Optimal patent val

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Given any collection of p-limit commas, there is a finite list of p-limit [[Patent val|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 [[Tenney-Euclidean temperament measures|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 N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [[http://www.huygens-fokker.org/scala/|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 [[Patent val|patent 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, the [[Normal lists|normal val list]] is given. Note that this differs from the method of [[Graham Breed]]'s [[http://x31eq.com/temper/|regular temperament finder]], which does not assume the numbers refer to patent vals.

==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
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
7629394531250/7625597484987: [[3501edo]] 17.191

==7-limit rank two== 
Name: Optimal patent val: Val name: 1000*badness
[[Trienstonic clan|Father]]: [[5edo]] [<1 0 4 -2|, <0 1 -1 3|] 21.312
[[Archytas clan|Mother]]: [[5edo]] 2&3 24.152
[[Meantone family|Sharptone]]: [[5edo]] [<1 0 -4 -2|, <0 1 4 3|] 24.848
[[Dicot family|Dicot]]: [[7edo]] 4&7 19.935
[[Bug family|Beep]]: [[9edo]] 4&5 18.638
[[Dicot family|Sharp]]: [[10edo]] [<1 1 2 1|, <0 2 1 6|] 28.942
[[Dicot family|Decimal]]: [[10edo]] 4&10 28.334
[[Meantone family|Dominant]]: [[12edo]] 5&7 20.690
[[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.401
[[Augmented family|August]]: [[12edo]] 9&12 26.459
[[Augmented family|Hexe]]: [[12edo]] 6&12 57.730
[[Archytas clan|Blacksmith]]: [[15edo]] 5&10 25.640
[[Trienstonic clan|Opossum]]: [[15edo]] [<1 2 3 4|, <0 -3 -5 -9|] 40.650
[[Gamelismic clan|Gorgo]]: [[16edo]] 5&16 46.385
[[Trienstonic clan|Octokaidecal]]: [[18edo]] 10&18 36.747
[[Marvel temperaments|Negri]]: [[19edo]] 1&9 26.483
[[Meantone family|Godzilla]]: [[19edo]] 5&19 26.747
[[Kleismic family|Keemun]]: [[19edo]] 4&15 27.408
[[Diaschismic family|Pajara]]: [[22edo]] 2&10 20.033
[[Jubilismic clan|Lemba]]: [[26edo]] 10&16 62.208
[[Augmented family|Augene]]: [[27edo]] 3&12 24.816
[[Archytas clan|Beatles]]: [[27edo]] 10&27 45.872
[[Meantone family|Mothra]]: [[31edo]] 5&26 37.146
[[Meantone family|Injera]]: [[38edo]] 12&26 31.130
[[Augmented family|Hemis]]: [[39edo]] 6&9 54.988
[[Magic family|Magic]]: [[41edo]] 19&41 18.918
[[Gamelismic clan|Superkleismic]]: [[41edo]] 15&26 47.932
[[Meantone family|Flattone]]: [[45edo]] 7&19 38.553
[[Starling temperaments|Sensi]]: [[46edo]] 19&27 25.622
[[Jubilismic clan|Doublewide]]: [[48edo]] 4&18 43.462
[[Archytas clan|Superpyth]]: [[49edo]] 5&17 32.318
[[Meantone family|Liese]]: [[55edo]] 19&55 46.706
[[Gamelismic clan|Miracle]]: [[72edo]] 10&21 16.742
[[Marvel temperaments|Wizard]]: [[72edo]] 22&50 40.846
[[Meantone family|Meantone]]: [[81edo]] 12&19 13.707
[[Starling temperaments|Myna]]: [[89edo]] 4&27 27.044
[[Marvel temperaments|Septimin]]: [[91edo]] 9&32 54.502
[[Schismatic family|Garibaldi]]: [[94edo]] 12&29 21.644
[[Hemimean clan|Misty]]: [[99edo]] 12&87 36.802
[[Starling temperaments|Nusecond]]: [[101edo]] 31&101 50.389
[[Tetracot family|Octacot]]: [[109edo]] 27&41 33.845
[[Hemimean clan|Semisept]]: [[111edo]] 18&31 50.472
[[Hemifamity temperaments|Buzzard]]: [[111edo]] 5&48 47.963
[[Gamelismic clan|Guiron]]: [[118edo]] 36&41 47.544
[[Marvel temperaments|Tritonic]]: [[122edo]] 2&29 47.578
[[Marvel temperaments|Slender]]: [[125edo]] 1&31 56.934
[[Gamelismic clan|Rodan]]: [[128edo]] 5&41 37.112
[[Semicomma family|Orwell]]: [[137edo]] 9&22 20.375
[[Breedsmic temperaments|Tertiaseptal]]: [[171edo]] 31&109 12.995
[[Schismatic family|Pontiac]]: [[171edo]] 53&118 14.133
[[Ragismic microtemperaments|Mitonic]]: [[171edo]] 46&125 25.184
[[Mutt family|Mutt]]: [[171edo]] 3&84 28.406
[[Starling temperaments|Valentine]]: [[185edo]] 15&31 31.056
[[Gamelismic clan|Unidec]]: [[190edo]] 26&46 38.393
[[Kleismic family|Catakleismic]]: [[197edo]] 19&53 21.501
[[Kleismic family|Quadritikleismic]]: [[212edo]] 4&68 39.231
[[Ragismic microtemperaments|Octoid]]: [[224edo]] 72&152 42.670
[[Pythagorean family|Compton]]: [[228edo]] 12&60 35.686
[[Kleismic family|Tritikleismic]]: [[231edo]] 15&57 56.337
[[Mirkwai clan|Mirkat]]: [[255edo]] 72&183 59.376
[[Breedsmic temperaments|Decoid]]: [[270edo]] 10&120 33.902
[[Wuerschmidt family|Hemiwuerschmidt]]: [[328edo]] 6&31 20.307
[[Breedsmic temperaments|Hemififths]]: [[338edo]] 41&58 22.243
[[Ragismic microtemperaments|Amity]]: [[350edo]] 7&39 23.649
[[Schismatic family|Bischismic]]: [[378edo]] 12&118 54.744
[[Ragismic microtemperaments|Parakleismic]]: [[415edo]] 19&80 27.431
[[Ragismic microtemperaments|Gamera]]: [[422edo]] 26&73 37.648
[[Kleismic family|Countercata]]: [[473edo]] 34&53 52.128
[[Breedsmic temperaments|Harry]]: [[534edo]] 58&72 34.077
[[Ragismic microtemperaments|Ennealimmal]]: [[612edo]] 27&45 3.610
[[Breedsmic temperaments|Quasiorwell]]: [[1111edo]] 31&177 35.832
[[Schismatic family|Sesquiquartififths]]: [[1498edo]] 41&89 11.244
[[Schismatic family|Term]]: [[1722edo]] 12&159 19.950
[[Gammic family|Neptune]]: [[1778edo]] 35&68 23.427
[[Ragismic microtemperaments|Abigail]]: [[1798edo]] 46&132 37.000
[[Ragismic microtemperaments|Enneadecal]]: [[2185edo]] 19&152 10.954
[[Ragismic microtemperaments|Supermajor]]: [[6214edo]] 80&171 10.836

==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
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
[[Marvel temperaments|Negric]]: [[9edo]] 1&9 30.617
[[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.132
[[Trienstonic clan|Opossum]]: [[15edo]] [<1 2 3 4 4|, <0 -3 -5 -9 -4|] 22.325
[[Marvel temperaments|Negri]]: [[19edo]] 9&10 26.190
[[Marvel temperaments|Meanenneadecal]]: [[19edo]] 12&19 21.182
[[Diaschismic family|Pajara]]: [[22edo]] 12&22 20.343 
[[Meantone family|Mohajira]]: [[31edo]] 7&24 26.064
[[Meantone family|Squares]]: [[31edo]] 31&45 56.826
[[Starling temperaments|Casablanca]]: [[31edo]] 31&42 67.291
[[Kleismic family|Keemun]]: [[34edo]] 4&15 27.410
[[Augmented family|Hemis]]: [[39edo]] 6&9 20.248
[[Schismatic family|Cassandra]]: [[41edo]] 12&29 23.556
[[Gamelismic clan|Superkleismic]]: [[41edo]] 15&26 25.659
[[Starling temperaments|Sensor]]: [[46edo]] 19&27 37.942
[[Starling temperaments|Sensus]]: [[46edo]] 46&73 29.486
[[Hemifamity temperaments|Leapday]]: [[46edo]] 29&46 38.624
[[Jubilismic clan|Doublewide]]: [[48edo]] 4&18 32.058
[[Semicomma family|Orwell]]: [[53edo]] 9&22 15.231
[[Diaschismic family|Diaschismic]]: [[58edo]] 12&46 25.034
[[Breedsmic temperaments|Hemififths]]: [[58edo]] 41&58 23.498
[[Porcupine family|Porcupine]]: [[59edo]] 7&15 21.562
[[Gamelismic clan|Miracle]]: [[72edo]] 10&31 10.684
[[Kleismic family|Catakleismic]]: [[72edo]] 19&53 21.849
[[Pythagorean family|Compton]]: [[72edo]] 12&72 22.235
[[Starling temperaments|Valentine]]: [[77edo]] 15&31 16.687
[[Diaschismic family|Keen]]: [[78edo]] 22&56 45.270
[[Diaschismic family|Echidna]]: [[80edo]] 22&58 25.987
[[Meantone family|Meanpop]]: [[81edo]] 19&31 21.543
[[Gamelismic clan|Rodan]]: [[87edo]] 5&41 23.093
[[Meantone family|Mothra]]: [[88edo]] 5&26 25.642
[[Starling temperaments|Myna]]: [[89edo]] 31&58 16.842
[[Marvel temperaments|Septimin]]: [[91edo]] 9&32 31.309
[[Schismatic family|Garibaldi]]: [[94edo]] 41&53 27.396
[[Starling temperaments|Nusecond]]: [[101edo]] 31&101 25.621
[[Magic family|Magic]]: [[104edo]] 19&22 25.621
[[Meantone family|Meantone]]: [[105edo]] 12&31 17.027
[[Hemifamity temperaments|Buzzard]]: [[111edo]] 53&58 34.484
[[Diaschismic family|Shrutar]]: [[114edo]] 22&46 26.489
[[Gamelismic clan|Hemithirds]]: [[118edo]] 31&87 19.003
[[Ragismic microtemperaments|Parakleismic]]: [[118edo]] 19&99 49.711
[[Marvel temperaments|Tritonic]]: [[122edo]] 31&91 45.456
[[Marvel temperaments|Slender]]: [[125edo]] 31&63 25.342
[[Wuerschmidt family|Hemiwuerschmidt]]: [[130edo]] 31&130 21.069
[[Hemifamity temperaments|Mystery]]: [[145edo]] 29&58 34.291
[[Diaschismic family|Echidnic]]: [[148edo]] 10&46 45.127
[[Mirkwai clan|Grendel]]: [[152edo]] 31&121 19.845
[[Mirkwai clan|Kwai]]: [[152edo]] 41&111 26.219
[[Schismatic family|Guiron]]: [[159edo]] 41&77 26.648
[[Marvel temperaments|Wizard]]: [[166edo]] 22&50 18.539
[[Gamelismic clan|Unidec]]: [[190edo]] 26&46 15.479
[[Breedsmic temperaments|Harry]]: [[202edo]] 58&72 15.867
[[Breedsmic temperaments|Tertiaseptal]]: [[202edo]] 31&171 35.576
[[Ragismic microtemperaments|Octoid]]: [[224edo]] 72&152 14.097
[[Kleismic family|Countercata]]: [[227edo]] 34&53 39.770
[[Kleismic family|Tritikleismic]]: [[231edo]] 15&57 19.333
[[Mirkwai clan|Mirkat]]: [[255edo]] 72&183 22.126
[[Breedsmic temperaments|Quasiorwell]]: [[270edo]] 31&208 17.540
[[Marvel temperaments|Marvo]]: [[281edo]] 72&281 31.685
[[Kleismic family|Quadritikleismic]]: [[284edo]] 4&68 23.406
[[Ragismic microtemperaments|Hemiamity]]: [[350edo]] 46&106 31.307
[[Ragismic microtemperaments|Abigail]]: [[764edo]] 46&132 12.860
[[Breedsmic temperaments|Decoid]]: [[1210edo]] 130&270 18.735
[[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283

==11-limit rank three== 
Name: Optimal patent val: Val name: 10^6 * badness
[[Archytas family|Vulcan]]: [[15edo]] 2&3&10 577.113
[[Marvel family|Potassium]]: [[19edo]] 2&9&10 463.665
[[Quartonic family|Nickel]]: [[28edo]] 7&9&12 463.463
[[Jubilismic family|Festival]]: [[26edo]] 2&10&26 688.510
[[Starling family|Thrasher]]: [[34edo]] 3&4&12 480.188
[[Semaphore family|Selenium]]: [[34edo]] 4&5&10 665.377
[[Didymus rank three family|Calliope]]: [[45edo]] 7&12&19 529.778
[[Jubilismic family|Jubilee]]: [[48edo]] 4&12&22 559.842
[[Archytas family|Ares]]: [[49edo]] 3&7&12 468.874
[[Porcupine rank three family|Sonic]]: [[51edo]] 7&15&51 523.158
[[Nuwell family|Big Brother]]: [[53edo]] 9&22&70 506.556
[[Marvel family|Prodigy]]: [[72edo]] 2&10&29 334.745 
[[Didymus rank three family|Erato]]: [[81edo]] 12&19&31 557.521
[[Didymus rank three family|Euterpe]]: [[88edo]] 5&12&26 535.809
[[Starling family|Thrush]]: [[89edo]] 3&12&31 352.591
[[Starling family|Aplonis]]: [[89edo]] 19&31&58 647.511
[[Porwell family|Zeus]]: [[99edo]] 6&7&9 399.882
[[Marvel family|Apollo]]: [[104edo]] 12&19&22 529.766
[[Orwellismic family|Guanyin]]: [[111edo]] 9&22&58 411.252
[[Marvel family|Minerva]]: [[127edo]] 1&9&12 381.064
[[Didymus rank three family|Clio]]: [[129edo]] 7&12&24 737.590
[[Hemifamity family|Pele]]: [[145edo]] 5&12&29 648.199
[[Hemifamity family|Laka]]: [[152edo]] 41&53&58 825.317
[[Marvel family|Marvel]]: [[166edo]] 9&10&22 255.033
[[Gamelismic family|Portent]]: [[190edo]] 5&10&26 234.092
[[Breed family|Jove]]: [[202edo]] 10&31&58 241.055
[[Marvel family|Spectacle]]: [[281edo]] 10&31&281 498.669
[[Porwell family|Jupiter]]: [[282edo]] 9&22&108 562.368
[[Breed family|Agni]]: [[284edo]] 4&6&27 493.790
[[Werckismic temperaments|Varuna]]: [[320edo]] 12&26&46 418.714
[[Werckismic temperaments|History]]: [[491edo]] 15&29&43 603.636
[[Swetismic temperaments|Hades]]: [[578edo]] 22&50&58 361.398
[[Cataharry family|Albus]]: [[640edo]] 19&53&58 621.100
[[Mirkwai family|Indra]]: [[703edo]] 10&31&152 266.020
[[Kalismic temperaments|Loki]]: [[1092edo]] 12&22&86 492.550
[[Lehmerismic temperaments|Ganesha]]: [[1502edo]] 31&87&152 390.462
[[Breed family|Freya]]: [[1566edo]] 10&31&167 169.702
[[Lehmerismic temperaments|Tyr]]: [[1566edo]] 15&57&183 258.374
[[Breed family|Baldur]]: [[1696edo]] 4&58&72 166.115
[[Lehmerismic temperaments|Skadi]]: [[2014edo]] 31&97&121 348.944
[[Ragisma family|Thor]]: [[2554edo]] 26&46&152 88.834
[[Breed family|Ennealimmic]]: [[2619edo]] 27&45&171 274.685
[[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 116.151

==13-limit rank two== 
Name: Optimal patent val: Val name: 1000*badness
[[Quartonic temperaments|Progression]]: [[9edo]] [<1 1 2 2 3 3|, <0 5 3 7 4 6|] 18.158
[[Marvel temperaments|Negric]]: [[9edo]] 1&9 20.205
[[Trienstonic clan|Opossum]]: [[15edo]] [<1 2 3 4 4 4|, <0 -3 -5 -9  -4 -2|] 19.389
[[Marvel temperaments|Negri]]: [[19edo]] 9&10 17.833
[[Starling temperaments|Sensis]]: [[19edo]] [<1 6 8 11 6 10|, <0 -8 -9 -13 -4 -10|] 20.017 
[[Starling temperaments|Keemun]]: [[19edo]] 4&19 29.749
[[Meantone family|Meanenneadecal]]: [[19edo]] [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] 21.182
[[Augmented family|August]]: [[21edo]] 9&21 18.448
[[Augmented family|Hemis]]: [[24edo]] 9&15 20.248
[[Avicennmic temperaments|Roman]]: [[29edo]] 3&23 30.043
[[Meantone family|Meantone]]: [[31edo]] [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] 18.048
[[Semicomma family|Winston]]: [[31edo]] 9&31 19.931
[[Starling temperaments|Lupercalia]]: [[31edo]] 15&31 21.328
[[Gamelismic clan|Miraculous]]: [[41edo]] 10&31 18.669
[[Tetracot family|Octacot]]: [[41edo]] [<1 1 1 2 2 4|, <0 8 18 11 20 -4|] 23.276
[[Gamelismic clan|Superkleismic]]: [[41edo]] 15&26 21.478
[[Schismatic family|Cassandra]]: [[41edo]] 29&41 20.749
[[Magic family|Magic]]: [[41edo]] 19&41 21.509
[[Starling temperaments|Sensor]]: [[46edo]] 19&27 25.474
[[Starling temperaments|Sensus]]: [[46edo]] 46&73 20.789
[[Starling temperaments|Dynwen]]: [[46edo]] 15&46 23.461
[[Hemifamity temperaments|Leapday]]: [[46edo]] 29&46 24.732
[[Diaschismic family|Diaschismic]]: [[58edo]] 46&58 18.926
[[Starling temperaments|Myna]]: [[58edo]] 31&58 17.125
[[Breedsmic temperaments|Hemififths]]: [[58edo]] 41&58 19.090
[[Gamelismic clan|Manna]]: [[72edo]] 41&72 17.012
[[Gamelismic clan|Hendec]]: [[72edo]] 26&46 17.707
[[Kleismic family|Catakleismic]]: [[72edo]] 19&53 16.883
[[Starling temperaments|Valentino]]: [[77edo]] 31&46 20.665
[[Diaschismic family|Echidna]]: [[80edo]] 22&58 23.679
[[Meantone family|Meanpop]]: [[81edo]] 19&31 20.883
[[Gamelismic clan|Rodan]]: [[87edo]] 41&46 18.448
[[Diaschismic family|Echidnic]]: [[102edo]] 10&46 28.874
[[Gamelismic clan|Benediction]]: [[103edo]] 31&72 15.715
[[Hemifamity temperaments|Buzzard]]: [[111edo]] 53&58 18.842
[[Mirkwai clan|Grendel]]: [[121edo]] 31&121 24.839
[[Breedsmic temperaments|Harry]]: [[130edo]] 58&72 13.046
[[Kleismic family|Countercata]]: [[140edo]] 34&53 20.156
[[Kleismic family|Tritikleismic]]: [[159edo]] 15&72 15.652
[[Mirkwai clan|Mirkat]]: [[183edo]] 72&111 18.632
[[Gamelismic clan|Ekadash]]: [[190edo]] 72&118 20.381
[[Kleismic family|Quadritikleismic]]: [[212edo]] 68&72 18.731
[[Ragismic microtemperaments|Octoid]]: [[224edo]] 72&224 15.274
[[Hemifamity temperaments|Mystery]]: [[232edo]] 29&58 18.591
[[Ragismic microtemperaments|Hemiennealimmal]]: [[270edo]] 72&198 12.505
[[Breedsmic temperaments|Decoid]]: [[940edo]] 130&270 13.475
[[Ragismic microtemperaments|Abigail]]: [[1258edo]] 46&178 8.856

==13-limit rank three== 
Name: Optimal patent val: Val name: 10^5 * badness
[[Starling family|Mockingbird]]: [[15edo]] 3&4&15 85.942
[[Starling family|Catbird]]: [[19edo]] [<1 0 0 -1 2 0|, <0 1 0 -2 -2 -5|, <0 0 1 3 2 5|] 90.475
[[Marvel family|Potassium]]: [[19edo]] 9&10&11 73.346
[[Quartonic family|Nickel]]: [[28edo]] 7&9&21 64.705
[[Starling family|Nightingale]]: [[31edo]] 3&15&28 83.714
[[Starling family|Thrasher]]: [[34edo]] 12&15&19 87.636
[[Semaphore family|Selenium]]: [[34edo]] 5&9&10 78.368
[[Sengic family|Krypton]]: [[36edo]] 9&10&36 72.673
[[Gamelismic family|Foreboding]]: [[41edo]] 5&10&26 87.267
[[Marvel family|Athene]]: [[53edo]] [<1 0 0 -5 -9 -4|, <0 1 0 2 2 -1|, <0 0 1 2 2 4|] 81.781
[[Starling family|Thrush]]: [[58edo]] 31&46&58 67.663
[[Starling family|Bluebird]]: [[58edo]] 12&15&31 91.548
[[Starling family|Aplonis]]: [[58edo]] 19&31&58 82.078 
[[Breed family|Jofur]]: [[58edo]] 10&31&58 74.932
[[Marvel family|Deecee]]: [[72edo]] 9&22&41 92.048
[[Marvel family|Tripod]]: [[91edo]] 9&10&31 74.546
[[Gamelismic family|Ominous]]: [[149edo]] 5&26&46 75.055
[[Gamelismic family|Portending]]: [[159edo]] 15&26&46 62.715
[[Gamelismic family|Momentous]]: [[164edo]] 10&31&46 83.202
[[Marvel family|Marvell]]: [[166edo]] 9&31&63 86.160
[[Gamelismic family|Portentous]]: [[190edo]] 15&31&56 66.119
[[Hemifamity family|Laka]]: [[205edo]] 41&53&58 82.186
[[Breed family|Agni]]: [[212edo]] 6&31&72 92.251
[[Hemifamity family|Pele]]: [[232edo]] 29&41&46 70.328
[[Breed family|Jovial]]: [[243edo]] 41&58&72 62.380
[[The Archipelago|Borneo]]: [[270edo]] 15&72&111 54.937
[[Werckismic temperaments|History]]: [[289edo]] 15&29&43 53.955
[[Cataharry family|Madagascar]]: [[313edo]] 19&53&58 56.006
[[Breed family|Jovis]]: [[363edo]] 31&58&72 54.221
[[Breed family|Vili]]: [[643edo]] 37&93&103 73.821
[[Breed family|Greenland]]: [[940edo]] 58&72&198 44.343
[[The Archipelago|Baffin]]: [[940edo]] 34&43&53 60.443
[[Breed family|Ennealimmic]]: [[981edo]] 27&72&171 75.488
[[Breed family|Zisa]]: [[1241edo]] 21&31&109 82.974
[[Ragisma family|Thunor]]: [[1258edo]] 26&46&178 34.135
[[Breed family|Frigg]]: [[1344edo]] 6&58&103 93.387
[[Ragisma family|Donar]]: [[1448edo]] 46&80&144 32.154
[[Ragisma family|Rym]]: [[1870edo]] 46&106&118 54.933
[[Breed family|Freya]]: [[2615edo]] 10&31&229 85.515

Original HTML content:

<html><head><title>Optimal patent val</title></head><body>Given any collection of p-limit commas, there is a finite list of p-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> 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 <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">TE error</a>; this is the <em>optimal (TE) patent val</em> 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.<br />
<br />
By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow">Scala</a> using the Quantize command: either type in &quot;Quantize/consistent N&quot; on the bottom, or use the pull-down menu under &quot;Modify&quot;, check the box saying &quot;Consistent&quot; and type N (without a decimal point) into &quot;Resolution&quot;.<br />
<br />
To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &lt;= p, if d is the absolute value in cents of the difference between the tuning of q given by the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a> and the POTE tuning rounded to the nearest N-edo value, then d &lt; 600/N, from which it follows that N &lt; 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e &lt; 600/N and so N &lt; 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 &lt; 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.<br />
<br />
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 <a class="wiki_link" href="/Patent%20val">patent vals</a> are given. Normally this is by way of integers conjoined by ampersands, such as 2&amp;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, the <a class="wiki_link" href="/Normal%20lists">normal val list</a> is given. Note that this differs from the method of <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">regular temperament finder</a>, which does not assume the numbers refer to patent vals.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-5-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit rank two</h2>
 Comma: Optimal patent val: 1000 * badness<br />
16/15: <a class="wiki_link" href="/8edo">8edo</a> 14.884<br />
648/625: <a class="wiki_link" href="/12edo">12edo</a> 47.231<br />
27/25: <a class="wiki_link" href="/14edo">14edo</a> 32.801<br />
25/24: <a class="wiki_link" href="/17edo">17edo</a> 13.028<br />
250/243: <a class="wiki_link" href="/22edo">22edo</a> 30.778<br />
135/128: <a class="wiki_link" href="/23edo">23edo</a> 39.556<br />
128/125: <a class="wiki_link" href="/39edo">39edo</a> 22.315<br />
3125/3072: <a class="wiki_link" href="/60edo">60edo</a> 39.163<br />
2048/2025: <a class="wiki_link" href="/80edo">80edo</a> 19.915<br />
81/80: <a class="wiki_link" href="/81edo">81edo</a> 7.381<br />
393216/390625: <a class="wiki_link" href="/164edo">164edo</a> 40.603<br />
2109375/2097152: <a class="wiki_link" href="/296edo">296edo</a> 40.807<br />
15625/15552: <a class="wiki_link" href="/458edo">458edo</a> 13.234;<br />
78732/78125: <a class="wiki_link" href="/539edo">539edo</a> 35.220<br />
32805/32768: <a class="wiki_link" href="/749edo">749edo</a> 4.259<br />
1600000/1594323: <a class="wiki_link" href="/873edo">873edo</a> 21.960<br />
6115295232/6103515625: <a class="wiki_link" href="/1400edo">1400edo</a> 31.181<br />
1224440064/1220703125: <a class="wiki_link" href="/1496edo">1496edo</a> 43.729<br />
274877906944/274658203125: <a class="wiki_link" href="/1559edo">1559edo</a> 20.576<br />
7629394531250/7625597484987: <a class="wiki_link" href="/3501edo">3501edo</a> 17.191<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-7-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit rank two</h2>
 Name: Optimal patent val: Val name: 1000*badness<br />
<a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a> [&lt;1 0 4 -2|, &lt;0 1 -1 3|] 21.312<br />
<a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a> 2&amp;3 24.152<br />
<a class="wiki_link" href="/Meantone%20family">Sharptone</a>: <a class="wiki_link" href="/5edo">5edo</a> [&lt;1 0 -4 -2|, &lt;0 1 4 3|] 24.848<br />
<a class="wiki_link" href="/Dicot%20family">Dicot</a>: <a class="wiki_link" href="/7edo">7edo</a> 4&amp;7 19.935<br />
<a class="wiki_link" href="/Bug%20family">Beep</a>: <a class="wiki_link" href="/9edo">9edo</a> 4&amp;5 18.638<br />
<a class="wiki_link" href="/Dicot%20family">Sharp</a>: <a class="wiki_link" href="/10edo">10edo</a> [&lt;1 1 2 1|, &lt;0 2 1 6|] 28.942<br />
<a class="wiki_link" href="/Dicot%20family">Decimal</a>: <a class="wiki_link" href="/10edo">10edo</a> 4&amp;10 28.334<br />
<a class="wiki_link" href="/Meantone%20family">Dominant</a>: <a class="wiki_link" href="/12edo">12edo</a> 5&amp;7 20.690<br />
<a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&amp;12 22.401<br />
<a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/12edo">12edo</a> 9&amp;12 26.459<br />
<a class="wiki_link" href="/Augmented%20family">Hexe</a>: <a class="wiki_link" href="/12edo">12edo</a> 6&amp;12 57.730<br />
<a class="wiki_link" href="/Archytas%20clan">Blacksmith</a>: <a class="wiki_link" href="/15edo">15edo</a> 5&amp;10 25.640<br />
<a class="wiki_link" href="/Trienstonic%20clan">Opossum</a>: <a class="wiki_link" href="/15edo">15edo</a> [&lt;1 2 3 4|, &lt;0 -3 -5 -9|] 40.650<br />
<a class="wiki_link" href="/Gamelismic%20clan">Gorgo</a>: <a class="wiki_link" href="/16edo">16edo</a> 5&amp;16 46.385<br />
<a class="wiki_link" href="/Trienstonic%20clan">Octokaidecal</a>: <a class="wiki_link" href="/18edo">18edo</a> 10&amp;18 36.747<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 1&amp;9 26.483<br />
<a class="wiki_link" href="/Meantone%20family">Godzilla</a>: <a class="wiki_link" href="/19edo">19edo</a> 5&amp;19 26.747<br />
<a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/19edo">19edo</a> 4&amp;15 27.408<br />
<a class="wiki_link" href="/Diaschismic%20family">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a> 2&amp;10 20.033<br />
<a class="wiki_link" href="/Jubilismic%20clan">Lemba</a>: <a class="wiki_link" href="/26edo">26edo</a> 10&amp;16 62.208<br />
<a class="wiki_link" href="/Augmented%20family">Augene</a>: <a class="wiki_link" href="/27edo">27edo</a> 3&amp;12 24.816<br />
<a class="wiki_link" href="/Archytas%20clan">Beatles</a>: <a class="wiki_link" href="/27edo">27edo</a> 10&amp;27 45.872<br />
<a class="wiki_link" href="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/31edo">31edo</a> 5&amp;26 37.146<br />
<a class="wiki_link" href="/Meantone%20family">Injera</a>: <a class="wiki_link" href="/38edo">38edo</a> 12&amp;26 31.130<br />
<a class="wiki_link" href="/Augmented%20family">Hemis</a>: <a class="wiki_link" href="/39edo">39edo</a> 6&amp;9 54.988<br />
<a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/41edo">41edo</a> 19&amp;41 18.918<br />
<a class="wiki_link" href="/Gamelismic%20clan">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</a> 15&amp;26 47.932<br />
<a class="wiki_link" href="/Meantone%20family">Flattone</a>: <a class="wiki_link" href="/45edo">45edo</a> 7&amp;19 38.553<br />
<a class="wiki_link" href="/Starling%20temperaments">Sensi</a>: <a class="wiki_link" href="/46edo">46edo</a> 19&amp;27 25.622<br />
<a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&amp;18 43.462<br />
<a class="wiki_link" href="/Archytas%20clan">Superpyth</a>: <a class="wiki_link" href="/49edo">49edo</a> 5&amp;17 32.318<br />
<a class="wiki_link" href="/Meantone%20family">Liese</a>: <a class="wiki_link" href="/55edo">55edo</a> 19&amp;55 46.706<br />
<a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a> 10&amp;21 16.742<br />
<a class="wiki_link" href="/Marvel%20temperaments">Wizard</a>: <a class="wiki_link" href="/72edo">72edo</a> 22&amp;50 40.846<br />
<a class="wiki_link" href="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/81edo">81edo</a> 12&amp;19 13.707<br />
<a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a> 4&amp;27 27.044<br />
<a class="wiki_link" href="/Marvel%20temperaments">Septimin</a>: <a class="wiki_link" href="/91edo">91edo</a> 9&amp;32 54.502<br />
<a class="wiki_link" href="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a> 12&amp;29 21.644<br />
<a class="wiki_link" href="/Hemimean%20clan">Misty</a>: <a class="wiki_link" href="/99edo">99edo</a> 12&amp;87 36.802<br />
<a class="wiki_link" href="/Starling%20temperaments">Nusecond</a>: <a class="wiki_link" href="/101edo">101edo</a> 31&amp;101 50.389<br />
<a class="wiki_link" href="/Tetracot%20family">Octacot</a>: <a class="wiki_link" href="/109edo">109edo</a> 27&amp;41 33.845<br />
<a class="wiki_link" href="/Hemimean%20clan">Semisept</a>: <a class="wiki_link" href="/111edo">111edo</a> 18&amp;31 50.472<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Buzzard</a>: <a class="wiki_link" href="/111edo">111edo</a> 5&amp;48 47.963<br />
<a class="wiki_link" href="/Gamelismic%20clan">Guiron</a>: <a class="wiki_link" href="/118edo">118edo</a> 36&amp;41 47.544<br />
<a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/122edo">122edo</a> 2&amp;29 47.578<br />
<a class="wiki_link" href="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/125edo">125edo</a> 1&amp;31 56.934<br />
<a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/128edo">128edo</a> 5&amp;41 37.112<br />
<a class="wiki_link" href="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/137edo">137edo</a> 9&amp;22 20.375<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Tertiaseptal</a>: <a class="wiki_link" href="/171edo">171edo</a> 31&amp;109 12.995<br />
<a class="wiki_link" href="/Schismatic%20family">Pontiac</a>: <a class="wiki_link" href="/171edo">171edo</a> 53&amp;118 14.133<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Mitonic</a>: <a class="wiki_link" href="/171edo">171edo</a> 46&amp;125 25.184<br />
<a class="wiki_link" href="/Mutt%20family">Mutt</a>: <a class="wiki_link" href="/171edo">171edo</a> 3&amp;84 28.406<br />
<a class="wiki_link" href="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/185edo">185edo</a> 15&amp;31 31.056<br />
<a class="wiki_link" href="/Gamelismic%20clan">Unidec</a>: <a class="wiki_link" href="/190edo">190edo</a> 26&amp;46 38.393<br />
<a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/197edo">197edo</a> 19&amp;53 21.501<br />
<a class="wiki_link" href="/Kleismic%20family">Quadritikleismic</a>: <a class="wiki_link" href="/212edo">212edo</a> 4&amp;68 39.231<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Octoid</a>: <a class="wiki_link" href="/224edo">224edo</a> 72&amp;152 42.670<br />
<a class="wiki_link" href="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/228edo">228edo</a> 12&amp;60 35.686<br />
<a class="wiki_link" href="/Kleismic%20family">Tritikleismic</a>: <a class="wiki_link" href="/231edo">231edo</a> 15&amp;57 56.337<br />
<a class="wiki_link" href="/Mirkwai%20clan">Mirkat</a>: <a class="wiki_link" href="/255edo">255edo</a> 72&amp;183 59.376<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Decoid</a>: <a class="wiki_link" href="/270edo">270edo</a> 10&amp;120 33.902<br />
<a class="wiki_link" href="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/328edo">328edo</a> 6&amp;31 20.307<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Hemififths</a>: <a class="wiki_link" href="/338edo">338edo</a> 41&amp;58 22.243<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Amity</a>: <a class="wiki_link" href="/350edo">350edo</a> 7&amp;39 23.649<br />
<a class="wiki_link" href="/Schismatic%20family">Bischismic</a>: <a class="wiki_link" href="/378edo">378edo</a> 12&amp;118 54.744<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Parakleismic</a>: <a class="wiki_link" href="/415edo">415edo</a> 19&amp;80 27.431<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Gamera</a>: <a class="wiki_link" href="/422edo">422edo</a> 26&amp;73 37.648<br />
<a class="wiki_link" href="/Kleismic%20family">Countercata</a>: <a class="wiki_link" href="/473edo">473edo</a> 34&amp;53 52.128<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/534edo">534edo</a> 58&amp;72 34.077<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Ennealimmal</a>: <a class="wiki_link" href="/612edo">612edo</a> 27&amp;45 3.610<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/1111edo">1111edo</a> 31&amp;177 35.832<br />
<a class="wiki_link" href="/Schismatic%20family">Sesquiquartififths</a>: <a class="wiki_link" href="/1498edo">1498edo</a> 41&amp;89 11.244<br />
<a class="wiki_link" href="/Schismatic%20family">Term</a>: <a class="wiki_link" href="/1722edo">1722edo</a> 12&amp;159 19.950<br />
<a class="wiki_link" href="/Gammic%20family">Neptune</a>: <a class="wiki_link" href="/1778edo">1778edo</a> 35&amp;68 23.427<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/1798edo">1798edo</a> 46&amp;132 37.000<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Enneadecal</a>: <a class="wiki_link" href="/2185edo">2185edo</a> 19&amp;152 10.954<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</a> 80&amp;171 10.836<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-7-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit rank three</h2>
 Comma: Optimal patent val: 10^6 * badness<br />
36/35: <a class="wiki_link" href="/12edo">12edo</a> 63.675<br />
49/48: <a class="wiki_link" href="/19edo">19edo</a> 115.514<br />
875/864: <a class="wiki_link" href="/41edo">41edo</a> 212.413<br />
525/512: <a class="wiki_link" href="/45edo">45edo</a> 476.454<br />
686/675: <a class="wiki_link" href="/46edo">46edo</a> 320.094<br />
50/49: <a class="wiki_link" href="/48edo">48edo</a> 127.208<br />
64/63: <a class="wiki_link" href="/49edo">49edo</a> 98.719<br />
1029/1000: <a class="wiki_link" href="/55edo">55edo</a> 678.164<br />
3125/3087: <a class="wiki_link" href="/94edo">94edo</a> 613.731<br />
1728/1715: <a class="wiki_link" href="/111edo">111edo</a> 187.341<br />
2430/2401: <a class="wiki_link" href="/137edo">137edo</a> 518.625<br />
65625/65536: <a class="wiki_link" href="/171edo">171edo</a> 172.601<br />
126/125: <a class="wiki_link" href="/185edo">185edo</a> 69.867<br />
1029/1024: <a class="wiki_link" href="/190edo">190edo</a> 176.105<br />
225/224: <a class="wiki_link" href="/197edo">197edo</a> 36.517<br />
4000/3969: <a class="wiki_link" href="/215edo">215edo</a> 215.550<br />
16875/16807: <a class="wiki_link" href="/224edo">224edo</a> 341.931<br />
245/243: <a class="wiki_link" href="/283edo">283edo</a> 121.191<br />
6144/6125: <a class="wiki_link" href="/381edo">381edo</a> 142.375<br />
3136/3125: <a class="wiki_link" href="/446edo">446edo</a> 160.039<br />
19683/19600: <a class="wiki_link" href="/587edo">587edo</a> 300.987<br />
10976/10935: <a class="wiki_link" href="/695edo">695edo</a> 199.139<br />
703125/702464: <a class="wiki_link" href="/2185edo">2185edo</a> 149.828<br />
2401/2400: <a class="wiki_link" href="/2749edo">2749edo</a> 15.288<br />
420175/419904: <a class="wiki_link" href="/4306edo">4306edo</a> 86.449<br />
4375/4374: <a class="wiki_link" href="/8419edo">8419edo</a> 11.665<br />
250047/250000: <a class="wiki_link" href="/12555edo">12555edo</a> 21.301<br />
78125000/78121827: <a class="wiki_link" href="/101654edo">101654edo</a> 20.457<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-11-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit rank two</h2>
 Name: Optimal patent val: Val name: 1000*badness<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negric</a>: <a class="wiki_link" href="/9edo">9edo</a> 1&amp;9 30.617<br />
<a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&amp;12 22.132<br />
<a class="wiki_link" href="/Trienstonic%20clan">Opossum</a>: <a class="wiki_link" href="/15edo">15edo</a> [&lt;1 2 3 4 4|, &lt;0 -3 -5 -9 -4|] 22.325<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10 26.190<br />
<a class="wiki_link" href="/Marvel%20temperaments">Meanenneadecal</a>: <a class="wiki_link" href="/19edo">19edo</a> 12&amp;19 21.182<br />
<a class="wiki_link" href="/Diaschismic%20family">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a> 12&amp;22 20.343 <br />
<a class="wiki_link" href="/Meantone%20family">Mohajira</a>: <a class="wiki_link" href="/31edo">31edo</a> 7&amp;24 26.064<br />
<a class="wiki_link" href="/Meantone%20family">Squares</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;45 56.826<br />
<a class="wiki_link" href="/Starling%20temperaments">Casablanca</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;42 67.291<br />
<a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/34edo">34edo</a> 4&amp;15 27.410<br />
<a class="wiki_link" href="/Augmented%20family">Hemis</a>: <a class="wiki_link" href="/39edo">39edo</a> 6&amp;9 20.248<br />
<a class="wiki_link" href="/Schismatic%20family">Cassandra</a>: <a class="wiki_link" href="/41edo">41edo</a> 12&amp;29 23.556<br />
<a class="wiki_link" href="/Gamelismic%20clan">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</a> 15&amp;26 25.659<br />
<a class="wiki_link" href="/Starling%20temperaments">Sensor</a>: <a class="wiki_link" href="/46edo">46edo</a> 19&amp;27 37.942<br />
<a class="wiki_link" href="/Starling%20temperaments">Sensus</a>: <a class="wiki_link" href="/46edo">46edo</a> 46&amp;73 29.486<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Leapday</a>: <a class="wiki_link" href="/46edo">46edo</a> 29&amp;46 38.624<br />
<a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&amp;18 32.058<br />
<a class="wiki_link" href="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/53edo">53edo</a> 9&amp;22 15.231<br />
<a class="wiki_link" href="/Diaschismic%20family">Diaschismic</a>: <a class="wiki_link" href="/58edo">58edo</a> 12&amp;46 25.034<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Hemififths</a>: <a class="wiki_link" href="/58edo">58edo</a> 41&amp;58 23.498<br />
<a class="wiki_link" href="/Porcupine%20family">Porcupine</a>: <a class="wiki_link" href="/59edo">59edo</a> 7&amp;15 21.562<br />
<a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a> 10&amp;31 10.684<br />
<a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/72edo">72edo</a> 19&amp;53 21.849<br />
<a class="wiki_link" href="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/72edo">72edo</a> 12&amp;72 22.235<br />
<a class="wiki_link" href="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/77edo">77edo</a> 15&amp;31 16.687<br />
<a class="wiki_link" href="/Diaschismic%20family">Keen</a>: <a class="wiki_link" href="/78edo">78edo</a> 22&amp;56 45.270<br />
<a class="wiki_link" href="/Diaschismic%20family">Echidna</a>: <a class="wiki_link" href="/80edo">80edo</a> 22&amp;58 25.987<br />
<a class="wiki_link" href="/Meantone%20family">Meanpop</a>: <a class="wiki_link" href="/81edo">81edo</a> 19&amp;31 21.543<br />
<a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/87edo">87edo</a> 5&amp;41 23.093<br />
<a class="wiki_link" href="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/88edo">88edo</a> 5&amp;26 25.642<br />
<a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a> 31&amp;58 16.842<br />
<a class="wiki_link" href="/Marvel%20temperaments">Septimin</a>: <a class="wiki_link" href="/91edo">91edo</a> 9&amp;32 31.309<br />
<a class="wiki_link" href="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a> 41&amp;53 27.396<br />
<a class="wiki_link" href="/Starling%20temperaments">Nusecond</a>: <a class="wiki_link" href="/101edo">101edo</a> 31&amp;101 25.621<br />
<a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/104edo">104edo</a> 19&amp;22 25.621<br />
<a class="wiki_link" href="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/105edo">105edo</a> 12&amp;31 17.027<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Buzzard</a>: <a class="wiki_link" href="/111edo">111edo</a> 53&amp;58 34.484<br />
<a class="wiki_link" href="/Diaschismic%20family">Shrutar</a>: <a class="wiki_link" href="/114edo">114edo</a> 22&amp;46 26.489<br />
<a class="wiki_link" href="/Gamelismic%20clan">Hemithirds</a>: <a class="wiki_link" href="/118edo">118edo</a> 31&amp;87 19.003<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Parakleismic</a>: <a class="wiki_link" href="/118edo">118edo</a> 19&amp;99 49.711<br />
<a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/122edo">122edo</a> 31&amp;91 45.456<br />
<a class="wiki_link" href="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/125edo">125edo</a> 31&amp;63 25.342<br />
<a class="wiki_link" href="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/130edo">130edo</a> 31&amp;130 21.069<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Mystery</a>: <a class="wiki_link" href="/145edo">145edo</a> 29&amp;58 34.291<br />
<a class="wiki_link" href="/Diaschismic%20family">Echidnic</a>: <a class="wiki_link" href="/148edo">148edo</a> 10&amp;46 45.127<br />
<a class="wiki_link" href="/Mirkwai%20clan">Grendel</a>: <a class="wiki_link" href="/152edo">152edo</a> 31&amp;121 19.845<br />
<a class="wiki_link" href="/Mirkwai%20clan">Kwai</a>: <a class="wiki_link" href="/152edo">152edo</a> 41&amp;111 26.219<br />
<a class="wiki_link" href="/Schismatic%20family">Guiron</a>: <a class="wiki_link" href="/159edo">159edo</a> 41&amp;77 26.648<br />
<a class="wiki_link" href="/Marvel%20temperaments">Wizard</a>: <a class="wiki_link" href="/166edo">166edo</a> 22&amp;50 18.539<br />
<a class="wiki_link" href="/Gamelismic%20clan">Unidec</a>: <a class="wiki_link" href="/190edo">190edo</a> 26&amp;46 15.479<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/202edo">202edo</a> 58&amp;72 15.867<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Tertiaseptal</a>: <a class="wiki_link" href="/202edo">202edo</a> 31&amp;171 35.576<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Octoid</a>: <a class="wiki_link" href="/224edo">224edo</a> 72&amp;152 14.097<br />
<a class="wiki_link" href="/Kleismic%20family">Countercata</a>: <a class="wiki_link" href="/227edo">227edo</a> 34&amp;53 39.770<br />
<a class="wiki_link" href="/Kleismic%20family">Tritikleismic</a>: <a class="wiki_link" href="/231edo">231edo</a> 15&amp;57 19.333<br />
<a class="wiki_link" href="/Mirkwai%20clan">Mirkat</a>: <a class="wiki_link" href="/255edo">255edo</a> 72&amp;183 22.126<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/270edo">270edo</a> 31&amp;208 17.540<br />
<a class="wiki_link" href="/Marvel%20temperaments">Marvo</a>: <a class="wiki_link" href="/281edo">281edo</a> 72&amp;281 31.685<br />
<a class="wiki_link" href="/Kleismic%20family">Quadritikleismic</a>: <a class="wiki_link" href="/284edo">284edo</a> 4&amp;68 23.406<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiamity</a>: <a class="wiki_link" href="/350edo">350edo</a> 46&amp;106 31.307<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/764edo">764edo</a> 46&amp;132 12.860<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Decoid</a>: <a class="wiki_link" href="/1210edo">1210edo</a> 130&amp;270 18.735<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 72&amp;198 6.283<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-11-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit rank three</h2>
 Name: Optimal patent val: Val name: 10^6 * badness<br />
<a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&amp;3&amp;10 577.113<br />
<a class="wiki_link" href="/Marvel%20family">Potassium</a>: <a class="wiki_link" href="/19edo">19edo</a> 2&amp;9&amp;10 463.665<br />
<a class="wiki_link" href="/Quartonic%20family">Nickel</a>: <a class="wiki_link" href="/28edo">28edo</a> 7&amp;9&amp;12 463.463<br />
<a class="wiki_link" href="/Jubilismic%20family">Festival</a>: <a class="wiki_link" href="/26edo">26edo</a> 2&amp;10&amp;26 688.510<br />
<a class="wiki_link" href="/Starling%20family">Thrasher</a>: <a class="wiki_link" href="/34edo">34edo</a> 3&amp;4&amp;12 480.188<br />
<a class="wiki_link" href="/Semaphore%20family">Selenium</a>: <a class="wiki_link" href="/34edo">34edo</a> 4&amp;5&amp;10 665.377<br />
<a class="wiki_link" href="/Didymus%20rank%20three%20family">Calliope</a>: <a class="wiki_link" href="/45edo">45edo</a> 7&amp;12&amp;19 529.778<br />
<a class="wiki_link" href="/Jubilismic%20family">Jubilee</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&amp;12&amp;22 559.842<br />
<a class="wiki_link" href="/Archytas%20family">Ares</a>: <a class="wiki_link" href="/49edo">49edo</a> 3&amp;7&amp;12 468.874<br />
<a class="wiki_link" href="/Porcupine%20rank%20three%20family">Sonic</a>: <a class="wiki_link" href="/51edo">51edo</a> 7&amp;15&amp;51 523.158<br />
<a class="wiki_link" href="/Nuwell%20family">Big Brother</a>: <a class="wiki_link" href="/53edo">53edo</a> 9&amp;22&amp;70 506.556<br />
<a class="wiki_link" href="/Marvel%20family">Prodigy</a>: <a class="wiki_link" href="/72edo">72edo</a> 2&amp;10&amp;29 334.745 <br />
<a class="wiki_link" href="/Didymus%20rank%20three%20family">Erato</a>: <a class="wiki_link" href="/81edo">81edo</a> 12&amp;19&amp;31 557.521<br />
<a class="wiki_link" href="/Didymus%20rank%20three%20family">Euterpe</a>: <a class="wiki_link" href="/88edo">88edo</a> 5&amp;12&amp;26 535.809<br />
<a class="wiki_link" href="/Starling%20family">Thrush</a>: <a class="wiki_link" href="/89edo">89edo</a> 3&amp;12&amp;31 352.591<br />
<a class="wiki_link" href="/Starling%20family">Aplonis</a>: <a class="wiki_link" href="/89edo">89edo</a> 19&amp;31&amp;58 647.511<br />
<a class="wiki_link" href="/Porwell%20family">Zeus</a>: <a class="wiki_link" href="/99edo">99edo</a> 6&amp;7&amp;9 399.882<br />
<a class="wiki_link" href="/Marvel%20family">Apollo</a>: <a class="wiki_link" href="/104edo">104edo</a> 12&amp;19&amp;22 529.766<br />
<a class="wiki_link" href="/Orwellismic%20family">Guanyin</a>: <a class="wiki_link" href="/111edo">111edo</a> 9&amp;22&amp;58 411.252<br />
<a class="wiki_link" href="/Marvel%20family">Minerva</a>: <a class="wiki_link" href="/127edo">127edo</a> 1&amp;9&amp;12 381.064<br />
<a class="wiki_link" href="/Didymus%20rank%20three%20family">Clio</a>: <a class="wiki_link" href="/129edo">129edo</a> 7&amp;12&amp;24 737.590<br />
<a class="wiki_link" href="/Hemifamity%20family">Pele</a>: <a class="wiki_link" href="/145edo">145edo</a> 5&amp;12&amp;29 648.199<br />
<a class="wiki_link" href="/Hemifamity%20family">Laka</a>: <a class="wiki_link" href="/152edo">152edo</a> 41&amp;53&amp;58 825.317<br />
<a class="wiki_link" href="/Marvel%20family">Marvel</a>: <a class="wiki_link" href="/166edo">166edo</a> 9&amp;10&amp;22 255.033<br />
<a class="wiki_link" href="/Gamelismic%20family">Portent</a>: <a class="wiki_link" href="/190edo">190edo</a> 5&amp;10&amp;26 234.092<br />
<a class="wiki_link" href="/Breed%20family">Jove</a>: <a class="wiki_link" href="/202edo">202edo</a> 10&amp;31&amp;58 241.055<br />
<a class="wiki_link" href="/Marvel%20family">Spectacle</a>: <a class="wiki_link" href="/281edo">281edo</a> 10&amp;31&amp;281 498.669<br />
<a class="wiki_link" href="/Porwell%20family">Jupiter</a>: <a class="wiki_link" href="/282edo">282edo</a> 9&amp;22&amp;108 562.368<br />
<a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/284edo">284edo</a> 4&amp;6&amp;27 493.790<br />
<a class="wiki_link" href="/Werckismic%20temperaments">Varuna</a>: <a class="wiki_link" href="/320edo">320edo</a> 12&amp;26&amp;46 418.714<br />
<a class="wiki_link" href="/Werckismic%20temperaments">History</a>: <a class="wiki_link" href="/491edo">491edo</a> 15&amp;29&amp;43 603.636<br />
<a class="wiki_link" href="/Swetismic%20temperaments">Hades</a>: <a class="wiki_link" href="/578edo">578edo</a> 22&amp;50&amp;58 361.398<br />
<a class="wiki_link" href="/Cataharry%20family">Albus</a>: <a class="wiki_link" href="/640edo">640edo</a> 19&amp;53&amp;58 621.100<br />
<a class="wiki_link" href="/Mirkwai%20family">Indra</a>: <a class="wiki_link" href="/703edo">703edo</a> 10&amp;31&amp;152 266.020<br />
<a class="wiki_link" href="/Kalismic%20temperaments">Loki</a>: <a class="wiki_link" href="/1092edo">1092edo</a> 12&amp;22&amp;86 492.550<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments">Ganesha</a>: <a class="wiki_link" href="/1502edo">1502edo</a> 31&amp;87&amp;152 390.462<br />
<a class="wiki_link" href="/Breed%20family">Freya</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 10&amp;31&amp;167 169.702<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments">Tyr</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 15&amp;57&amp;183 258.374<br />
<a class="wiki_link" href="/Breed%20family">Baldur</a>: <a class="wiki_link" href="/1696edo">1696edo</a> 4&amp;58&amp;72 166.115<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments">Skadi</a>: <a class="wiki_link" href="/2014edo">2014edo</a> 31&amp;97&amp;121 348.944<br />
<a class="wiki_link" href="/Ragisma%20family">Thor</a>: <a class="wiki_link" href="/2554edo">2554edo</a> 26&amp;46&amp;152 88.834<br />
<a class="wiki_link" href="/Breed%20family">Ennealimmic</a>: <a class="wiki_link" href="/2619edo">2619edo</a> 27&amp;45&amp;171 274.685<br />
<a class="wiki_link" href="/Kalismic%20temperaments">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</a> 12&amp;42&amp;72 116.151<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-13-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit rank two</h2>
 Name: Optimal patent val: Val name: 1000*badness<br />
<a class="wiki_link" href="/Quartonic%20temperaments">Progression</a>: <a class="wiki_link" href="/9edo">9edo</a> [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|] 18.158<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negric</a>: <a class="wiki_link" href="/9edo">9edo</a> 1&amp;9 20.205<br />
<a class="wiki_link" href="/Trienstonic%20clan">Opossum</a>: <a class="wiki_link" href="/15edo">15edo</a> [&lt;1 2 3 4 4 4|, &lt;0 -3 -5 -9  -4 -2|] 19.389<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10 17.833<br />
<a class="wiki_link" href="/Starling%20temperaments">Sensis</a>: <a class="wiki_link" href="/19edo">19edo</a> [&lt;1 6 8 11 6 10|, &lt;0 -8 -9 -13 -4 -10|] 20.017 <br />
<a class="wiki_link" href="/Starling%20temperaments">Keemun</a>: <a class="wiki_link" href="/19edo">19edo</a> 4&amp;19 29.749<br />
<a class="wiki_link" href="/Meantone%20family">Meanenneadecal</a>: <a class="wiki_link" href="/19edo">19edo</a> [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|] 21.182<br />
<a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/21edo">21edo</a> 9&amp;21 18.448<br />
<a class="wiki_link" href="/Augmented%20family">Hemis</a>: <a class="wiki_link" href="/24edo">24edo</a> 9&amp;15 20.248<br />
<a class="wiki_link" href="/Avicennmic%20temperaments">Roman</a>: <a class="wiki_link" href="/29edo">29edo</a> 3&amp;23 30.043<br />
<a class="wiki_link" href="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/31edo">31edo</a> [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|] 18.048<br />
<a class="wiki_link" href="/Semicomma%20family">Winston</a>: <a class="wiki_link" href="/31edo">31edo</a> 9&amp;31 19.931<br />
<a class="wiki_link" href="/Starling%20temperaments">Lupercalia</a>: <a class="wiki_link" href="/31edo">31edo</a> 15&amp;31 21.328<br />
<a class="wiki_link" href="/Gamelismic%20clan">Miraculous</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&amp;31 18.669<br />
<a class="wiki_link" href="/Tetracot%20family">Octacot</a>: <a class="wiki_link" href="/41edo">41edo</a> [&lt;1 1 1 2 2 4|, &lt;0 8 18 11 20 -4|] 23.276<br />
<a class="wiki_link" href="/Gamelismic%20clan">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</a> 15&amp;26 21.478<br />
<a class="wiki_link" href="/Schismatic%20family">Cassandra</a>: <a class="wiki_link" href="/41edo">41edo</a> 29&amp;41 20.749<br />
<a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/41edo">41edo</a> 19&amp;41 21.509<br />
<a class="wiki_link" href="/Starling%20temperaments">Sensor</a>: <a class="wiki_link" href="/46edo">46edo</a> 19&amp;27 25.474<br />
<a class="wiki_link" href="/Starling%20temperaments">Sensus</a>: <a class="wiki_link" href="/46edo">46edo</a> 46&amp;73 20.789<br />
<a class="wiki_link" href="/Starling%20temperaments">Dynwen</a>: <a class="wiki_link" href="/46edo">46edo</a> 15&amp;46 23.461<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Leapday</a>: <a class="wiki_link" href="/46edo">46edo</a> 29&amp;46 24.732<br />
<a class="wiki_link" href="/Diaschismic%20family">Diaschismic</a>: <a class="wiki_link" href="/58edo">58edo</a> 46&amp;58 18.926<br />
<a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/58edo">58edo</a> 31&amp;58 17.125<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Hemififths</a>: <a class="wiki_link" href="/58edo">58edo</a> 41&amp;58 19.090<br />
<a class="wiki_link" href="/Gamelismic%20clan">Manna</a>: <a class="wiki_link" href="/72edo">72edo</a> 41&amp;72 17.012<br />
<a class="wiki_link" href="/Gamelismic%20clan">Hendec</a>: <a class="wiki_link" href="/72edo">72edo</a> 26&amp;46 17.707<br />
<a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/72edo">72edo</a> 19&amp;53 16.883<br />
<a class="wiki_link" href="/Starling%20temperaments">Valentino</a>: <a class="wiki_link" href="/77edo">77edo</a> 31&amp;46 20.665<br />
<a class="wiki_link" href="/Diaschismic%20family">Echidna</a>: <a class="wiki_link" href="/80edo">80edo</a> 22&amp;58 23.679<br />
<a class="wiki_link" href="/Meantone%20family">Meanpop</a>: <a class="wiki_link" href="/81edo">81edo</a> 19&amp;31 20.883<br />
<a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/87edo">87edo</a> 41&amp;46 18.448<br />
<a class="wiki_link" href="/Diaschismic%20family">Echidnic</a>: <a class="wiki_link" href="/102edo">102edo</a> 10&amp;46 28.874<br />
<a class="wiki_link" href="/Gamelismic%20clan">Benediction</a>: <a class="wiki_link" href="/103edo">103edo</a> 31&amp;72 15.715<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Buzzard</a>: <a class="wiki_link" href="/111edo">111edo</a> 53&amp;58 18.842<br />
<a class="wiki_link" href="/Mirkwai%20clan">Grendel</a>: <a class="wiki_link" href="/121edo">121edo</a> 31&amp;121 24.839<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/130edo">130edo</a> 58&amp;72 13.046<br />
<a class="wiki_link" href="/Kleismic%20family">Countercata</a>: <a class="wiki_link" href="/140edo">140edo</a> 34&amp;53 20.156<br />
<a class="wiki_link" href="/Kleismic%20family">Tritikleismic</a>: <a class="wiki_link" href="/159edo">159edo</a> 15&amp;72 15.652<br />
<a class="wiki_link" href="/Mirkwai%20clan">Mirkat</a>: <a class="wiki_link" href="/183edo">183edo</a> 72&amp;111 18.632<br />
<a class="wiki_link" href="/Gamelismic%20clan">Ekadash</a>: <a class="wiki_link" href="/190edo">190edo</a> 72&amp;118 20.381<br />
<a class="wiki_link" href="/Kleismic%20family">Quadritikleismic</a>: <a class="wiki_link" href="/212edo">212edo</a> 68&amp;72 18.731<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Octoid</a>: <a class="wiki_link" href="/224edo">224edo</a> 72&amp;224 15.274<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Mystery</a>: <a class="wiki_link" href="/232edo">232edo</a> 29&amp;58 18.591<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/270edo">270edo</a> 72&amp;198 12.505<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Decoid</a>: <a class="wiki_link" href="/940edo">940edo</a> 130&amp;270 13.475<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/1258edo">1258edo</a> 46&amp;178 8.856<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-13-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit rank three</h2>
 Name: Optimal patent val: Val name: 10^5 * badness<br />
<a class="wiki_link" href="/Starling%20family">Mockingbird</a>: <a class="wiki_link" href="/15edo">15edo</a> 3&amp;4&amp;15 85.942<br />
<a class="wiki_link" href="/Starling%20family">Catbird</a>: <a class="wiki_link" href="/19edo">19edo</a> [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|] 90.475<br />
<a class="wiki_link" href="/Marvel%20family">Potassium</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10&amp;11 73.346<br />
<a class="wiki_link" href="/Quartonic%20family">Nickel</a>: <a class="wiki_link" href="/28edo">28edo</a> 7&amp;9&amp;21 64.705<br />
<a class="wiki_link" href="/Starling%20family">Nightingale</a>: <a class="wiki_link" href="/31edo">31edo</a> 3&amp;15&amp;28 83.714<br />
<a class="wiki_link" href="/Starling%20family">Thrasher</a>: <a class="wiki_link" href="/34edo">34edo</a> 12&amp;15&amp;19 87.636<br />
<a class="wiki_link" href="/Semaphore%20family">Selenium</a>: <a class="wiki_link" href="/34edo">34edo</a> 5&amp;9&amp;10 78.368<br />
<a class="wiki_link" href="/Sengic%20family">Krypton</a>: <a class="wiki_link" href="/36edo">36edo</a> 9&amp;10&amp;36 72.673<br />
<a class="wiki_link" href="/Gamelismic%20family">Foreboding</a>: <a class="wiki_link" href="/41edo">41edo</a> 5&amp;10&amp;26 87.267<br />
<a class="wiki_link" href="/Marvel%20family">Athene</a>: <a class="wiki_link" href="/53edo">53edo</a> [&lt;1 0 0 -5 -9 -4|, &lt;0 1 0 2 2 -1|, &lt;0 0 1 2 2 4|] 81.781<br />
<a class="wiki_link" href="/Starling%20family">Thrush</a>: <a class="wiki_link" href="/58edo">58edo</a> 31&amp;46&amp;58 67.663<br />
<a class="wiki_link" href="/Starling%20family">Bluebird</a>: <a class="wiki_link" href="/58edo">58edo</a> 12&amp;15&amp;31 91.548<br />
<a class="wiki_link" href="/Starling%20family">Aplonis</a>: <a class="wiki_link" href="/58edo">58edo</a> 19&amp;31&amp;58 82.078 <br />
<a class="wiki_link" href="/Breed%20family">Jofur</a>: <a class="wiki_link" href="/58edo">58edo</a> 10&amp;31&amp;58 74.932<br />
<a class="wiki_link" href="/Marvel%20family">Deecee</a>: <a class="wiki_link" href="/72edo">72edo</a> 9&amp;22&amp;41 92.048<br />
<a class="wiki_link" href="/Marvel%20family">Tripod</a>: <a class="wiki_link" href="/91edo">91edo</a> 9&amp;10&amp;31 74.546<br />
<a class="wiki_link" href="/Gamelismic%20family">Ominous</a>: <a class="wiki_link" href="/149edo">149edo</a> 5&amp;26&amp;46 75.055<br />
<a class="wiki_link" href="/Gamelismic%20family">Portending</a>: <a class="wiki_link" href="/159edo">159edo</a> 15&amp;26&amp;46 62.715<br />
<a class="wiki_link" href="/Gamelismic%20family">Momentous</a>: <a class="wiki_link" href="/164edo">164edo</a> 10&amp;31&amp;46 83.202<br />
<a class="wiki_link" href="/Marvel%20family">Marvell</a>: <a class="wiki_link" href="/166edo">166edo</a> 9&amp;31&amp;63 86.160<br />
<a class="wiki_link" href="/Gamelismic%20family">Portentous</a>: <a class="wiki_link" href="/190edo">190edo</a> 15&amp;31&amp;56 66.119<br />
<a class="wiki_link" href="/Hemifamity%20family">Laka</a>: <a class="wiki_link" href="/205edo">205edo</a> 41&amp;53&amp;58 82.186<br />
<a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/212edo">212edo</a> 6&amp;31&amp;72 92.251<br />
<a class="wiki_link" href="/Hemifamity%20family">Pele</a>: <a class="wiki_link" href="/232edo">232edo</a> 29&amp;41&amp;46 70.328<br />
<a class="wiki_link" href="/Breed%20family">Jovial</a>: <a class="wiki_link" href="/243edo">243edo</a> 41&amp;58&amp;72 62.380<br />
<a class="wiki_link" href="/The%20Archipelago">Borneo</a>: <a class="wiki_link" href="/270edo">270edo</a> 15&amp;72&amp;111 54.937<br />
<a class="wiki_link" href="/Werckismic%20temperaments">History</a>: <a class="wiki_link" href="/289edo">289edo</a> 15&amp;29&amp;43 53.955<br />
<a class="wiki_link" href="/Cataharry%20family">Madagascar</a>: <a class="wiki_link" href="/313edo">313edo</a> 19&amp;53&amp;58 56.006<br />
<a class="wiki_link" href="/Breed%20family">Jovis</a>: <a class="wiki_link" href="/363edo">363edo</a> 31&amp;58&amp;72 54.221<br />
<a class="wiki_link" href="/Breed%20family">Vili</a>: <a class="wiki_link" href="/643edo">643edo</a> 37&amp;93&amp;103 73.821<br />
<a class="wiki_link" href="/Breed%20family">Greenland</a>: <a class="wiki_link" href="/940edo">940edo</a> 58&amp;72&amp;198 44.343<br />
<a class="wiki_link" href="/The%20Archipelago">Baffin</a>: <a class="wiki_link" href="/940edo">940edo</a> 34&amp;43&amp;53 60.443<br />
<a class="wiki_link" href="/Breed%20family">Ennealimmic</a>: <a class="wiki_link" href="/981edo">981edo</a> 27&amp;72&amp;171 75.488<br />
<a class="wiki_link" href="/Breed%20family">Zisa</a>: <a class="wiki_link" href="/1241edo">1241edo</a> 21&amp;31&amp;109 82.974<br />
<a class="wiki_link" href="/Ragisma%20family">Thunor</a>: <a class="wiki_link" href="/1258edo">1258edo</a> 26&amp;46&amp;178 34.135<br />
<a class="wiki_link" href="/Breed%20family">Frigg</a>: <a class="wiki_link" href="/1344edo">1344edo</a> 6&amp;58&amp;103 93.387<br />
<a class="wiki_link" href="/Ragisma%20family">Donar</a>: <a class="wiki_link" href="/1448edo">1448edo</a> 46&amp;80&amp;144 32.154<br />
<a class="wiki_link" href="/Ragisma%20family">Rym</a>: <a class="wiki_link" href="/1870edo">1870edo</a> 46&amp;106&amp;118 54.933<br />
<a class="wiki_link" href="/Breed%20family">Freya</a>: <a class="wiki_link" href="/2615edo">2615edo</a> 10&amp;31&amp;229 85.515</body></html>