Zeta peak index: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Fredg999 moved page ZPI to Zeta peak index: WP:NCA
Contribution (talk | contribs)
2,880 edits
No edit summary
Line 6: Line 6:


ZPIs are a kind of [[equal-step tuning]].
ZPIs are a kind of [[equal-step tuning]].
{|class="wikitable sortable"
!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]]
[[Category:Edonoi]][[Category:Zeta]]

Revision as of 01:58, 28 March 2024

A zeta peak index (ZPI or zpi) is a tuning obtained from one of the peaks of the Riemann zeta function.

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.

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.

Tuning Strength Closest EDO Odd-limit 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
Retrieved from "https://en.xen.wiki/index.php?title=Zeta_peak_index&oldid=140270"