Optimal patent val
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[edit]
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[edit]
Name: Optimal patent val: Val name: 1000*badness
Progression: 9edo 9&17c 48.356
Diminished: 12edo 4&12 22.401
Blacksmith: 15edo 5&10 25.640
Inflated: 15edo 15&48bc 54.729
Octokaidecal: 18edo 10&18 36.747
Superkleismic: 41edo 15&26 47.932
Doublewide: 48edo 4&18 43.462
Diaschismic: 58edo 12&34 37.914
Nusecond: 101edo 31&101 50.389
Hemithirds: 118edo 25&31 44.284
Hemikleismic: 121edo 15&38 52.054
Subpental: 130edo 19&111 54.303
Tertiaseptal: 171edo 31&109 12.995
Septidiasemi: 171edo 10&151 44.115
Fifthplus: 171edo 22&149 25.840
Seniority: 171edo 26&145 44.877
Valentine: 185edo 15&31 31.056
Catakleismic: 197edo 19&53 21.501
Quadritikleismic: 212edo 4&68 39.231
Tritikleismic: 231edo 15&57 56.337
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
Bischismic: 378edo 12&118 54.744
Septiquarter: 391edo 5&94 53.760
Parakleismic: 415edo 19&80 27.431
Countercata: 473edo 34&53 52.128
Ennealimmal: 612edo 27&45 3.610
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
Abigail: 1798edo 46&132 37.000
Enneadecal: 2185edo 19&152 10.954
Subneutral: 2236edo 31&348 45.792
Supermajor: 6214edo 80&171 10.836
Semidimi: 8419edo 171&863 15.075
Domain: 22038edo 171&1164 13.979
7-limit rank three[edit]
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[edit]
Name: Optimal patent val: Val name: 1000*badness
Progression: 9edo 8d&9 26.050
Superpelog: 9edo 9&14c 28.535
Octokaidecal: 10edo 10&18e 30.235
Diminished: 12edo 4&12 22.132
Domineering: 12edo 7&12 21.978
Injerous: 12edo 12e&14c 38.577
Kangaroo: 15edo 2cde&15 43.195
Darjeeling: 15edo 15&19e 27.648
Inflated: 15edo 15&48bce 31.171
Domination: 17c 5e&12e 36.562
Meanenneadecal: 19edo 12&19 21.182
Undevigintone: 19edo 19&38d 36.387
Meanundeci: 19e 12e&19e 31.539
Rhinoceros: 19edo 1ce&19 59.319
Cataleptic: 19edo 19&34d 44.335
Telepathy: 22edo 19e&22 27.109
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
Crepuscular: 26edo 26&34d 40.758
Demolished: 28edo 12&28 26.574
Cuboctahedra: 31edo 31&45 56.826
Migration: 31edo 31&100de 25.516
Casablanca: 31edo 31&42 67.291
Cypress: 31edo 20cde&31 42.719
Revelation: 31edo 21&31 32.946
Superkleismic: 41edo 15&26 25.659
Hemimiracle: 41edo 20&21 59.232
Witchcraft: 41edo 41&60e 30.706
Trismegistus: 41edo 16&41 45.623
Twothirdtonic: 46edo 9&37 40.768
Doublewide: 48edo 4&18 32.058
Bimeantone: 50edo 12&50 38.122
Cataclysmic: 53edo 53&140d 39.954
Diaschismic: 58edo 12&46 25.034
Hemififths: 58edo 41&58 23.498
Infraorwell: 58edo 9&49 40.721
Worschmidt: 65edo 31&65 33.436
Hemisensi: 65edo 27e&65 48.714
Hemikleismic: 68edo 15&38 38.023
Catakleismic: 72edo 19&53 21.849
Bikleismic: 72edo 72&106 29.319
Ennealimnic: 72edo 27e&72 20.697
Ennealiminal: 72edo 27&45 31.123
Catalytic: 72edo 53e&72 30.422
Enneaportent: 72edo 9&54 30.426
Paradigmic: 80edo 19&80 41.720
Interpental: 96edo 43&53 51.806
[[Würschmidt_family#Hemiw�rschmidt|Hemiwur]]: 99edo 6&31 29.270
Hendecatonic: 99edo 22&55 46.088
Nusecond: 101edo 31&101 25.621
Bisemidim: 108edo 50&58 41.190
Hemithirds: 118edo 31&87 19.003
Parakleismic: 118edo 19&99 49.711
Subfourth: 121edo 58&121 45.323
[[Würschmidt_family#W�rschmidt|Würschmidt]]: 127edo 31&96 24.413
Hemiwürschmidt: 130edo 31&130 21.069
Hemimaquila: 130edo 9&121 47.440
Monocerus: 154edo 58&154 52.757
Polypyth: 167edo 46&121 51.131
Parkleismic: 179edo 80&179 55.844
Widefourth: 190edo 16&71 40.785
Semihemi: 198edo 58&140 42.487
Tertiaseptal: 202edo 31&171 35.576
Swetneus: 207edo 58&149 55.163
Metakleismic: 208edo 87&121 48.570
Kleischismic: 212edo 24&94 36.749
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
Quasiorwell: 270edo 31&208 17.540
Counteracro: 270edo 270&1061e 42.572
Septisuperfourth: 282edo 22&86 24.619
Quadritikleismic: 284edo 4&68 23.406
Commatic: 294edo 58&118 30.461
Neominor: 305edo 72&161 27.958
Unthirds: 311edo 72&167 22.926
Hemischis: 313edo 53&130 36.289
Semiparakleismic: 316edo 80&118 34.208
Octowerck: 320edo 72&176 30.159
Hemitert: 342edo 31&280 15.633
Hemiamity: 350edo 46&106 31.307
Stearnscape: 354edo 72&282 32.096
Catafourth: 363edo 103&130 36.785
Alphaquarter: 391edo 87&152 29.638
Novemkleismic: 405edo 72&261 51.730
Subsemifourth: 407edo 49&103 34.276
Semiennealimmal: 441edo 72&441 34.196
Quatracot: 638edo 190&224 41.043
Hemigamera: 646edo 26&198 40.955
Brahmagupta: 665edo 7&217 52.190
Sextile: 1092edo 12&270 29.677
Decoid: 1210edo 130&270 18.735
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
Semisupermajor: 2554edo 80&342 12.773
11-limit rank three[edit]
Name: Optimal patent val: Val name: 10^5 * badness
Potassium: 19edo 2&9&10 46.3665
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
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
Oxpecker: 77edo 8d&15&31 69.945
Starling: 77edo 4&15&27 67.680
Sensawer: 87edo 41&46&73 79.632
Euterpe: 88edo 5&12&26 53.5809
Aplonis: 89edo 19&31&58 64.7511
Malcolm: 94edo 19e&41&53 124.957
Shrusus: 95edo 22&46&95 87.6795
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
Marvel: 166edo 9&10&22 25.5033
Fantastic: 166edo 12&22&50 74.3439
Portent: 190edo 5&10&26 23.4092
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
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[edit]
Name: Optimal patent val: Val name: 1000*badness
Diminished: 4edo 4&8d 19.509
Dominatrix: 7edo 7&12f 18.289
Progression: 9edo 8d&9 18.158
Kangaroo: 15edo 2cde&15 32.653
Blacksmith: 15edo 5&10 20.498
Archytas clan: 15edo 5e&15 22.325
Belauensis: 15edo 14c&15 29.816
Darjeeling: 15edo 15&19e 21.445
Domination: 17c 5e&12e 27.435
Progressive: 17c 2f&15f 32.721
Meanenneadecal: 19edo 12f&19 21.182
Godzilla: 19edo 14cf&19 22.503
Undevigintone: 19edo 19&38df 22.933
Cataleptic: 19edo 19&34d 27.343
Rhinoceros: 19edo 1ce&19 39.343
Telepathy: 22edo 19e&22 25.522
Divination: 22edo 22f&60e 34.551
Ceratitid: 22edo 1ce&22 44.739
Fleetwood: 22edo 22&84bdf 31.835
Quasisupra: 22edo 17c&22 30.219
Injera: 26edo 26&104bcf 21.565
Crepuscular: 26edo 26&34d 24.368
Revelation: 31edo 21&31 29.452
Lupercalia: 31edo 15&31 21.328
Cypress: 31edo 11cdeef&31 37.849
Prajapati: 31edo 31&93f 37.885
Würschmidt: 31edo 31&65d 23.593
Worseschmidt: 31edo 3def&31 34.382
Miraculous: 41edo 10&31 18.669
Superkleismic: 41edo 15&26 21.478
Hemimiracle: 41edo 20&21 43.151
Witchcraft: 41edo 41&60e 23.547
Trismegistus: 41edo 16&41 33.081
Meridetone: 43edo 43&117df 26.421
Twothirdtonic: 46edo 9&37 25.941
Aerodactyl: 46edo 5&46 33.986
Bimeantone: 50edo 12f&50 28.817
Cataclysmic: 53edo 53&140d 22.554
Hemikleismic: 53edo 15&53 26.005
Doublethink: 53edo 9&44 27.120
Diaschismic: 58edo 46&58 18.926
Hemififths: 58edo 41&58 19.090
Infraorwell: 58edo 9&58 23.683
Semivalentine: 62edo 16&30 32.749
Hemimeantone: 62edo 43&62 31.433
Catakleismic: 72edo 19&53 16.883
Neominor: 72edo 72&161f 26.942
Bikleismic: 72edo 72&106 21.814
Semimiracle: 72edo 10&62 24.622
Catalytic: 72edo 19e&72 22.337
Enneaportent: 72edo 9&54 22.322
Ennealiminal: 72edo 27&72 30.325
Paradigmic: 80edo 19&80 35.781
Soothsaying: 82edo 22&60 55.443
Interpental: 96edo 43&53 29.680
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
Hemithirds: 118edo 31&56 21.738
Kleischismic: 118edo 24&94 37.640
Subfourth: 121edo 58&121 23.800
Septisuperquad: 130edo 22&108 33.038
Hemimaquila: 130edo 9&121 24.445
Countercata: 140edo 34&53 20.156
Omicronbeta: 144edo 72&144 29.956
Monocerus: 154edo 58&154 28.795
Tritikleismic: 159edo 15&72 15.652
Polypyth: 167edo 46&121 30.292
Tertiaseptal: 171edo 31&171 36.876
Parkleismic: 179edo 80&179 36.559
Widefourth: 190edo 16&71 21.636
Hemimist: 198edo 87&111 22.923
Semihemi: 198edo 58&140 21.188
Swetneus: 207edo 58&149 28.876
Metakleismic: 208edo 87&121 24.371
Quadritikleismic: 212edo 68&72 18.731
Quanharuk: 224edo 41&142 21.392
Subpental: 241edo 19&111 23.940
Ennealimnic: 243edo 72&171 20.250
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
Sextilififths: 289edo 29&130 25.276
Hemiwürschmidt: 291edo 31&130 23.074
Hemischis: 313edo 53&130 20.816
Octowerck: 320edo 72&248 27.632
Novemkleismic: 333edo 72&261 39.072
Catafourth: 363edo 103&130 21.694
Bischismic: 378edo 12&118 28.722
Hemigamera: 422edo 26&198 20.416
Semiennealimmal: 441edo 72&441 26.122
Fermionic: 546edo 130&286 43.581
Quatracot: 638edo 190&224 22.643
Unthirds: 694edo 72&311 20.888
Quasithird: 836edo 164&224 29.501
Semihemiennealimmal 954edo 126&144 13.104
Quasiorwell: 1111edo 31&239 17.921
Ananta: 1468edo 118&152 23.678
13-limit rank three[edit]
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
Nightingale: 31edo 3&15&28 83.714
Thrasher: 34edo 12&15&19 87.636
Foreboding: 41edo 5&10&26 87.267
Prodigy: 41edo 10&29&31 73.639
Athene: 53edo 12f&22&53 81.781
Bluebird: 58edo 12&15&31 91.548
Aplonis: 58edo 19&31&58 82.078
Parahemif: 58edo 17&24&58 119.366
Prodigious: 72edo 2&29&41 89.974
Marvelcat: 72edo 9&10&44 99.966
Sensawer: 87edo 41&46&73 92.807
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
Vibhu: 183edo 31&72&111 82.598
Portentous: 190edo 15&31&56 66.119
Namaka: 198edo 29&53&58 78.124
Kujuku 208edo 24&29&58 105.976
Tolerant: 208edo 41&46&80 102.099
Spectacle: 209edo 31^72&209 100.924
Yajna: 224edo 31&72&121 69.986
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
Madagascar: 313edo 19&53&58 56.006
Hanuman: 535edo 15&72&137 65.278
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