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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.
From the original Riemann zeta: 12edo, (14edo), (15edo), (17edo), 19edo, 22edo, 24edo, (26edo), 27edo, (29edo), 31edo, 34edo, 36edo, 41edo, 46edo, 50edo, 53edo, 58edo, 60edo, 65edo, 68edo, 72edo, 77edo, 80edo, 84edo, 87edo, 94edo, 99edo, 103edo, 111edo, 118edo, 130edo, 140edo, 152edo, 171edo


From the no-2 Riemann zeta: 39edt, 56edt, 69edt, 71edt, 75edt, 78edt, 82edt, 88edt, 99edt, 101edt, 105edt, 110edt, 131edt, 140edt, 144edt, 153edt, 170edt, 183edt, 185edt, 202edt, 209edt, 213edt, 215edt, 219edt, 245edt
From the no-2 Riemann zeta: 39edt, 56edt, 69edt, 71edt, 75edt, 78edt, 82edt, 88edt, 99edt, 101edt, 105edt, 110edt, 131edt, 140edt, 144edt, 153edt, 170edt, 183edt, 185edt, 202edt, 209edt, 213edt, 215edt, 219edt, 245edt
Line 22: Line 20:
From the no-2 no-3 Riemann zeta:
From the no-2 no-3 Riemann zeta:


From the alpha-beta-gamma set: 5ed2/1, 7ed2/1, 12ed2/1, 7ed5/3, 9ed5/3, 16ed5/3, 9ed3/2, 11ed3/2, 20ed3/2, 11ed7/5, 13ed7/5, 24ed7/5, 13ed4/3, 15ed4/3, 28ed4/3
=== Notable Local Maxima of the Riemann Zeta Function ===
 
{| class="wikitable sortable"
{| class="wikitable sortable"
|+ style="font-size: 105%;" | Notable Local Maxima of the Riemann Zeta Function
|+ style="font-size: 105%;" | Notable Local Maxima of the Riemann Zeta Function
Line 429: Line 426:
|}
|}


=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product ===
=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product ===
=== Notable Local Maxima of the Riemann Zeta Function after removing the primes 2 and 3 from the zeta product ===
=== The α–β–γ family ===
{| class="wikitable sortable"
{| class="wikitable sortable"
|+ style="font-size: 105%;" | α–β–γ family
|+ style="font-size: 105%;" | α–β–γ family
Line 547: Line 551:
|Dave Benson
|Dave Benson
|[[28ed4/3]]
|[[28ed4/3]]
|}
=== Equal divisions of a ratio & optimization ===
{| class="wikitable sortable"
|+ style="font-size: 105%;" | EDRs collection & optimization
|- style="white-space: nowrap;"
! colspan="3" |EDRs
! colspan="3" |Optimization
!Comments
|- style="white-space: nowrap;"
!EDR
!Steps per octave
!Cents
!Optimization
!Steps per octave
!Cents
!Why it matters
|-
|[[7ed5/3]]
|9.49840814199707
|126.336958999921
|[[Benson Alpha 5/3]]
|9.50583353877785
|126.238272015258
|Alpha 5/3
|-
|[[10edo]]
|10
|120.
| colspan="3" |None
|EDO ≤ 29
|-
|[[11edo]]
|11
|109.090909090909
| colspan="3" |None
|EDO ≤ 29
|-
|[[12edo]]
|12
|100.
|[[34zpi]]
|12.0231830072926
|99.8071807833375
|EDO ≤ 29, strong zeta peak
|-
|[[9ed5/3]]
|12.2122390397105
|98.2620792221608
|[[Benson Beta 5/3]]
|12.2053823008782
|98.3172808862904
|Beta 5/3
|-
|[[13edo]]
|13
|92.3076923076923
| colspan="3" |None
|EDO ≤ 29
|-
|[[14edo]]
|14
|85.7142857142857
|[[42zpi]]
|13.9002525327005
|86.3293668353859
|EDO ≤ 29, medium zeta peak
|-
|[[15edo]]
|15
|80.
|[[47zpi]]
|15.0534898676781
|79.7157343943591
|EDO ≤ 29, medium zeta peak
|-
|[[9edf|9ed3/2]]
|15.3856016221631
|77.9950000961542
|[[Benson Alpha 3/2]]
|15.3915238996928
|77.9649895501219
|Alpha 3/2
|-
|[[16edo]]
|16
|75.
| colspan="3" |None
|EDO ≤ 29
|-
|[[17edo]]
|17
|70.5882352941176
|[[56zpi]]
|17.0445886606675
|70.4035764012981
|EDO ≤ 29, medium zeta peak
|-
|[[18edo]]
|18
|66.6666666666667
| colspan="3" |None
|EDO ≤ 29
|-
|[[11edf|11ed3/2]]
|18.8046242048660
|63.8140909877625
|[[Benson Beta 3/2]]
|18.7990736394111
|63.8329325698408
|Beta 3/2
|-
|[[19edo]]
|19
|63.1578947368421
|[[65zpi]]
|18.9480867166984
|63.3309324546460
|EDO ≤ 29, strong zeta peak
|-
|[[20edo]]
|20
|60.
| colspan="3" |None
|EDO ≤ 29
|-
|[[21edo]]
|21
|57.1428571428571
| colspan="3" |None
|EDO ≤ 29
|-
|[[16ed5/3]]
|21.7106471817076
|55.2724195624655
|[[Benson Gamma 5/3]]
|21.7094399215509
|55.2754932571412
|Gamma 5/3
|-
|[[22edo]]
|22
|54.5454545454545
|[[80zpi]]
|22.0251467420146
|54.4831784348982
|EDO ≤ 29, strong zeta peak
|-
|[[11ed7/5]]
|22.6604698881676
|52.9556538731173
|[[Benson Alpha 7/5]]
|22.6653911133366
|52.9441558718088
|Alpha 7/5
|-
|[[23edo]]
|23
|52.1739130434783
| colspan="3" |None
|EDO ≤ 29
|-
|[[24edo]]
|24
|50.
|[[90zpi]]
|24.0057421830853
|49.9880399800983
|EDO ≤ 29, medium zeta peak
|-
|[[39edt]]
|24.6062603892868
|48.7680769452663
|[[93zpi no-2 analogue]]
|24.5738316304204
|48.8324335434323
|strong no-2 zeta peak
|-
|[[25edo]]
|25
|48.
| colspan="3" |None
|EDO ≤ 29
|-
|[[26edo]]
|26
|46.1538461538462
|[[100zpi]]
|25.9356996537225
|46.2682717652372
|EDO ≤ 29, medium zeta peak
|-
|[[13ed7/5]]
|26.7805553223799
|44.8086302003300
|[[Benson Beta 7/5]]
|26.7758951088566
|44.8164289231577
|Beta 7/5
|-
|[[27edo]]
|27
|44.4444444444444
|[[106zpi]]
|27.0866140827635
|44.3023257293579
|EDO ≤ 29, strong zeta peak
|-
|[[28edo]]
|28
|42.8571428571429
| colspan="3" |None
|EDO ≤ 29
|-
|[[29edo]]
|29
|41.3793103448276
|[[116zpi]]
|28.9399661541990
|41.4651487014917
|EDO ≤ 29, medium zeta peak
|-
|[[31edo]]
|31
|38.7096774193548
|[[127zpi]]
|30.9783816349790
|38.7366910944446
|strong zeta peak
|-
|[[13ed4/3]]
|31.3224709154917
|38.3111537795856
|[[Benson Alpha 4/3]]
|31.3266790320926
|38.3060074376432
|Alpha 4/3
|-
|[[34edo]]
|34
|35.2941176470588
|[[144zpi]]
|34.0448410043159
|35.2476312005063
|strong zeta peak
|-
|[[20edf|20ed3/2]]
|34.1902258270291
|35.0977500432694
|[[Benson Gamma 3/2]]
|34.1894540921914
|35.0985422804417
|Gamma 3/2
|-
|[[56edt]]
|35.3320662000016
|33.9634821583105
|[[151zpi no-2 analogue]]
|35.3059427335609
|33.9886123153798
|strong no-2 zeta peak
|-
|[[36edo]]
|36
|33.3333333333333
|[[155zpi no-5 analogue]]
|35.9775957344990
|33.3540909419168
|strong no-5 zeta peak
|-
|[[15ed4/3]]
|36.1413125947981
|33.2029999423075
|[[Benson Beta 4/3]]
|36.1372975038827
|33.2066890135066
|Beta 4/3
|-
|[[37edo]]
|37
|32.4324324324324
|[[161zpi no-3 analogue]]
|37.0117501336435
|32.4221360964286
|strong no-3 zeta peak
|-
|[[41edo]]
|41
|29.2682926829268
|[[184zpi]]
|40.9880783925993
|29.2768055263764
|strong zeta peak
|-
|[[96ed5]]
|41.3449495750457
|29.0241011860920
|[[186zpi no-2 no-3 analogue]]
|41.3477989230936
|29.0221010852836
|strong no-2 no-3 zeta peak
|-
|[[66edt]]
|41.6413637357162
|28.8175000131119
|[[188zpi no-2 no-5 analogue]]
|41.6281274155763
|28.8266629920756
|strong no-2 no-5 zeta peak
|-
|[[46edo]]
|46
|26.0869565217391
|[[214zpi]]
|46.0089748051542
|26.0818678330031
|strong zeta peak
|-
|[[24ed7/5]]
|49.4410252105475
|24.2713413585121
|[[Benson Gamma 7/5]]
|49.4404896216012
|24.2716042900130
|Gamma 7/5
|-
|[[50edo]]
|50
|24.0
|[[238zpi]]
|49.9385162652878
|24.0295485277387
|medium zeta peak
|-
|[[53edo]]
|53
|22.6415094339623
|[[257zpi]]
|52.9968291550147
|22.6428640945673
|strong zeta peak
|-
|[[57edo]]
|57
|21.0526315789474
|[[282zpi no-3 no-5 analogue]]
|56.9949885079207
|21.0544827083040
|strong no-3 no-5 zeta peak
|-
|[[58edo]]
|58
|20.6896551724138
|[[289zpi]]
|58.0667185533159
|20.6658827964969
|strong zeta peak
|-
|[[60edo]]
|60
|20.
|[[301zpi]]
|59.9201656607655
|20.0266469020418
|medium zeta peak
|-
|[[65edo]]
|65
|18.4615384615385
|[[334zpi]]
|65.0158450885860
|18.4570391781413
|strong zeta peak
|-
|[[28ed4/3]]
|67.4637835102899
|17.7873213976647
|[[Benson Gamma 4/3]]
|67.4633901646646
|17.7874251067289
|Gamma 4/3
|-
|[[68edo]]
|68
|17.6470588235294
|[[354zpi]]
|68.0493056282519
|17.6342725163943
|strong zeta peak
|-
|[[72edo]]
|72
|16.6666666666667
|[[380zpi]]
|71.9506065993786
|16.6781081733140
|strong zeta peak
|-
|[[77edo]]
|77
|15.5844155844156
|[[414zpi]]
|76.9918536925042
|15.5860645308353
|strong zeta peak
|-
|[[80edo]]
|80
|15.
|[[435zpi]]
|80.0731374302484
|14.9862992572924
|medium zeta peak
|-
|[[131edt]]
|82.6517977178609
|14.5187404646213
|[[453zpi no-2 analogue]]
|82.6705208991009
|14.5154522670130
|strong no-2 zeta peak
|-
|[[83edo]]
|83
|14.4578313253012
|[[455zpi no-3 no-5 analogue]]
|82.9585473728587
|14.4650555970632
|strong no-3 no-5 zeta peak
|-
|[[84edo]]
|84
|14.2857142857143
|[[462zpi]]
|83.9972142607288
|14.2861880666087
|medium zeta peak
|-
|[[87edo]]
|87
|13.7931034482759
|[[483zpi]]
|87.0139255957575
|13.7908960178956
|strong zeta peak
|-
|[[94edo]]
|94
|12.7659574468085
|[[532zpi]]
|93.9836761074943
|12.7681747480009
|strong zeta peak
|-
|[[99edo]]
|99
|12.1212121212121
|[[568zpi]]
|99.0473345956631
|12.1154194093028
|strong zeta peak
|-
|[[103edo]]
|103
|11.6504854368932
|[[596zpi]]
|102.936629522070
|11.6576577800491
|medium zeta peak
|-
|[[111edo]]
|111
|10.8108108108108
|[[655zpi]]
|111.059577998833
|10.8050113427643
|medium zeta peak
|-
|[[327ed7]]
|116.479750184323
|10.3022198974591
|[[695zpi no-2 no-3 no-5 analogue]]
|116.481879086492
|10.3020316070705
|strong no-2 no-3 no-5 zeta peak
|-
|[[118edo]]
|118
|10.1694915254237
|[[706zpi]]
|117.969513574257
|10.1721195895637
|strong zeta peak
|-
|[[130edo]]
|130
|9.23076923076923
|[[796zpi]]
|130.003910460506
|9.23049157328654
|strong zeta peak
|-
|[[140edo]]
|140
|8.57142857142857
|[[872zpi]]
|139.990541024216
|8.57200773152536
|strong zeta peak
|-
|[[152edo]]
|152
|7.89473684210526
|[[965zpi]]
|152.052848107925
|7.89199291517551
|strong zeta peak
|-
|[[171edo]]
|171
|7.01754385964912
|[[1114zpi]]
|170.995891689006
|7.01771246166817
|exceptionally strong zeta peak
|-
|[[270edo]]
|270
|4.44444444444444
|[[1936zpi]]
|270.017794631965
|4.44415154799558
|exceptionally strong zeta peak
|-
|[[311edo]]
|311
|3.85852090032154
|[[2293zpi]]
|311.004029926555
|3.85847090239759
|exceptionally strong zeta peak
|-
|[[342edo]]
|342
|3.50877192982456
| colspan="3" |None
|171*2^n family
|-
|[[684edo]]
|684
|1.75438596491228
| colspan="3" |None
|171*2^n family
|}
|}


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