User:Contribution/Gallery of Zeta Peak Indexes at sigma = 1
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Presentation
This page lists Zeta Peak Indexes (ZPIs) computed at σ = 1. Each ZPI corresponds to the largest value of ∣ζ(σ+iτ)∣ found between two successive non-trivial zeros of the Riemann zeta function, mapped to musical units using a base-2 transform. The index nzpi simply means “the peak found in the n-th zero interval”.
What the columns mean
- ZPI (σ = 1): Name and link to the entry for that peak.
- Steps per octave: Real-valued EDO implied by the peak.
- Step size (cents): The size of that step.
- Height: The peak magnitude of ∣ζ(1+iτ)∣.
- EDO: The nearest equal division of the octave obtained by rounding steps per octave to an integer.
- Octave (cents): The shrinked / stretched octave size that this rounded EDO implies for the given step size.
Integer limit: For every reduced fraction a/b with 1 ≤ b < a ≤ N, we compare two step mappings: direct mapping and prime-factor mapping.
- Consistent: highest N for which the two mappings agree for every such a/b.
- Distinct: highest N for which they agree and all mapped step numbers are pairwise distinct (no collisions).
Filters and scope
The table is filtered to show musically relevant entries only: height ≥ 1.6 and step size ≥ 12.0 cents. Every EDO from 2edo through 99edo has at least one ZPI represented. Values depend on the chosen σ; different choices will yield different selections.
| Tuning | Strength | Closest EDO | Integer limit | ||||
|---|---|---|---|---|---|---|---|
| ZPI (σ = 1) | Steps per octave | Step size (cents) | Height | EDO | Octave (cents) | Consistent | Distinct |
| 2zpi (σ = 1) | 1.96786068448 | 609.799265499 | 1.85479 | 2edo | 1219.59853100 | 4 | 3 |
| 4zpi (σ = 1) | 3.05757511990 | 392.467871743 | 2.06097 | 3edo | 1177.40361523 | 6 | 4 |
| 6zpi (σ = 1) | 3.90427837366 | 307.355133306 | 2.03662 | 4edo | 1229.42053322 | 8 | 3 |
| 9zpi (σ = 1) | 5.03333981877 | 238.410288835 | 2.39426 | 5edo | 1192.05144418 | 10 | 4 |
| 12zpi (σ = 1) | 6.03429281600 | 198.863402322 | 1.88601 | 6edo | 1193.18041393 | 3 | 3 |
| 15zpi (σ = 1) | 6.95688550773 | 172.490980147 | 2.55384 | 7edo | 1207.43686103 | 6 | 5 |
| 19zpi (σ = 1) | 8.13534227567 | 147.504549820 | 2.19888 | 8edo | 1180.03639856 | 7 | 4 |
| 22zpi (σ = 1) | 8.94981568027 | 134.080973605 | 2.36401 | 9edo | 1206.72876245 | 8 | 6 |
| 26zpi (σ = 1) | 10.0089746115 | 119.892401228 | 2.57426 | 10edo | 1198.92401228 | 8 | 5 |
| 29zpi (σ = 1) | 10.7609124972 | 111.514706611 | 1.73344 | 11edo | 1226.66177273 | 3 | 3 |
| 30zpi (σ = 1) | 11.0361501117 | 108.733569936 | 1.63570 | 1196.06926930 | 4 | 4 | |
| 34zpi (σ = 1) | 12.0220488259 | 99.8165967700 | 2.85866 | 12edo | 1197.79916124 | 10 | 6 |
| 38zpi (σ = 1) | 12.9702736722 | 92.5192505826 | 1.80339 | 13edo | 1202.75025757 | 4 | 4 |
| 39zpi (σ = 1) | 13.2304682673 | 90.6997375873 | 1.63605 | 1179.09658863 | 4 | 4 | |
| 42zpi (σ = 1) | 13.9020220557 | 86.3183783764 | 2.50514 | 14edo | 1208.45729727 | 7 | 5 |
| 47zpi (σ = 1) | 15.0534708836 | 79.7158349246 | 2.69313 | 15edo | 1195.73752387 | 8 | 7 |
| 51zpi (σ = 1) | 15.9438120096 | 75.2643093932 | 2.26766 | 16edo | 1204.22895029 | 6 | 6 |
| 56zpi (σ = 1) | 17.0432556931 | 70.4090827252 | 2.65741 | 17edo | 1196.95440633 | 4 | 4 |
| 61zpi (σ = 1) | 18.1175033636 | 66.2342915530 | 1.91404 | 18edo | 1192.21724795 | 3 | 3 |
| 65zpi (σ = 1) | 18.9489976130 | 63.3278880767 | 3.02387 | 19edo | 1203.22987346 | 10 | 7 |
| 70zpi (σ = 1) | 19.9849819349 | 60.0450880520 | 1.85716 | 20edo | 1200.90176104 | 4 | 4 |
| 71zpi (σ = 1) | 20.2189188724 | 59.3503543674 | 1.88061 | 1187.00708735 | 6 | 6 | |
| 74zpi (σ = 1) | 20.7934810727 | 57.7103947052 | 1.61754 | 21edo | 1211.91828881 | 4 | 4 |
| 75zpi (σ = 1) | 21.0247432519 | 57.0756078027 | 2.15264 | 1198.58776386 | 4 | 4 | |
| 80zpi (σ = 1) | 22.0251749360 | 54.4831086920 | 2.99601 | 22edo | 1198.62839122 | 12 | 8 |
| 84zpi (σ = 1) | 22.8100982503 | 52.6082784401 | 1.93341 | 23edo | 1209.99040412 | 3 | 3 |
| 90zpi (σ = 1) | 24.0053572889 | 49.9888414723 | 2.82476 | 24edo | 1199.73219533 | 6 | 6 |
| 95zpi (σ = 1) | 24.9683144487 | 48.0609134617 | 2.01987 | 25edo | 1201.52283654 | 6 | 6 |
| 100zpi (σ = 1) | 25.9356337472 | 46.2683893402 | 2.71167 | 26edo | 1202.97812285 | 14 | 9 |
| 106zpi (σ = 1) | 27.0853383248 | 44.3044124320 | 2.90524 | 27edo | 1196.21913566 | 10 | 8 |
| 110zpi (σ = 1) | 27.8188648928 | 43.1361956940 | 1.68295 | 28edo | 1207.81347943 | 6 | 6 |
| 111zpi (σ = 1) | 28.0282687206 | 42.8139180469 | 1.90453 | 1198.78970531 | 6 | 6 | |
| 116zpi (σ = 1) | 28.9431579907 | 41.4605759463 | 2.68561 | 29edo | 1202.35670244 | 8 | 7 |
| 122zpi (σ = 1) | 30.0624012664 | 39.9169710152 | 1.71886 | 30edo | 1197.50913046 | 6 | 6 |
| 127zpi (σ = 1) | 30.9779815456 | 38.7371913897 | 3.23190 | 31edo | 1200.85293308 | 12 | 9 |
| 133zpi (σ = 1) | 32.0697555104 | 37.4184330658 | 2.21331 | 32edo | 1197.38985810 | 4 | 4 |
| 137zpi (σ = 1) | 32.7688376176 | 36.6201576633 | 1.84757 | 33edo | 1208.46520289 | 4 | 4 |
| 138zpi (σ = 1) | 32.9701792271 | 36.3965264409 | 1.76149 | 1201.08537255 | 4 | 4 | |
| 144zpi (σ = 1) | 34.0437506778 | 35.2487600839 | 3.07414 | 34edo | 1198.45784285 | 6 | 6 |
| 149zpi (σ = 1) | 34.9249367119 | 34.3594037092 | 2.03484 | 35edo | 1202.57912982 | 3 | 3 |
| 151zpi (σ = 1) | 35.2920666399 | 34.0019759184 | 1.67293 | 1190.06915714 | 3 | 3 | |
| 155zpi (σ = 1) | 35.9827898689 | 33.3492762616 | 2.80355 | 36edo | 1200.57394542 | 8 | 8 |
| 161zpi (σ = 1) | 37.0298101647 | 32.4063233017 | 2.46975 | 37edo | 1199.03396216 | 8 | 8 |
| 166zpi (σ = 1) | 37.8928300093 | 31.6682601882 | 2.68249 | 38edo | 1203.39388715 | 6 | 6 |
| 173zpi (σ = 1) | 39.1204815298 | 30.6744690524 | 2.58738 | 39edo | 1196.30429304 | 7 | 7 |
| 178zpi (σ = 1) | 39.9671419065 | 30.0246638303 | 1.95959 | 40edo | 1200.98655321 | 4 | 4 |
| 184zpi (σ = 1) | 40.9880790756 | 29.2768050385 | 3.32966 | 41edo | 1200.34900658 | 16 | 10 |
| 191zpi (σ = 1) | 42.1844137726 | 28.4465254506 | 1.78057 | 42edo | 1194.75406892 | 3 | 3 |
| 196zpi (σ = 1) | 43.0234004818 | 27.8917981043 | 2.78019 | 43edo | 1199.34731849 | 8 | 8 |
| 202zpi (σ = 1) | 44.0173438504 | 27.2619811881 | 2.19686 | 44edo | 1199.52717228 | 6 | 6 |
| 207zpi (σ = 1) | 44.8428900827 | 26.7600950293 | 2.39414 | 45edo | 1204.20427632 | 7 | 7 |
| 214zpi (σ = 1) | 46.0106419996 | 26.0809227572 | 3.25119 | 46edo | 1199.72244683 | 14 | 11 |
| 220zpi (σ = 1) | 47.0043481568 | 25.5295530532 | 2.01996 | 47edo | 1199.88899350 | 2 | 2 |
| 226zpi (σ = 1) | 47.9873892126 | 25.0065698445 | 2.64954 | 48edo | 1200.31535253 | 6 | 6 |
| 233zpi (σ = 1) | 49.1378005687 | 24.4211174719 | 2.53180 | 49edo | 1196.63475612 | 7 | 7 |
| 238zpi (σ = 1) | 49.9382924730 | 24.0296562132 | 2.90274 | 50edo | 1201.48281066 | 10 | 9 |
| 244zpi (σ = 1) | 50.8939583740 | 23.5784371729 | 1.73567 | 51edo | 1202.50029582 | 4 | 4 |
| 245zpi (σ = 1) | 51.0772688312 | 23.4938168673 | 2.26424 | 1198.18466023 | 4 | 4 | |
| 249zpi (σ = 1) | 51.7001700683 | 23.2107553692 | 1.61354 | 52edo | 1206.95927920 | 3 | 3 |
| 251zpi (σ = 1) | 52.0406940559 | 23.0588777066 | 1.92556 | 1199.06164074 | 4 | 4 | |
| 257zpi (σ = 1) | 52.9969882711 | 22.6427961125 | 3.46399 | 53edo | 1200.06819396 | 10 | 10 |
| 264zpi (σ = 1) | 54.1168355338 | 22.1742455589 | 1.72387 | 54edo | 1197.40926018 | 4 | 4 |
| 269zpi (σ = 1) | 54.8977887530 | 21.8588039201 | 2.38189 | 55edo | 1202.23421561 | 4 | 4 |
| 270zpi (σ = 1) | 55.0788120696 | 21.7869622621 | 1.74011 | 1198.28292441 | 6 | 6 | |
| 276zpi (σ = 1) | 56.0105455489 | 21.4245369018 | 2.65718 | 56edo | 1199.77406650 | 8 | 8 |
| 281zpi (σ = 1) | 56.7866988983 | 21.1317090671 | 1.61804 | 57edo | 1204.50741683 | 6 | 6 |
| 282zpi (σ = 1) | 56.9640637187 | 21.0659128170 | 2.21228 | 1200.75703057 | 8 | 8 | |
| 289zpi (σ = 1) | 58.0645692462 | 20.6666477609 | 3.25823 | 58edo | 1198.66557013 | 16 | 12 |
| 295zpi (σ = 1) | 58.9938251043 | 20.3411119363 | 1.87842 | 59edo | 1200.12560424 | 5 | 5 |
| 301zpi (σ = 1) | 59.9223835273 | 20.0259056693 | 2.98826 | 60edo | 1201.55434016 | 10 | 10 |
| 308zpi (σ = 1) | 61.0095132211 | 19.6690636697 | 1.79038 | 61edo | 1199.81288385 | 6 | 6 |
| 309zpi (σ = 1) | 61.1838523056 | 19.6130180559 | 2.06261 | 1196.39410141 | 6 | 6 | |
| 314zpi (σ = 1) | 61.9395458904 | 19.3737293800 | 2.69409 | 62edo | 1201.17122156 | 8 | 8 |
| 321zpi (σ = 1) | 63.0197888699 | 19.0416378969 | 2.87513 | 63edo | 1199.62318750 | 8 | 8 |
| 326zpi (σ = 1) | 63.7779204673 | 18.8152889152 | 1.92975 | 64edo | 1204.17849057 | 3 | 3 |
| 328zpi (σ = 1) | 64.0989938281 | 18.7210426925 | 1.71163 | 1198.14673232 | 2 | 2 | |
| 334zpi (σ = 1) | 65.0145858034 | 18.4573966776 | 3.23462 | 65edo | 1199.73078404 | 6 | 6 |
| 340zpi (σ = 1) | 65.9189579111 | 18.2041712737 | 1.99235 | 66edo | 1201.47530407 | 2 | 2 |
| 342zpi (σ = 1) | 66.2425799721 | 18.1152364613 | 1.84689 | 1195.60560644 | 4 | 4 | |
| 346zpi (σ = 1) | 66.8216441694 | 17.9582531217 | 1.75197 | 67edo | 1203.20295915 | 4 | 4 |
| 347zpi (σ = 1) | 66.9945453666 | 17.9119060132 | 2.36700 | 1200.09770288 | 4 | 4 | |
| 354zpi (σ = 1) | 68.0496579343 | 17.6341812204 | 3.14200 | 68edo | 1199.12432299 | 10 | 10 |
| 359zpi (σ = 1) | 68.7887294987 | 17.4447181791 | 1.74330 | 69edo | 1203.68555435 | 6 | 6 |
| 360zpi (σ = 1) | 68.9566524714 | 17.4022368690 | 1.94551 | 1200.75434396 | 6 | 6 | |
| 367zpi (σ = 1) | 70.0057147532 | 17.1414577257 | 2.50896 | 70edo | 1199.90204080 | 10 | 10 |
| 368zpi (σ = 1) | 70.1765489892 | 17.0997294293 | 1.65227 | 1196.98106005 | 7 | 7 | |
| 374zpi (σ = 1) | 71.0590727060 | 16.8873580009 | 1.79434 | 71edo | 1199.00241807 | 6 | 6 |
| 380zpi (σ = 1) | 71.9512656175 | 16.6779554147 | 3.61665 | 72edo | 1200.81278986 | 18 | 13 |
| 387zpi (σ = 1) | 72.9913942566 | 16.4402942596 | 1.67798 | 73edo | 1200.14148095 | 4 | 4 |
| 388zpi (σ = 1) | 73.1571361202 | 16.4030477906 | 2.14656 | 1197.42248871 | 6 | 6 | |
| 394zpi (σ = 1) | 74.0470091225 | 16.2059212684 | 2.21069 | 74edo | 1199.23817386 | 6 | 6 |
| 400zpi (σ = 1) | 74.9121965725 | 16.0187533526 | 1.99703 | 75edo | 1201.40650145 | 6 | 6 |
| 401zpi (σ = 1) | 75.0853553740 | 15.9818115533 | 2.56694 | 1198.63586650 | 6 | 6 | |
| 406zpi (σ = 1) | 75.8047568970 | 15.8301411299 | 1.79862 | 76edo | 1203.09072587 | 3 | 3 |
| 409zpi (σ = 1) | 76.2674414726 | 15.7341058888 | 1.67201 | 1195.79204755 | 3 | 3 | |
| 414zpi (σ = 1) | 76.9924672555 | 15.5859403235 | 3.28825 | 77edo | 1200.11740491 | 10 | 10 |
| 421zpi (σ = 1) | 78.0208663564 | 15.3805008332 | 2.29670 | 78edo | 1199.67906499 | 8 | 8 |
| 427zpi (σ = 1) | 78.8946639244 | 15.2101541512 | 2.49755 | 79edo | 1201.60217795 | 6 | 6 |
| 435zpi (σ = 1) | 80.0733926855 | 14.9862514845 | 3.14833 | 80edo | 1198.90011876 | 12 | 12 |
| 441zpi (σ = 1) | 80.9458566656 | 14.8247241975 | 2.24505 | 81edo | 1200.80266000 | 8 | 8 |
| 448zpi (σ = 1) | 81.9570350248 | 14.6418181140 | 2.76665 | 82edo | 1200.62908535 | 8 | 8 |
| 453zpi (σ = 1) | 82.6799133832 | 14.5138033036 | 1.82605 | 83edo | 1204.64567420 | 3 | 3 |
| 455zpi (σ = 1) | 82.9731036327 | 14.4625179421 | 1.78766 | 1200.38898919 | 5 | 5 | |
| 462zpi (σ = 1) | 83.9950884037 | 14.2865496400 | 3.19687 | 84edo | 1200.07016976 | 10 | 10 |
| 470zpi (σ = 1) | 85.1557457841 | 14.0918265579 | 2.13929 | 85edo | 1197.80525742 | 4 | 4 |
| 475zpi (σ = 1) | 85.8578619456 | 13.9765884312 | 2.10884 | 86edo | 1201.98660509 | 4 | 4 |
| 483zpi (σ = 1) | 87.0139579095 | 13.7908908965 | 3.44872 | 87edo | 1199.80750799 | 16 | 14 |
| 489zpi (σ = 1) | 87.8681343292 | 13.6568280317 | 1.74922 | 88edo | 1201.80086679 | 3 | 3 |
| 492zpi (σ = 1) | 88.3138811137 | 13.5878979031 | 1.64240 | 1195.73501547 | 3 | 3 | |
| 497zpi (σ = 1) | 89.0215260329 | 13.4798857476 | 3.02681 | 89edo | 1199.70983154 | 12 | 12 |
| 504zpi (σ = 1) | 90.0112315448 | 13.3316696084 | 2.11530 | 90edo | 1199.85026476 | 8 | 8 |
| 505zpi (σ = 1) | 90.1696009943 | 13.3082545200 | 1.72137 | 1197.74290680 | 7 | 7 | |
| 510zpi (σ = 1) | 90.8575221337 | 13.2074920361 | 2.69164 | 91edo | 1201.88177528 | 10 | 10 |
| 511zpi (σ = 1) | 91.0190832461 | 13.1840484127 | 1.69295 | 1199.74840556 | 6 | 6 | |
| 518zpi (σ = 1) | 91.9996980234 | 13.0435210743 | 2.06133 | 92edo | 1200.00393884 | 6 | 6 |
| 519zpi (σ = 1) | 92.1623477354 | 13.0205016418 | 2.24497 | 1197.88615104 | 6 | 6 | |
| 525zpi (σ = 1) | 93.0001476743 | 12.9032053175 | 2.30725 | 93edo | 1199.99809453 | 8 | 8 |
| 532zpi (σ = 1) | 93.9843698073 | 12.7680805059 | 3.39762 | 94edo | 1200.19956756 | 24 | 15 |
| 540zpi (σ = 1) | 95.1175548504 | 12.6159677032 | 2.22020 | 95edo | 1198.51693180 | 7 | 7 |
| 546zpi (σ = 1) | 95.9558568688 | 12.5057504477 | 2.93099 | 96edo | 1200.55204298 | 6 | 6 |
| 553zpi (σ = 1) | 96.9320641382 | 12.3798044607 | 1.90133 | 97edo | 1200.84103269 | 5 | 5 |
| 554zpi (σ = 1) | 97.0862691382 | 12.3601412502 | 1.80558 | 1198.93370127 | 4 | 4 | |
| 560zpi (σ = 1) | 97.9216718532 | 12.2546927283 | 1.95488 | 98edo | 1200.95988737 | 4 | 4 |
| 568zpi (σ = 1) | 99.0456175574 | 12.1156294402 | 3.56676 | 99edo | 1199.44731458 | 12 | 12 |