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5-limit rank two | 7-limit rank two | 7-limit rank three | 11-limit rank two | 11-limit rank three | 13-limit rank two | 13-limit rank three
Given any collection of p-limit commas, there is a finite list of p-limit patent vals tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest TE error; this is the optimal (TE) patent val for the temperament defined by the commas. Note that other definitions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.

By tempering a JI scale using the N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in Scala using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution".

To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q ≤ p, if d is the absolute value in cents of the difference between the tuning of q given by the POTE tuning and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N.

Below are tabulated some values. In each case an identifier which uniquely identifies the temperament in question is given. In the codimension one case, where the temperament is defined by a single comma, the comma is given and used as a name. In other cases, for a temperament of rank n, n independent vals are given. Normally this is by way of integers conjoined by ampersands, such as 2&10 for 7-limit pajara. This tells us we can use the 7-limit patent vals for 2 and 10 to define the temperament. In case n independent patent vals cannot be found, vals using the Keenan notation are given; this adjusts the nth prime mapping to its second-best value by appending the nth lower-case letter in alphabetical order. Thus, "12f" adjusts a patent val for 12 in the 13-limit or above, for instance <12 19 28 34 42 44|, to <12 19 28 34 42 45| (which is actually a better mapping, and hence more useful for this purpose.)

5-limit rank two

Comma: Optimal patent val: 1000 * badness
16/15: 8edo 14.884
648/625: 12edo 47.231
27/25: 14edo 32.801
25/24: 17edo 13.028
250/243: 22edo 30.778
135/128: 23edo 39.556
128/125: 39edo 22.315
3125/3072: 60edo 39.163
2048/2025: 80edo 19.915
81/80: 81edo 7.381
20000/19683: 109edo 48.518
393216/390625: 164edo 40.603
2109375/2097152: 296edo 40.807
15625/15552: 458edo 13.234;
78732/78125: 539edo 35.220
32805/32768: 749edo 4.259
1600000/1594323: 873edo 21.960
6115295232/6103515625: 1400edo 31.181
1224440064/1220703125: 1496edo 43.729
274877906944/274658203125: 1559edo 20.576
31381059609/31250000000: 2760edo 82.423
7629394531250/7625597484987: 3501edo 17.191

7-limit rank two

Name: Optimal patent val: Val name: 1000*badness
Pater: 3edo 3&11b 53.001
Flat: 4edo 3&4 25.381
Father: 5edo 3d&5 21.312
Mother: 5edo 2&3 24.152
Sharptone: 5edo 5&7d 24.848
Baba: 5edo 1&5 44.321
Quint: 5edo 5&15cd 48.312
Mite: 5edo 1cdd&5 54.770
Walid: 6edo 2&6 48.978
Dicot: 7edo 4&7 19.935
Jamesbond: 7edo 7&14c 41.714
Oxygen: 8edo 7d&8 59.866
Beep: 9edo 4&5 18.638
Progression: 9edo 9&17c 48.356
Deflated: 9edo 3&9 59.079
Fluorine: 9edo 9&20bd 55.623
Sharp: 10edo 3d&7d 28.942
Decimal: 10edo 4&10 28.334
Sodium: 11edo 4&11 55.814
Dominant: 12edo 5&7 20.690
Diminished: 12edo 4&12 22.401
August: 12edo 9&12 26.459
Hexe: 12edo 6&12 57.730
Schism: 12edo 12&29de 56.648
Ripple: 12edo 12&23 59.735
Blacksmith: 15edo 5&10 25.640
Opossum: 15edo 7d&8d 40.650
Inflated: 15edo 15&48bc 54.729
Gorgo: 16edo 5&16 46.385
Armodue: 16edo 7&9 49.038
Dichotic: 17edo 3&7 37.565
Octokaidecal: 18edo 10&18 36.747
Uncle: 18edo 5&18 72.653
Negri: 19edo 1&9 26.483
Godzilla: 19edo 5&19 26.747
Keemun: 19edo 4&15 27.408
Wallaby: 19ccdd 2d&5c 58.468
Pajara: 22edo 2&10 20.033
Hedgehog: 22edo 14c&22 43.983
Lemba: 26edo 10&16 62.208
Augene: 27edo 3&12 24.816
Niner: 27edo 9&18 67.157
Beatles: 27edo 10&27 45.872
Nautilus: 29edo 15&29 57.420
Mothra: 31edo 5&26 37.146
Squares: 31edo 31&45 45.993
Immunity: 34edo 5&29 77.631
Injera: 38edo 12&26 31.130
Triforce: 39edo 6&9 54.988
Magic: 41edo 19&41 18.918
Superkleismic: 41edo 15&26 47.932
Quasitemp: 41edo 4&37 60.269
Flattone: 45edo 7&19 38.553
Sensi: 46edo 19&27 25.622
Vines: 46edo 4&42 78.049
Doublewide: 48edo 4&18 43.462
Superpyth: 49edo 5&17 32.318
Passion: 49edo 12&37 62.327
Porky: 51edo 22&51 54.389
Submajor: 53edo 10&33 60.533
Liese: 55edo 19&55 46.706
Sycamore: 56edo 18&19 62.018
Diaschismic: 58edo 12&34 37.914
Porcupine: 59edo 7&15 41.057
Kumonga: 70edo 16&27 87.500
Miracle: 72edo 10&21 16.742
Wizard: 72edo 22&50 40.846
Alicorn 77edo 19&58 40.913
Bidia: 80edo 12&56 56.474
Quartonic: 80edo 26&27 42.632
Meantone: 81edo 12&19 13.707
Myna: 89edo 4&27 27.044
Septimin: 91edo 9&32 54.502
Garibaldi: 94edo 12&29 21.644
Misty: 99edo 12&87 36.802
Nusecond: 101edo 31&101 50.389
Octacot: 109edo 27&41 33.845
Semisept: 111edo 18&31 50.472
Buzzard: 111edo 5&48 47.963
Guiron: 118edo 36&41 47.544
Hemithirds: 118edo 25&31 44.284
Hemikleismic: 121edo 15&38 52.054
Pluto: 121edo 41&80 57.514
Tritonic: 122edo 2&29 47.578
Gravid: 123edo 58&123 47.283
Slender: 125edo 1&31 56.934
Merman: 125edo 2&41 55.078
Rodan: 128edo 5&41 37.112
Subpental: 130edo 19&111 54.303
Bison: 130edo 8&38 70.375
Orwell: 137edo 9&22 20.375
Tertiaseptal: 171edo 31&109 12.995
Septidiasemi: 171edo 10&151 44.115
Pontiac: 171edo 53&118 14.133
Tsaharuk: 171edo 17&77 30.697
Mitonic: 171edo 46&125 25.184
Mutt: 171edo 3&84 28.406
Fifthplus: 171edo 22&149 25.840
Pnict: 171edo 15&141 45.660
Tokko: 171edo 5&166 44.417
Seniority: 171edo 26&145 44.877
Valentine: 185edo 15&31 31.056
Sensei: 187edo 19&84 59.218
Unidec: 190edo 26&46 38.393
Catakleismic: 197edo 19&53 21.501
Quadritikleismic: 212edo 4&68 39.231
Hemiseven: 221edo 5&72 56.557
Octoid: 224edo 72&152 42.670
Compton: 228edo 12&60 35.686
Tritikleismic: 231edo 15&57 56.337
Mirkat: 255edo 72&183 59.376
Decoid: 270edo 10&120 33.902
Acrokleismic 270edo 19&251 56.184
Septisuperfourth: 282edo 22&86 59.241
Hemischis: 313edo 24&53 45.817
Hemiwürschmidt: 328edo 6&31 20.307
Hemififths: 338edo 41&58 22.243
Amity: 350edo 7&39 23.649
Bischismic: 378edo 12&118 54.744
Nessafof: 381edo 15&69 45.478
Septiquarter: 391edo 5&94 53.760
Parakleismic: 415edo 19&80 27.431
Gamera: 422edo 26&73 37.648
Sengagen: 446edo 49&50 57.978
Chromat: 456edo 60&99 57.499
Countercata: 473edo 34&53 52.128
Harry: 534edo 58&72 34.077
Ennealimmal: 612edo 27&45 3.610
Quinmite: 701edo 4&95 37.322
Semidimfourth: 875edo 91&99 55.245
Quasiorwell: 1111edo 31&177 35.832
Amicable: 1131edo 99&212 45.473
Vulture: 1133edo 53&164 36.985
Emmthird 1255edo 58&113 16.736
septichrome 1308edo 60&111 16.814
Sesquiquartififths: 1498edo 41&89 11.244
Brahmagupta: 1547edo 7&217 29.122
Maviloid: 1614edo 76&99 57.632
Osiris: 1696edo 157&171 28.307
Term: 1722edo 12&159 19.950
Neptune: 1778edo 35&68 23.427
Abigail: 1798edo 46&132 37.000
Qak: 1831edo 50&121 29.267
Enneadecal: 2185edo 19&152 10.954
Newt: 2201edo 41&188 41.878
Subneutral: 2236edo 31&348 45.792
Orga: 2482edo 26&244 40.236
Supermajor: 6214edo 80&171 10.836
Semidimi: 8419edo 171&863 15.075
Domain: 22038edo 171&1164 13.979

7-limit rank three

Comma: Optimal patent val: 10^6 * badness
36/35: 12edo 63.675
49/48: 19edo 115.514
875/864: 41edo 212.413
525/512: 45edo 476.454
686/675: 46edo 320.094
50/49: 48edo 127.208
64/63: 49edo 98.719
1029/1000: 55edo 678.164
3125/3087: 94edo 613.731
1728/1715: 111edo 187.341
2430/2401: 137edo 518.625
65625/65536: 171edo 172.601
126/125: 185edo 69.867
1029/1024: 190edo 176.105
225/224: 197edo 36.517
4000/3969: 215edo 215.550
16875/16807: 224edo 341.931
245/243: 283edo 121.191
6144/6125: 381edo 142.375
5120/5103: 391edo 153.052
3136/3125: 446edo 160.039
19683/19600: 587edo 300.987
10976/10935: 695edo 199.139
703125/702464: 2185edo 149.828
2401/2400: 2749edo 15.288
420175/419904: 4306edo 86.449
4375/4374: 8419edo 11.665
250047/250000: 12555edo 21.301
78125000/78121827: 101654edo 20.457

11-limit rank two

Name: Optimal patent val: Val name: 1000*badness
Fervor: 2edo 2&27e 52.054
Decimated: 4edo 4&10e 31.456
Coleto: 4edo 4&27e 48.687
Semafour: 5edo 5&14c 28.510
Maqamic: 7edo 7&10c 40.240
Arnold: 7edo 5&7 26.141
Hystrix: 7edo 7&15d 44.944
Dicot: 7edo 4e&7 19.854
Jamesbond: 7edo 7&14c 23.524
Modus: 7edo 7&27e 35.149
Ptolemy: 7edo 7&38d 58.785
Negric: 9edo 1&9 30.617
Progression: 9edo 8d&9 26.050
Superpelog: 9edo 9&14c 28.535
Octokaidecal: 10edo 10&18e 30.235
Decimal: 10edo 10&14c 26.712
Sharp: 10edo 10&17d 22.366
Ringo: 10edo 10&17c 32.863
Dichosis: 10edo 3&10 41.361
Decibel: 10edo 4&6 32.385
Diminished: 12edo 4&12 22.132
Pajaric: 12edo 2&10 23.798
Dominant: 12edo 5&12 24.180
Domineering: 12edo 7&12 21.978
Lahoh: 12edo 12&14ce 43.062
Duodecim: 12edo 12&24d 30.536
Schism: 12edo 12&41d 37.482
Injerous: 12edo 12e&14c 38.577
Hexe: 12edo 6&12 38.412
Ripple: 12edo 1ce&12 38.811
Hemidim: 12edo 12&20b 54.965
Smate: 14edo 14&17c 42.518
Opossum: 15edo 7d&8d 22.325
Kangaroo: 15edo 2cde&15 43.195
Augene: 15edo 3&12 19.613
Progress: 15edo 2&13 31.036
Darjeeling: 15edo 15&19e 27.648
Ferrier: 15edo 5e&15 29.200
Inflated: 15edo 15&48bce 31.171
Armodue: 16edo 7&9 27.211
Bipelog: 16edo 2&16 35.694
Dichotic: 17edo 7&10 30.680
Sidi: 17edo 14c&17 32.957
Domination: 17c 5e&12e 36.562
Negri: 19edo 9&10 26.190
Meanenneadecal: 19edo 12&19 21.182
Flattone 19edo 7&19 33.839
Godzilla: 19edo 14c&19 28.947
Undevigintone: 19edo 19&38d 36.387
Meanundeci: 19e 12e&19e 31.539
Lisa: 19edo 2cde&19 54.829
Horcrux: 19edo 19&60e 39.282
Spell: 19edo 6&19 59.791
Rhinoceros: 19edo 1ce&19 59.319
Triton: 19edo 2&19 45.675
Qeema: 19edo 4e&19 40.056
Cataleptic: 19edo 19&34d 44.335
Oodako: 21edo 6&21 59.192
Pajara: 22edo 12&22 20.343
Pajarous: 22edo 10&22 28.349
Telepathy: 22edo 19e&22 27.109
Hedgehog: 22edo 14c&22 23.095
Ceratitid: 22edo 1ce&22 51.319
Vigintiduo: 22edo 22&66de 48.372
Astrology: 22edo 22&60de 39.151
Divination: 22edo 22&38d 35.864
Fleetwood: 22edo 22&26e 35.202
Quasisupra: 22edo 17c&22 32.203
Comic: 22edo 2cde&22 45.052
Hemipaj: 22edo 2&20 38.890
Mohamaq: 24edo 17c&24 36.207
Baragon: 24edo 5&19 35.673
Varan: 24edo 19e&24 39.647
Hemiaug: 24edo 24&27e 38.232
Hemidim: 24edo 4e&24 56.576
Semidim: 24edo 24&32c 30.597
Injera: 26edo 12&26 23.124
Lemba: 26edo 10&16 41.563
Secund: 26edo 9&17 42.562
Crepuscular: 26edo 26&34d 40.758
Quarto: 26edo 26&27e 41.786
Cynder: 26edo 5e&26 55.706
Cavalier: 26edo 4e&26 52.899
Sensa: 27edo 19e&27 36.835
Demolished: 28edo 12&28 26.574
Nautilus: 29edo 15&29 26.023
Negril: 29edo 19&29 38.679
Negroni: 29edo 10&29 35.296
Coendou: 29edo 7&29 49.669
Wilsec: 29edo 9&20 41.886
Impunity: 29edo 5e&29 55.243
Immunity: 29edo 5&29 52.661
Mohajira: 31edo 7&24 26.064
Squares: 31edo 31&45e 21.636
Cuboctahedra: 31edo 31&45 56.826
Migration: 31edo 31&100de 25.516
Tritonic: 31edo 2&29 23.659
Casablanca: 31edo 31&42 67.291
Myno: 31edo 4&27 33.434
Cypress: 31edo 20cde&31 42.719
Revelation: 31edo 21&31 32.946
Diesic: 31edo 1ce&31 44.186
Slurpee: 33edo 16&33 50.514
Keemun: 34edo 4&15 27.410
Wollemia: 34edo 27e&34 37.551
Anguirus: 34edo 10&24 49.253
Muggles: 35edo 16&19 48.038
Greenwood: 35edo 21&35 57.471
Liesel: 36edo 19&36 40.721
Elisa: 36e 19e&36e 41.592
Catcall: 36edo 12&24 34.478
Niner: 36edo 9&36 34.861
Beatles: 37edo 27e&37 45.639
Triforce: 39edo 6&9 20.248
Cassandra: 41edo 12&29 23.556
Superkleismic: 41edo 15&26 25.659
Hemimiracle: 41edo 20&21 59.232
Monkey: 41edo 7&34 38.836
Bunya: 41edo 7d&41 31.332
Hocus: 41edo 38d&41 38.519
Witchcraft: 41edo 41&60e 30.706
Trismegistus: 41edo 16&41 45.623
Salsa: 41edo 17&24 39.444
Varan: 41edo 41&46e 44.937
Bohpier: 41edo 41&49 33.949
Octacot: 41edo 27e&41 24.078
Quato: 41edo 4e&41 41.170
Plutino: 41edo 2cde&41 57.966
Eugene: 42edo 15&27 35.554
Amavil: 43edo 9&43 42.649
Kumonga: 43edo 16&43 43.336
Flattone: 45edo 7&19 33.839
Sensor: 46edo 19&27 37.942
Sensus: 46edo 46&73 29.486
Leapday: 46edo 29&46 38.624
Twothirdtonic: 46edo 9&37 40.768
Aerodino: 46edo 5e&46 54.294
Bamity: 46edo 14c&46 35.504
Accord: 46edo 7d&46 42.468
Vines: 46edo 4&42 44.499
Doublewide: 48edo 4&18 32.058
Superpyth: 49edo 22&49 24.976
Passion: 49edo 12&37 40.809
Catalan: 49edo 15&49 36.894
Bimeantone: 50edo 12&50 38.122
Porky: 51edo 22&51 27.268
Orwell: 53edo 9&22 15.231
Tricot: 53edo 53&70 56.134
Cataclysmic: 53edo 53&140d 39.954
Hamity: 53edo 14c&53 42.947
Kleiboh: 53edo 49&53 52.805
Maja: 53edo 45e&53 49.991
Helenus: 53edo 12&53 35.637
Hemigari: 53edo 29&53 50.681
Quartz: 53edo 26&53 53.285
Sycamore: 56edo 18&19 55.940
Diaschismic: 58edo 12&46 25.034
Hemififths: 58edo 41&58 23.498
Thuja: 58edo 15&43 33.078
Alicorn: 58edo 19&58 39.156
Secant: 58edo 26&58 46.373
Infraorwell: 58edo 9&49 40.721
Sruti: 58edo 24&58 41.459
Qilin: 58edo 58&77 41.426
Gorgik: 58edo 21&37 59.260
Porcupine: 59edo 7&15 21.562
Worschmidt: 65edo 31&65 33.436
Hemisensi: 65edo 27e&65 48.714
Ammonite: 66edo 29&37 45.694
Hemikleismic: 68edo 15&38 38.023
Miracle: 72edo 10&31 10.684
Catakleismic: 72edo 19&53 21.849
Compton: 72edo 12&72 22.235
Hours: 72edo 24&48 36.248
Bikleismic: 72edo 72&106 29.319
Ennealimnic: 72edo 27e&72 20.697
Ennealiminal: 72edo 27&45 31.123
Marvolo: 72edo 29&43 28.965
Catalytic: 72edo 53e&72 30.422
Enneaportent: 72edo 9&54 30.426
Marrakesh: 73edo 31&73 40.539
Valentine: 77edo 15&31 16.687
Camahueto 77edo 19&77 65.940
Keen: 78edo 22&56 45.270
Sentinel: 79edo 14&17 39.595
Echidna: 80edo 22&58 25.987
Paradigmic: 80edo 19&80 41.720
Bidia: 80edo 12&68 40.191
Quartonic: 80edo 53&80 34.031
Meanpop: 81edo 19&31 21.543
Newspeak: 81edo 31&84 31.438
Merman: 84edo 2&41 36.383
Rodan: 87edo 5&41 23.093
Clyde: 87edo 19&68 47.41686
Misty: 87edo 12&87 35.932
Emka: 87edo 37&50 54.744
Mothra: 88edo 5&26 25.642
Myna: 89edo 31&58 16.842
Amigo: 89edo 43&46 43.438
Septimin: 91edo 9&32 31.309
Prajapati: 93edo 31&93 42.959
Garibaldi: 94edo 41&53 27.396
Sanjaab: 94edo 29&94 58.040
Magus: 95edo 46&95 45.108
Interpental: 96edo 43&53 51.806
Hitchcock: 99edo 7&39 35.187
Hemiwur: 99edo 6&31 29.270
Hendecatonic: 99edo 22&55 46.088
Nusecond: 101edo 31&101 25.621
Magic: 104edo 19&22 25.621
Meantone: 105edo 12&31 17.027
Bisemidim: 108edo 50&58 41.190
Leapweek: 109edo 17&46 50.679
Buzzard: 111edo 53&58 34.484
Shrutar: 114edo 22&46 26.489
Submajor: 116edo 10&53 50.582
Hemithirds: 118edo 31&87 19.003
Parakleismic: 118edo 19&99 49.711
Quasitemp: 119edo 4&37 43.209
Pluto: 121edo 41&80 29.844
Subfourth: 121edo 58&121 45.323
Tritoni: 122edo 31&91 45.456
Slender: 125edo 31&63 25.342
Srutal: 126edo 46&80 35.315
Würschmidt: 127edo 31&96 24.413
Mosura: 129edo 31&129 31.334
Hemiwürschmidt: 130edo 31&130 21.069
Sesquart: 130edo 41&89 29.306
Bosonic: 130edo 26&130 65.219
Hemimaquila: 130edo 9&121 47.440
Borwell: 137edo 31&106 38.377
Tertia: 140edo 31&109 30.171
Fibo: 140edo 37&103 56.514
Ketchup: 140edo 46&94 39.555
Mystery: 145edo 29&58 34.291
Echidnic: 148edo 10&46 45.127
Heinz: 149edo 46&103 42.412
Hemiseven: 149edo 5&72 28.467
Grendel: 152edo 31&121 19.845
Kwai: 152edo 41&111 26.219
Supers: 152edo 58&94 28.240
Amity: 152edo 53&152 31.506
Whoops: 152edo 15&137 43.743
Monocerus: 154edo 58&154 52.757
Guiron: 159edo 41&77 26.648
Mintone: 161edo 58&103 39.962
Wizard: 166edo 22&50 18.539
Polypyth: 167edo 46&121 51.131
Parkleismic: 179edo 80&179 55.844
Unidec: 190edo 26&46 15.479
Triwell: 190edo 31&97 29.807
Widefourth: 190edo 16&71 40.785
Semihemi: 198edo 58&140 42.487
Harry: 202edo 58&72 15.867
Tertiaseptal: 202edo 31&171 35.576
Swetneus: 207edo 58&149 55.163
Metakleismic: 208edo 87&121 48.570
Marvo: 209edo 72&137 26.882
Kleischismic: 212edo 24&94 36.749
Slithy: 215edo 94&121 45.895
Arch: 217edo 43&87 36.541
Octoid: 224edo 72&152 14.097
Quanharuk: 224edo 41&142 31.549
Countercata: 227edo 34&53 39.770
Tritikleismic: 231edo 15&57 19.333
Subpental: 241edo 19&111 45.352
Ennealimnic: 243edo 72&171 20.347
Bischismic: 248edo 12&118 28.160
Birds: 248edo 31&217 39.921
Essence: 248edo 58&190 46.447
Mirkat: 255edo 72&183 22.126
Sqrtphi: 265edo 49&72 25.515
Quasiorwell: 270edo 31&208 17.540
Counteracro: 270edo 270&1061e 42.572
Marvo: 281edo 72&281 31.685
Septisuperfourth: 282edo 22&86 24.619
Sensa: 283edo 22&65 35.844
Quadritikleismic: 284edo 4&68 23.406
Quincy: 289edo 72&217 30.875
Commatic: 294edo 58&118 30.461
Neominor: 305edo 72&161 27.958
Bison: 306edo 46&84 37.132
Unthirds: 311edo 72&167 22.926
Hemischis: 313edo 53&130 36.289
Semiparakleismic: 316edo 80&118 34.208
Octowerck: 320edo 72&176 30.159
Decal: 320edo 130&190 40.633
Hemitert: 342edo 31&280 15.633
Hemiamity: 350edo 46&106 31.307
Degrees: 360edo 80&140 46.770
Stearnscape: 354edo 72&282 32.096
Catafourth: 363edo 103&130 36.785
Neusec: 388edo 190&388 59.127
Alphaquarter: 391edo 87&152 29.638
Novemkleismic: 405edo 72&261 51.730
Subsemifourth: 407edo 49&103 34.276
Semiennealimmal: 441edo 72&441 34.196
Emkay: 535edo 87&137 35.586
Pogo: 578edo 36&94 31.857
Quatracot: 638edo 190&224 41.043
Hemigamera: 646edo 26&198 40.955
Brahmagupta: 665edo 7&217 52.190
Eris: 703edo 31&193 27.621
Abigail: 764edo 46&132 12.860
Sextile: 1092edo 12&270 29.677
Orga: 1106edo 26&244 16.188
Octant: 1168edo 24&224 44.778
Decoid: 1210edo 130&270 18.735
Deca: 1270edo 80&190 24.329
Trinealimmal: 1323edo 27&243 29.812
Quasithird: 1448edo 164&224 21.125
Bisesqui: 1498edo 130&212 16.968
Vishnu: 1502edo 118&152 14.180
Hemiennealimmal: 1566edo 72&198 6.283
Acrokleismic: 1639edo 19&251 36.878
Quadraennealimmal: 1737edo 342&1053 21.320
Newt: 1972edo 41&188 19.461
Semisupermajor: 2554edo 80&342 12.773

11-limit rank three

Name: Optimal patent val: Val name: 10^5 * badness
Vulcan: 15edo 2&3&10 57.7113
Potassium: 19edo 2&9&10 46.3665
Nickel: 28edo 7&9&12 46.3463
Festival: 26edo 2&10&26 68.8510
Thrasher: 34edo 3&4&12 48.0188
Selenium: 34edo 4&5&10 66.5377
Calliope: 45edo 7&12&19 52.9778
Jubilee: 48edo 4&12&22 55.9842
Ares: 49edo 3&7&12 46.8874
Sonic: 51edo 7&15&51 52.3158
Big Brother: 53edo 9&22&70 50.6556
Orwellian: 53edo 4&5&17 71.513
Prodigy: 72edo 2&10&29 33.4745
Mirwomo: 72edo 7&10&24 64.0943
Gamel: 72edo 5e&31&72 85.266
Oxpecker: 77edo 8d&15&31 69.945
Starling: 77edo 4&15&27 67.680
Srutal: 80edo 46&80 25.286
Erato: 81edo 12&19&31 55.7521
Sensawer: 87edo 41&46&73 79.632
Euterpe: 88edo 5&12&26 53.5809
Thrush: 89edo 3&12&31 35.2591
Aplonis: 89edo 19&31&58 64.7511
Mandos: 89edo 7&24&58 75.1090
Malcolm: 94edo 19e&41&53 124.957
Shrusus: 95edo 22&46&95 87.6795
Zeus: 99edo 6&7&9 39.9882
Apollo: 104edo 12&19&22 52.9766
Supermagic: 104edo 4&7&15 64.0789
Supernatural: 104edo 19&22&82e 88.8164
Octarod: 104edo 19&22&49 58.0664
Guanyin: 111edo 9&22&58 41.1252
Manwe: 111edo 15&31&80 67.9150
Bisector : 114edo 14c&22&46 108.9026
Zelda: 118edo: 9&22&87 64.1882
Parahemif: 17&24&58 134.547
Varda: 126edo 12&22&46 72.8125
Minerva: 127edo 1&9&12 38.1064
Clio: 129edo 7&12&24 73.7590
Pele: 145edo 5&12&29 64.8199
Laka: 152edo 41&53&58 82.5317
Marvel: 166edo 9&10&22 25.5033
Fantastic: 166edo 12&22&50 74.3439
Portent: 190edo 5&10&26 23.4092
Beyla: 190edo 7&19&46 72.097
Jove: 202edo 10&31&58 24.1055
Tolerant: 208edo 5&41&80 103.923
Belobog: 248edo 12&31&87 60.9402
Spectacle: 281edo 10&31&281 49.8669
Jupiter: 282edo 9&22&108 56.2368
Sensamagic: 283edo 5&17&19 72.231
Agni: 284edo 4&6&27 49.3790
Enlil: 284edo 4&15&38 79.8062
Trimyna: 294edo 2&29&58 81.4047
Triglav: 316edo 31&80&87 81.896
Varuna: 320edo 12&26&46 41.8714
History: 491edo 15&29&43 60.3636
Hades: 578edo 22&50&58 36.1398
Albus: 640edo 19&53&58 62.1100
Indra: 703edo 10&31&152 26.6020
Hanuman: 998edo 15&57&152 49.965
Loki: 1092edo 12&22&86 49.2550
Ganesha: 1502edo 31&87&152 39.0462
Freya: 1566edo 10&31&167 16.9702
Tyr: 1566edo 15&57&183 25.8374
Baldur: 1696edo 4&58&72 16.6115
Skadi: 2014edo 31&97&121 34.8944
Thor: 2554edo 26&46&152 8.8834
Ennealimmic: 2619edo 27&45&171 27.4685
Van Gogh: 6992edo 22&58&284 29.7565
Odin: 7464edo 12&42&72 11.6151

13-limit rank two

Name: Optimal patent val: Val name: 1000*badness
Diminished: 4edo 4&8d 19.509
Maqamic: 7edo 7&10c 27.214
Modus: 7edo 7&27e 23.806
Arnold: 7edo 5&7 23.300
Dominatrix: 7edo 7&12f 18.289
Jamesbond: 7edo 7&14c 23.003
Septimal: 7edo 7&14cf 22.569
Ptolemy: 7edo 7&38df 34.316
Progression: 9edo 8d&9 18.158
Negric: 9edo 1&9 20.205
Pentoid: 9edo 5&9 21.159
Ringo: 10edo 10&17c 22.619
Pajaric: 10edo 10&12f 20.461
Pajaro: 10edo 10&22f 27.355
Augustus: 12edo 9&12 22.236
Dominion: 12edo 5&12 27.295
Hexe: 12edo 6f&12f 35.946
Ripple: 12f 1ce&12f 31.639
Smate: 14edo 14&17c 36.836
Opossum: 15edo 7d&8d 19.389
Kangaroo: 15edo 2cde&15 32.653
Blacksmith: 15edo 5&10 20.498
Archytas clan: 15edo 5e&15 22.325
Progress: 15edo 2&15 26.214
Porcupine: 15edo 7&15 21.276
Belauensis: 15edo 14c&15 29.816
Kumbaya: 15edo 4&15 31.633
Darjeeling: 15edo 15&19e 21.445
Augene: 15edo 3&15 20.785
Ogene: 15edo 12&15 22.890
Armodue: 16edo 7&9 19.351
Domination: 17c 5e&12e 27.435
Dominant: 17c 12f&17c 24.108
Progressive: 17c 2f&15f 32.721
Negri: 19edo 9&10 17.833
Sensis: 19edo 8d&19 20.017
Meanenneadecal: 19edo 12f&19 21.182
Meanplop: 19edo 12e&19 27.666
Godzilla: 19edo 14cf&19 22.503
Undevigintone: 19edo 19&38df 22.933
Lisa: 19edo 2cde&19 36.144
Keemun: 19edo 4&19 29.749
Qeema: 19edo 4ef&19 29.419
Cataleptic: 19edo 19&34d 27.343
Rhinoceros: 19edo 1ce&19 39.343
Spell: 19edo 6&19 45.591
Cantrip: 19edo 6f&19 41.603
Muggles: 19edo 16&19 30.386
August: 21edo 9&21 18.448
Oodako: 21edo 6&21 42.323
Telepathy: 22edo 19e&22 25.522
Sorcery: 22edo 19&22 25.829
Divination: 22edo 22f&60e 34.551
Blair: 22edo 9&22 23.086
Hedgehog: 22edo 14cf&22
Pajarous: 22edo 10&22 25.176
Pajara: 22edo 12&22 27.642
Porkpie: 22edo 7&22 26.043
Ceratitid: 22edo 1ce&22 44.739
Urchin: 22edo 14c&22f 25.233
Fleetwood: 22edo 22&84bdf 31.835
Suprapyth: 22edo 17&22 36.336
Comic: 22edo 2cde&22 41.470
Astrology: 22edo 6&16 34.376
Quasisupra: 22edo 17c&22 30.219
Catcall: 24edo 12f&24 28.363
Triforce: 24edo 9&15 20.248
Hemiaug: 24edo 3de&24 30.159
Hemidim: 24edo 4ef&24 39.030
Semidim: 24edo 24&32c 30.597
Mohamaq: 24edo 17c&24 28.738
Varan: 24edo 19e&24 25.676
Injera: 26edo 26&104bcf 21.565
Flattone: 26edo 7&19 22.260
Lemba: 26edo 10&16 25.477
Crepuscular: 26edo 26&34d 24.368
Quarto: 26edo 26&27e 27.692
Cynder: 26edo 5e&26 34.124
Secundly: 26edo 9&17 26.212
Cavalier: 26edo 4ef&26 35.040
Sensa: 27edo 19e&27 23.258
Ponens: 27edo 7&27 39.043
Fervor: 27edo 2f&27e 39.705
Octocat: 27e 14cf&27e 27.601
Minah: 27e 4ef&27e 27.568
Roman: 29edo 3&23 30.043
Porky: 29edo 22&29 26.543
Coendou: 29edo 7&29 30.233
Nautilus: 29edo 15&29 22.285
Negril: 29edo 19&29 24.383
Negroni: 29edo 10&29 21.559
Wilsec: 29edo 9&20 25.192
Quindecic: 30edo 15&30 28.944
Impunity: 29edo 5e&29 33.196
Immunity: 29edo 5&29 37.078
Meantone: 31edo 12f&31 18.048
Squares: 31edo 17c&31 25.514
Agora: 31edo 31&76e 24.522
Mohajira: 31edo 7&24 23.388
Winston: 31edo 9&31 19.931
Revelation: 31edo 21&31 29.452
Tritonic: 31edo 29&31 22.993
Lupercalia: 31edo 15&31 21.328
Murakuc: 31edo 31&73f 41.394
Nusecond: 31edo 31&70f 23.323
Cypress: 31edo 11cdeef&31 37.849
Maneh: 31edo 4ef&31 29.868
Prajapati: 31edo 31&93f 37.885
Würschmidt: 31edo 31&65d 23.593
Worseschmidt: 31edo 3def&31 34.382
Hemithir: 31edo 31&68e 31.199
Hemiwar: 31edo 6f&31 44.886
Sentinel: 31edo 14&17 37.408
Diesic: 31edo 1ce&31 38.116
Semishly: 31edo 31&49f 28.408
Slurpee: 33edo 16&33 33.102
Kema: 34edo 15&19 22.749
Wollemia: 34edo 27e&34 31.219
Anguirus: 34edo 10&24 30.829
Secund: 35edo 9&35 49.684
Greenwood: 35edo 21&35 54.009
Liesel: 36edo 19&36 27.304
Niner: 36edo 9&36 24.143
Beatles: 37edo 27e&37 30.161
Enjera: 38edo 12f&26f 26.542
Horoscope: 38edo 16&38 35.284
Miraculous: 41edo 10&31 18.669
Superkleismic: 41edo 15&26 21.478
Varan: 41edo 5e&41 32.284
Hemimiracle: 41edo 20&21 43.151
Cassandra: 41edo 29&41 20.749
Octacot: 41edo 27e&41 23.276
Octopod: 41edo 14cf&41 28.326
Monkey: 41edo 7&34 28.410
Bunya: 41edo 7d&41 24.886
Magic: 41edo 19&41 21.509
Hocus: 41edo 41&79d 30.280
Witchcraft: 41edo 41&60e 23.547
Trismegistus: 41edo 16&41 33.081
Bohpier: 41edo 41&90ef 24.864
Salsa: 41edo 17&24 30.793
Plutino: 41edo 2cde&41 40.182
Quato: 41edo 4ef&41 30.081
Slendi: 41edo 41&124ef 43.644
Jerome: 43edo 26&43 29.285
Amavil: 43edo 9&43 25.791
Meridetone: 43edo 43&117df 26.421
Kumonga: 43edo 16&43 28.920
Vigin: 44edo 22&44 29.849
Sensor: 46edo 19&27 25.474
Sensus: 46edo 46&73 20.789
Dwynwen: 46edo 15&46 23.461
Leapday: 46edo 29&46 24.732
Twothirdtonic: 46edo 9&37 25.941
Bamity: 46edo 14cf&46 30.885
Magus: 46edo 3de&46 43.024
Aerodactyl: 46edo 5&46 33.986
Aerodino: 46edo 5e&46 35.836
Vines: 46edo 4&42 29.693
Superpyth: 49edo 22&49 24.673
Bimeantone: 50edo 12f&50 28.817
Cataclysmic: 53edo 53&140d 22.554
Hemikleismic: 53edo 15&53 26.005
Orwell: 53edo 22&31 19.718
Doublethink: 53edo 9&44 27.120
Tricot: 53edo 53&70 32.102
Hamity: 53edo 14cf&53 29.753
Kleiboh: 53edo 4ef&53 31.074
Maja: 53edo 45ef&53 27.532
Helenus: 53edo 53&118d 26.284
Hemigari: 53edo 29&53 27.464
Quartz: 53edo 26&53 28.818
Sycamore: 56edo 18&19 34.295
Diaschismic: 58edo 46&58 18.926
Myna: 58edo 31&58 17.125
Hemififths: 58edo 41&58 19.090
Infraorwell: 58edo 9&58 23.683
Thuja: 58edo 15&43 22.838
Alicorn: 58edo 19&58 23.667
Bisemidim: 58edo 50&58 23.877
Sruti: 58edo 24&58 23.791
Qilin: 58edo 58&77 22.842
Gorgik: 58edo 21&37 32.205
Gallium: 62edo 31&62 25.484
Semivalentine: 62edo 16&30 32.749
Hemimeantone: 62edo 43&62 31.433
Ammonite: 66edo 29&37 27.168
Hemiwur: 68edo 6&31 28.432
Manna: 72edo 41&72 17.012
Hendec: 72edo 26&46 17.707
Catakleismic: 72edo 19&53 16.883
Lizard: 72edo 22&50 21.781
Neominor: 72edo 72&161f 26.942
Compton: 72edo 72&228f 21.852
Comptone: 72edo 12&72 25.144
Hours: 72edo 24&72 26.931
Bikleismic: 72edo 72&106 21.814
Marvolo: 72edo 29&43 21.470
Semimiracle: 72edo 10&62 24.622
Catalytic: 72edo 19e&72 22.337
Enneaportent: 72edo 9&54 22.322
Ennealim: 72edo 27e&72 20.697
Ennealiminal: 72edo 27&72 30.325
Marrakesh: 73edo 31&73 40.774
Grosstone: 74edo 12&31 25.899
Valentino: 77edo 31&46 20.665
Camahueto 77edo 19&77 36.155
Echidna: 80edo 22&58 23.679
Bidia: 80edo 12&68 41.137
Paradigmic: 80edo 19&80 35.781
Quartonic: 80edo 53&80 23.875
Meanpop: 81edo 19&31 20.883
Soothsaying: 82edo 22&60 55.443
Secant: 84edo 26&58 46.373
Rodan: 87edo 41&46 18.448
Clyde: 87edo 19&68 26.842
Misty: 87edo 12f&87 27.820
Emka: 87edo 37&50 29.741
Mothra: 88edo 5&26 23.954
Amigo: 89edo 43&46 30.666
Septimin: 91edo 9&41 23.117
Kumhar: 93edo 31&93 48.582
Slender: 94edo 31&63 25.913
Sanjaab: 94edo 29&94 33.849
Interpental: 96edo 43&53 29.680
Supers: 94edo 58&94 21.644
Mosura: 98edo 31&67 36.857
Hitchcock: 99edo 7&39 22.448
Echidnic: 102edo 10&46 28.874
Benediction: 103edo 31&72 15.715
Phicordial: 103edo 10&103 33.198
Hemisecordite: 103edo 41&62 25.589
Necromancy: 104edo 22&41 25.275
Leapweek: 109edo 17&46 32.727
Buzzard: 111edo 53&58 18.842
Semisept: 111edo 31&80 25.204
Kwai: 111edo 41&111 24.555
Shrutar: 114edo 46&68 28.057
Submajor: 116edo 10&53 27.689
Guiron: 118edo 41&77 28.444
Hemithirds: 118edo 31&56 21.738
Kleischismic: 118edo 24&94 37.640
Grendel: 121edo 31&121 24.839
Pluto: 121edo 41&80 25.717
Subfourth: 121edo 58&121 23.800
Harry: 130edo 58&72 13.046
Bosonic: 130edo 26&130 32.946
Bison: 130edo 46&84 23.504
Sesquart: 130edo 41&89 22.396
Septisuperquad: 130edo 22&108 33.038
Hemimaquila: 130edo 9&121 24.445
Countercata: 140edo 34&53 20.156
Degrees: 140edo 80&140 32.718
Ketchup: 140edo 46&94 24.824
Fibo: 140edo 37&103 27.429
Omicronbeta: 144edo 72&144 29.956
Heinz: 149edo 46&103 25.779
Hemiseven: 149edo 5&72 21.900
Octopus: 152edo 72&152 21.679
Monocerus: 154edo 58&154 28.795
Tritikleismic: 159edo 15&72 15.652
Mintone: 161edo 58&103 21.849
Gizzard: 166edo 72&166 20.252
Polypyth: 167edo 46&121 30.292
Tertiaseptal: 171edo 31&171 36.876
Parkleismic: 179edo 80&179 36.559
Mirkat: 183edo 72&111 18.632
Ekadash: 190edo 72&118 20.381
Widefourth: 190edo 16&71 21.636
Sqrtphi: 193edo 72&121 20.040
Hemimist: 198edo 87&111 22.923
Semihemi: 198edo 58&140 21.188
Amity: 205edo 53&205 28.008
Quanic: 205edo 94&111 32.481
Swetneus: 207edo 58&149 28.876
Metakleismic: 208edo 87&121 24.371
Quadritikleismic: 212edo 68&72 18.731
Slithy: 215edo 94&121 28.492
Arch: 217edo 43&87 19.504
Octoid: 224edo 72&224 15.274
Quanharuk: 224edo 41&142 21.392
Mystery: 232edo 29&58 18.591
Subpental: 241edo 19&111 23.940
Ennealimnic: 243edo 72&171 20.250
Essence: 248edo 58&190 26.107
Subsemifourth: 255edo 103&255 28.387
Hemiennealimmal: 270edo 72&198 12.505
Acrokleismic: 270edo 19&251 26.818
Counteracro: 270edo 270&1331c 26.028
Septisuperfourth: 282edo 130&282 22.887
Sensa: 283edo 22&65 31.366
Sextilififths: 289edo 29&130 25.276
Quincy: 289edo 72&145 23.862
Hemiwürschmidt: 291edo 31&130 23.074
Hemischis: 313edo 53&130 20.816
Octowerck: 320edo 72&248 27.632
Decal: 320edo 130&190 23.948
Novemkleismic: 333edo 72&261 39.072
Catafourth: 363edo 103&130 21.694
Bischismic: 378edo 12&118 28.722
Neusec: 388edo 190&198 30.941
Hemigamera: 422edo 26&198 20.416
Semiennealimmal: 441edo 72&441 26.122
Birds: 465edo 31&217 35.680
Emkay: 535edo 87&137 17.853
Fermionic: 546edo 130&286 43.581
Pogo: 578edo 36&94 17.514
Quatracot: 638edo 190&224 22.643
Unthirds: 694edo 72&311 20.888
Octant: 696edo 24&224 30.425
Quasithird: 836edo 164&224 29.501
Newt: 851edo 41&229 13.830
Decoid: 940edo 130&270 13.475
Semihemiennealimmal 954edo 126&144 13.104
Acyuta: 962edo 270&692 15.805
Deca: 1000edo 80&190 16.810
Quasiorwell: 1111edo 31&239 17.921
Abigail: 1258edo 46&178 8.856
Ananta: 1468edo 118&152 23.678

13-limit rank three

Name: Optimal patent val: Val name: 10^5 * badness
Mockingbird: 15edo 3&4&15 85.942
Catbird: 19edo 4&12f&19 90.475
Potassium: 19edo 9&10&11 73.346
Nickel: 28edo 7&9&21 64.705
Negric: 29edo 1&9&10 80.756
Nightingale: 31edo 3&15&28 83.714
Thrasher: 34edo 12&15&19 87.636
Selenium: 34edo 5&9&10 78.368
Krypton: 36edo 9&10&36 72.673
Foreboding: 41edo 5&10&26 87.267
Prodigy: 41edo 10&29&31 73.639
Veery: 46edo 12&15&46 99.122
Athene: 53edo 12f&22&53 81.781
Thrush: 58edo 31&46&58 67.663
Bluebird: 58edo 12&15&31 91.548
Aplonis: 58edo 19&31&58 82.078
Jofur: 58edo 10&31&58 74.932
Parahemif: 58edo 17&24&58 119.366
Deecee: 72edo 9&22&41 92.048
Prodigious: 72edo 2&29&41 89.974
Marvelcat: 72edo 9&10&44 99.966
Beyla: 72edo 7&19&46 83.962
Varda: 80edo 22&46&58 97.041
Sensawer: 87edo 41&46&73 92.807
Tripod: 91edo 9&10&31 74.546
Isis: 94edo 10&22&31 86.583
Malcolm: 94edo 19e&41&53 107.452
Marvel: 103edo 19&22&31 68.977
Prodigal: 103edo 31&72&84 88.910
Mirage: 103edo 10&31&62 73.850
Apollo: 104edo 22&29&41 103.106
Cerberus: 130edo 22&50&58 76.512
Ominous: 149edo 5&26&46 75.055
Portending: 159edo 15&26&46 62.715
Momentous: 164edo 10&31&46 83.202
Hecate: 166edo 19&41&53 72.113
Marvell: 166edo 9&31&63 86.160
Ulmo: 179edo 7&39&80 93.162
Vibhu: 183edo 31&72&111 82.598
Portentous: 190edo 15&31&56 66.119
Vish: 193edo 41&72&121 92.976
Minos: 193edo 9&54&58 72.381
Namaka: 198edo 29&53&58 78.124
Laka: 205edo 41&53&58 82.186
Kujuku 208edo 24&29&58 105.976
Tolerant: 208edo 41&46&80 102.099
Spectacle: 209edo 31^72&209 100.924
Agni: 212edo 6&31&72 92.251
Yajna: 224edo 31&72&121 69.986
Pele: 232edo 29&41&46 70.328
Trimyna: 232edo 29&31&56 81.405
Jovial: 243edo 41&58&72 62.380
Borneo: 270edo 15&72&111 54.937
History: 289edo 15&29&43 53.955
Enlil: 299edo 15&19&53 68.005
Madagascar: 313edo 19&53&58 56.006
Akea: 321edo 7&34&46 82.191
Jovis: 363edo 31&58&72 54.221
Hanuman: 535edo 15&72&137 65.278
Hades: 578edo 36&58&72 53.531
Vili: 643edo 37&93&103 73.821
Shibi: 703edo 41&72&111 72.094
Greenland: 940edo 58&72&198 44.343
Baffin: 940edo 34&43&53 60.443
Ennealimmic: 981edo 27&72&171 75.488
Zisa: 1241edo 21&31&109 82.974
Thunor: 1258edo 26&46&178 34.135
Frigg: 1344edo 6&58&103 93.387
Donar: 1448edo 46&80&144 32.154
Rym: 1870edo 46&106&118 54.933
Freya: 2615edo 10&31&229 85.515