 233zpi | |[[no-2 233zpi (σ = 1)]] | ||
|49. | |49.1657210129 | ||
|24. | |24.4072491012 | ||
| | |2.07714 | ||
|[[78edt]] | |[[78edt]] | ||
|1903. | |1903.76542989 | ||
|21 | |21 | ||
|21 | |21 | ||
|- | |- | ||
|[[no-2 273zpi (σ = 1)]] | |||
|55.5359583782 | |||
|21.6076220712 | |||
|2.19450 | |||
|[[no-2 273zpi | |||
|55. | |||
|21. | |||
| | |||
|[[88edt]] | |[[88edt]] | ||
|1901. | |1901.47074227 | ||
|11 | |11 | ||
|11 | |11 | ||
|- | |- | ||
|[[no-2 | |[[no-2 363zpi (σ = 1)]] | ||
| | |69.4191721809 | ||
| | |17.2862908372 | ||
| | |2.08043 | ||
|[[110edt]] | |||
|1901.49199210 | |||
|[[ | |23 | ||
| | |||
| | |||
|23 | |23 | ||
|- | |- | ||
|[[no-2 | |[[no-2 380zpi (σ = 1)]] | ||
| | |71.9200195089 | ||
| | |16.6852012582 | ||
| | |2.07565 | ||
|[[114edt]] | |||
|1902.11294344 | |||
|[[ | |||
| | |||
|17 | |17 | ||
|17 | |17 | ||
|- | |- | ||
|[[no-2 453zpi (σ = 1)]] | |||
|82.6700405439 | |||
|14.5155366092 | |||
|2.38406 | |||
|[[no-2 453zpi | |||
|82. | |||
|14. | |||
| | |||
|[[131edt]] | |[[131edt]] | ||
|1901. | |1901.53529581 | ||
|27 | |27 | ||
|27 | |27 | ||
|- | |- | ||
|[[no-2 492zpi | |[[no-2 492zpi (σ = 1)]] | ||
|88. | |88.3238806401 | ||
|13. | |13.5863595587 | ||
| | |2.12238 | ||
|[[140edt]] | |[[140edt]] | ||
|1902. | |1902.09033822 | ||
|9 | |9 | ||
|9 | |9 | ||
|- | |- | ||
|[[no-2 510zpi | |[[no-2 510zpi (σ = 1)]] | ||
|90. | |90.8334979880 | ||
|13. | |13.2109852266 | ||
| | |2.23067 | ||
|[[144edt]] | |[[144edt]] | ||
|1902. | |1902.38187263 | ||
|39 | |39 | ||
|27 | |27 | ||
|- | |- | ||
|[[no-2 550zpi | |[[no-2 550zpi (σ = 1)]] | ||
|96. | |96.5187261015 | ||
|12. | |12.4328205362 | ||
| | |2.24293 | ||
|[[153edt]] | |[[153edt]] | ||
|1902. | |1902.22154203 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 627zpi | |[[no-2 627zpi (σ = 1)]] | ||
|107. | |107.244021785 | ||
|11. | |11.1894348983 | ||
| | |2.29774 | ||
|[[170edt]] | |[[170edt]] | ||
|1902. | |1902.20393272 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 687zpi | |[[no-2 687zpi (σ = 1)]] | ||
|115. | |115.412802617 | ||
|10. | |10.3974600113 | ||
| | |2.18983 | ||
|[[183edt]] | |[[183edt]] | ||
|1902. | |1902.73518207 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 697zpi | |[[no-2 697zpi (σ = 1)]] | ||
|116. | |116.734850378 | ||
|10. | |10.2797064983 | ||
| | |2.15793 | ||
|[[185edt]] | |[[185edt]] | ||
|1901. | |1901.74570218 | ||
|29 | |29 | ||
|29 | |29 | ||
|- | |- | ||
|[[no-2 777zpi | |[[no-2 777zpi (σ = 1)]] | ||
|127. | |127.486291223 | ||
|9. | |9.41277676594 | ||
| | |2.21095 | ||
|[[202edt]] | |[[202edt]] | ||
|1901. | |1901.38090672 | ||
|17 | |17 | ||
|17 | |17 | ||
|- | |- | ||
|[[no-2 810zpi | |[[no-2 810zpi (σ = 1)]] | ||
|131. | |131.822840677 | ||
|9. | |9.10312654342 | ||
| | |2.25360 | ||
|[[209edt]] | |[[209edt]] | ||
|1902. | |1902.55344758 | ||
|21 | |21 | ||
|21 | |21 | ||
|- | |- | ||
|[[no-2 829zpi | |[[no-2 829zpi (σ = 1)]] | ||
|134. | |134.373782790 | ||
|8. | |8.93031345169 | ||
| | |2.13475 | ||
|[[213edt]] | |[[213edt]] | ||
|1902. | |1902.15676521 | ||
|29 | |29 | ||
|29 | |29 | ||
|- | |- | ||
|[[no-2 839zpi | |[[no-2 839zpi (σ = 1)]] | ||
|135. | |135.657892938 | ||
|8. | |8.84578091263 | ||
| | |2.11125 | ||
|[[215edt]] | |[[215edt]] | ||
|1901. | |1901.84289622 | ||
|15 | |15 | ||
|15 | |15 | ||
|- | |- | ||
|[[no-2 858zpi | |[[no-2 858zpi (σ = 1)]] | ||
|138. | |138.196070465 | ||
|8. | |8.68331491602 | ||
| | |2.20051 | ||
|[[219edt]] | |[[219edt]] | ||
|1901. | |1901.64596661 | ||
|11 | |||
|11 | |||
|- | |||
|[[no-2 902zpi (σ = 1)]] | |||
|143.873905513 | |||
|8.34063686336 | |||
|2.09948 | |||
|[[228edt]] | |||
|1901.66520485 | |||
|11 | |11 | ||
|11 | |11 | ||
|- | |- | ||
|[[no-2 | |[[no-2 965zpi (σ = 1)]] | ||
| | |152.075713777 | ||
|7. | |7.89080629768 | ||
| | |2.10893 | ||
| | |[[241edt]] | ||
| | |1901.68431774 | ||
|15 | |||
|15 | |||
|- | |||
|[[no-2 985zpi (σ = 1)]] | |||
|154.604034485 | |||
|7.76176381166 | |||
|2.40811 | |||
|[[245edt]] | |[[245edt]] | ||
|1901. | |1901.63213386 | ||
|21 | |21 | ||
|21 | |21 | ||
|- | |- | ||
|[[no-2 1029zpi (σ = 1)]] | |||
|160.260260060 | |||
|7.48782012177 | |||
|2.17192 | |||
|[[254edt]] | |||
|1901.90631093 | |||
|9 | |||
|9 | |||
|- | |- | ||
|[[no- | |[[no-2 1049zpi (σ = 1)]] | ||
| | |162.750022676 | ||
| | |7.37327086209 | ||
| | |2.14738 | ||
|[[258edt]] | |||
|1902.30388242 | |||
|[[ | |17 | ||
| | |17 | ||
| | |||
| | |||
|- | |- | ||
|[[no- | |[[no-2 1069zpi (σ = 1)]] | ||
| | |165.332187903 | ||
| | |7.25811480039 | ||
| | |2.19607 | ||
|[[262edt]] | |||
|1901.62607770 | |||
|[[ | |17 | ||
| | |||
|17 | |17 | ||
|- | |- | ||
|[[no- | |[[no-2 1134zpi (σ = 1)]] | ||
| | |173.506549648 | ||
|6.91616542681 | |||
|6. | |2.26764 | ||
|[[275edt]] | |||
|1901.94549237 | |||
|29 | |||
|29 | |||
| | |||
|[[ | |||
| | |||
| | |||
| | |||
|- | |- | ||
|[[no- | |[[no-2 1159zpi (σ = 1)]] | ||
|176.625850825 | |||
|6.79402247404 | |||
|2.14379 | |||
|[[280edt]] | |||
|1902.32629273 | |||
| | |||
|6. | |||
| | |||
|[[ | |||
| | |||
|11 | |11 | ||
|11 | |11 | ||
|- | |- | ||
|[[no- | |[[no-2 1179zpi (σ = 1)]] | ||
| | |179.167803205 | ||
|6.69763193238 | |||
|6. | |2.29964 | ||
|[[284edt]] | |||
| | |1902.12746880 | ||
|[[ | |15 | ||
| | |15 | ||
| | |||
| | |||
|- | |- | ||
|[[no- | |[[no-2 1245zpi (σ = 1)]] | ||
| | |187.354933401 | ||
| | |6.40495544056 | ||
| | |2.28021 | ||
|[[297edt]] | |||
|1902.27176585 | |||
|[[ | |21 | ||
| | |21 | ||
| | |||
| | |||
|- | |- | ||
|[[no- | |[[no-2 1266zpi (σ = 1)]] | ||
| | |189.909845446 | ||
|6.31878772364 | |||
| | |2.17116 | ||
| | |[[301edt]] | ||
|1901.95510482 | |||
|[[ | |17 | ||
| | |||
| | |||
|17 | |17 | ||
|- | |- | ||
|[[no- | |[[no-2 1297zpi (σ = 1)]] | ||
| | |193.736743714 | ||
|6.19397217583 | |||
|2.12380 | |||
| | |[[307edt]] | ||
| | |1901.54945798 | ||
|[[ | |21 | ||
| | |21 | ||
| | |||
| | |||
|- | |- | ||
|[[no- | |[[no-2 1343zpi (σ = 1)]] | ||
|199.415414525 | |||
|6.01758897555 | |||
|2.36503 | |||
|[[316edt]] | |||
|1901.55811627 | |||
|39 | |||
| | |39 | ||
| | |||
| | |||
|[[ | |||
| | |||
| | |||
| | |||
|} | |} | ||
| Line 826: | Line 866: | ||
{|class="wikitable sortable" | {|class="wikitable sortable" | ||
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 1.725 and cents ≥ 6.0) | |||
!colspan="3"|Tuning | !colspan="3"|Tuning | ||
!colspan=" | !colspan="1"|Strength | ||
!colspan="2"|Closest ED5 | !colspan="2"|Closest ED5 | ||
!colspan="2"|No-2 No-3 Integer limit | !colspan="2"|No-2 No-3 Integer limit | ||
| Line 835: | Line 876: | ||
!Cents | !Cents | ||
!Height | !Height | ||
!ED5 | !ED5 | ||
!Pentave | !Pentave | ||
| Line 842: | Line 881: | ||
!Distinct | !Distinct | ||
|- | |- | ||
|[[no-2 no-3 186zpi (σ = 1)]] | |||
|41.3464998527 | |||
|29.0230129340 | |||
|1.75534 | |||
|[[no-2 no-3 186zpi | |||
|41. | |||
|29. | |||
| | |||
|[[96ed5]] | |[[96ed5]] | ||
|2786. | |2786.20924167 | ||
|35 | |35 | ||
|23 | |23 | ||
|- | |- | ||
|[[no-2 no-3 565zpi (σ = 1)]] | |||
|98.6253027359 | |||
|12.1672630320 | |||
|1.74188 | |||
|[[no-2 no-3 565zpi | |||
|98. | |||
|12. | |||
| | |||
|[[229ed5]] | |[[229ed5]] | ||
|2786. | |2786.30323433 | ||
|29 | |29 | ||
|29 | |29 | ||
|- | |- | ||
|[[no-2 no-3 671zpi (σ = 1)]] | |||
|113.258011095 | |||
|10.5952769998 | |||
|1.77217 | |||
|[[no-2 no-3 671zpi | |||
|113. | |||
|10. | |||
| | |||
|[[263ed5]] | |[[263ed5]] | ||
|2786. | |2786.55785095 | ||
|19 | |19 | ||
|19 | |19 | ||
|- | |- | ||
|[[no-2 no-3 764zpi | |[[no-2 no-3 764zpi (σ = 1)]] | ||
|125. | |125.745000550 | ||
|9. | |9.54312294522 | ||
| | |1.75634 | ||
|[[292ed5]] | |[[292ed5]] | ||
|2786. | |2786.59190001 | ||
|37 | |37 | ||
|37 | |37 | ||
|- | |- | ||
|[[no-2 no-3 905zpi | |[[no-2 no-3 905zpi (σ = 1)]] | ||
|144. | |144.297529480 | ||
|8. | |8.31615069448 | ||
| | |1.73926 | ||
|[[335ed5]] | |[[335ed5]] | ||
|2785. | |2785.91048265 | ||
|43 | |43 | ||
|41 | |41 | ||
|- | |- | ||
|[[no-2 no-3 938zpi | |[[no-2 no-3 938zpi (σ = 1)]] | ||
|148. | |148.562870929 | ||
|8. | |8.07738833059 | ||
| | |1.79949 | ||
|[[345ed5]] | |[[345ed5]] | ||
|2786. | |2786.69897405 | ||
|25 | |||
|25 | |||
|- | |||
|[[no-2 no-3 1046zpi (σ = 1)]] | |||
|162.414291729 | |||
|7.38851234841 | |||
|1.73251 | |||
|[[377ed5]] | |||
|2785.46915535 | |||
|23 | |||
|23 | |||
|- | |||
|[[no-2 no-3 1145zpi (σ = 1)]] | |||
|174.880594782 | |||
|6.86182478678 | |||
|1.74084 | |||
|[[406ed5]] | |||
|2785.90086343 | |||
|25 | |25 | ||
|25 | |25 | ||
|- | |||
|[[no-2 no-3 1196zpi (σ = 1)]] | |||
|181.292147244 | |||
|6.61915046096 | |||
|1.77770 | |||
|[[421ed5]] | |||
|2786.66234406 | |||
|35 | |||
|35 | |||
|- | |||
|[[no-2 no-3 1280zpi (σ = 1)]] | |||
|191.632570168 | |||
|6.26198353937 | |||
|1.75036 | |||
|[[445ed5]] | |||
|2786.58267502 | |||
|29 | |||
|29 | |||
|} | |} | ||