Optimal patent val

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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

[[Würschmidt_family#Hemiw�rschmidt|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_family#W�rschmidt|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

[[Gamelismic_clan#Miracle-Phicordial[|Phicordial]]: 103edo 10&103 33.198

[[Gamelismic_clan#Miracle-Hemisecordite[|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