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A '''zeta peak index''' ('''ZPI''' or '''zpi''') is a [[tuning]] obtained from one of the peaks of the [[The Riemann zeta function and tuning|Riemann zeta function]].
A '''zeta peak index''' ('''ZPI''' or '''zpi''') is an [[equal-step tuning]] obtained from one of the peaks of the [[The Riemann zeta function and tuning|Riemann zeta function]]. The peaks provided are for the common value of σ = 0.5.


For instance, the closest zeta peak of 12edo, which has a value of 12.023edo, is the 34th peak of the Riemann zeta function: this tuning is 34zpi.  
For instance, the closest zeta peak of 12edo, which has a value of 12.023edo, is the 34th peak of the Riemann zeta function: this tuning is 34zpi.  
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ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.
ZPIs are particularly useful when dealing with zeta peak tunings that are not closely associated with an integer [[EDO]]. For example, 22.597edo is 83zpi, 22.807edo is 84zpi, 23.026edo is 85zpi, 23.232edo is 86zpi, and 23.437edo is 87zpi.


ZPIs are a kind of [[equal-step tuning]].
== What are zeta peaks? ==
The Riemann zeta function is a mathematical function known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem in mathematics. However, it also has a musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size.


{|class="wikitable sortable"
Zeta is not an objective metric: There are plenty of other metrics besides zeta for how "well" JI is approximated by an equal tuning, which you can find in: [[:Category:Regular temperament tuning|optimised regular temperament tunings]].
!colspan="3"|Tuning
!colspan="3"|Strength
!colspan="2"|Closest EDO
!colspan="2"|Odd-limit
!colspan="2"|Integer limit
|-
!ZPI
!Steps per octave
!Cents
!Height
!Integral
!Gap
!EDO
!Octave
!Consistent
!Distinct
!Consistent
!Distinct
|-
|[[2zpi]]
|1.972767114412
|608.282646
|2.340551
|1.103823
|10.222388
|[[2edo]]
|1216.565292
|3
|1
|4
|3
|-
|[[3zpi]]
|2.548854231382
|470.799776
|1.459266
|0.414716
|7.471444
|[[3edo]]
|1412.399327
|1
|1
|3
|3
|-
|[[4zpi]]
|3.059761627805
|392.187414
|2.847473
|1.044063
|11.592757
|[[3edo]]
|1176.562242
|5
|3
|6
|4
|-
|[[5zpi]]
|3.496845919785
|343.166393
|0.925523
|0.167718
|5.858780
|[[3edo]]
|1029.499178
|1
|1
|2
|2
|-
|[[6zpi]]
|3.904448124107
|307.341771
|2.942394
|0.927921
|11.574256
|[[4edo]]
|1229.367083
|7
|1
|8
|3
|-
|[[7zpi]]
|4.322093246475
|277.643246
|1.812834
|0.423656
|8.808621
|[[4edo]]
|1110.572985
|1
|1
|3
|3
|-
|[[8zpi]]
|4.652876066087
|257.905000
|1.129621
|0.195040
|6.611021
|[[5edo]]
|1289.524998
|1
|1
|2
|2
|-
|[[9zpi]]
|5.034475598603
|238.356503
|3.664837
|1.131648
|13.386581
|[[5edo]]
|1191.782517
|9
|3
|10
|4
|-
|[[10zpi]]
|5.391231348573
|222.583659
|0.713345
|0.091351
|5.235220
|[[5edo]]
|1112.918295
|1
|1
|2
|2
|-
|[[11zpi]]
|5.683417253069
|211.140577
|2.061177
|0.454332
|9.689889
|[[6edo]]
|1266.843464
|1
|1
|3
|3
|-
|[[12zpi]]
|6.034923687967
|198.842614
|2.913512
|0.699239
|10.852507
|[[6edo]]
|1193.055683
|1
|1
|3
|3
|-
|[[13zpi]]
|6.373110628934
|188.291098
|1.816095
|0.364080
|9.293895
|[[6edo]]
|1129.746590
|1
|1
|3
|3
|-
|[[14zpi]]
|6.632178173869
|180.936032
|0.603289
|0.064947
|4.836586
|[[7edo]]
|1266.552222
|3
|3
|4
|4
|-
|[[15zpi]]
|6.956687656588
|172.495886
|4.166936
|1.162332
|14.234171
|[[7edo]]
|1207.471201
|5
|3
|6
|5
|-
|[[16zpi]]
|7.285924823948
|164.701123
|1.134191
|0.159745
|6.678867
|[[7edo]]
|1152.907860
|1
|1
|2
|2
|-
|[[17zpi]]
|7.541342085555
|159.122870
|1.551068
|0.268585
|8.491473
|[[8edo]]
|1272.982964
|1
|1
|3
|3
|-
|[[18zpi]]
|7.819480070537
|153.462889
|2.004530
|0.355575
|8.808327
|[[8edo]]
|1227.703110
|1
|1
|2
|2
|-
|[[19zpi]]
|8.137425327401
|147.466791
|3.641859
|0.881068
|12.934091
|[[8edo]]
|1179.734328
|3
|3
|7
|4
|-
|[[20zpi]]
|8.427502201950
|142.390945
|0.632316
|0.065792
|5.190978
|[[8edo]]
|1139.127558
|3
|3
|4
|4
|-
|[[21zpi]]
|8.644750943874
|138.812559
|1.368228
|0.209799
|7.977229
|[[9edo]]
|1249.313031
|3
|3
|4
|4
|-
|[[22zpi]]
|8.949991971429
|134.078333
|3.998567
|0.954565
|13.186387
|[[9edo]]
|1206.704993
|7
|5
|8
|6
|-
|[[23zpi]]
|9.242995389543
|129.828043
|1.238064
|0.161912
|6.821862
|[[9edo]]
|1168.452384
|1
|1
|2
|2
|-
|[[24zpi]]
|9.492267674926
|126.418685
|1.952783
|0.359829
|10.156929
|[[9edo]]
|1137.768168
|1
|1
|3
|3
|-
|[[25zpi]]
|9.724186586529
|123.403638
|0.740985
|0.074196
|5.272217
|[[10edo]]
|1234.036379
|1
|1
|2
|2
|-
|[[26zpi]]
|10.008456337259
|119.898610
|4.477141
|1.082282
|14.181485
|[[10edo]]
|1198.986097
|7
|3
|8
|5
|-
|[[27zpi]]
|10.307582490254
|116.419151
|1.505698
|0.225586
|8.414283
|[[10edo]]
|1164.191508
|3
|3
|4
|4
|-
|[[28zpi]]
|10.511042552717
|114.165650
|0.519217
|0.045875
|4.782443
|[[11edo]]
|1255.822145
|1
|1
|3
|3
|-
|[[29zpi]]
|10.757239444987
|111.552783
|2.933506
|0.582845
|11.704948
|[[11edo]]
|1227.080616
|1
|1
|3
|3
|-
|[[30zpi]]
|11.037364857955
|108.721603
|2.698327
|0.469089
|9.930302
|[[11edo]]
|1195.937633
|3
|3
|4
|4
|-
|[[31zpi]]
|11.301192518802
|106.183484
|2.126243
|0.355179
|9.698860
|[[11edo]]
|1168.018329
|1
|1
|3
|3
|-
|[[32zpi]]
|11.535009008294
|104.031128
|1.023117
|0.125398
|6.982530
|[[12edo]]
|1248.373537
|1
|1
|2
|2
|-
|[[33zpi]]
|11.736684783825
|102.243523
|1.198408
|0.146516
|7.026753
|[[12edo]]
|1226.922275
|3
|3
|4
|4
|-
|[[34zpi]]
|12.023183007293
|99.807181
|5.193290
|1.269599
|15.899282
|[[12edo]]
|1197.686169
|9
|5
|10
|6
|}


[[Category:Edonoi]][[Category:Zeta]]
Zeta peaks are those equal-step tunings which the zeta function suggests should "well" approximate JI for this particular (not objective) definition of "well approximating". See the page [[The Riemann zeta function and tuning]] for a fuller explanation of how zeta peaks are arrived at.
 
== Gallery of ZPIs ==
 
=== ZPIs with dedicated pages ===
* [[:Category:Zeta peak indexes]]''
 
=== Table of ZPIs up to 100 steps/octave ===
{{User:Contribution/Gallery of Zeta Peak Indexes (1 - 574)}}
 
=== Table of the first 10 000 ZPIs ===
* [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)]] (may take a long time to load)
 
[[Category:Zeta peak indexes| ]] <!-- main article -->
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