Optimal patent val

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Revision as of 18:48, 1 December 2011 by Wikispaces>genewardsmith (**Imported revision 281311752 - Original comment: **)
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This revision was by author genewardsmith and made on 2011-12-01 18:48:23 UTC.
The original revision id was 281311752.
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[[toc|flat]]

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 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
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]] 3d&5 21.312
[[Archytas clan|Mother]]: [[5edo]] 2&3 24.152
[[Meantone family|Sharptone]]: [[5edo]] 5&7d 24.848
[[Dicot family|Dicot]]: [[7edo]] 4&7 19.935
[[Bug family|Beep]]: [[9edo]] 4&5 18.638
[[Dicot family|Sharp]]: [[10edo]] 3d&7d 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]] 7d&8d 40.650
[[Gamelismic clan|Gorgo]]: [[16edo]] 5&16 46.385
[[Trienstonic clan|Octokaidecal]]: [[18edo]] 10&18 36.747
[[Trienstonic clan|Uncle]]: [[18edo]] 5&18 72.653
[[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
[[Porcupine family|Nautilus]]: [[29edo]] 15&29 57.420
[[Meantone family|Mothra]]: [[31edo]] 5&26 37.146
[[Meantone family|Squares]]: [[31edo]] 31&45 45.993
[[Meantone family|Injera]]: [[38edo]] 12&26 31.130
[[Augmented family|Triforce]]: [[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
[[Porcupine family#Porky|Porky]]: [[51edo]] 22&51 54.389
[[Meantone family|Liese]]: [[55edo]] 19&55 46.706
[[Diaschismic family|Diaschismic]]: [[58edo]] 12&34 37.914
[[Gamelismic clan|Miracle]]: [[72edo]] 10&21 16.742
[[Marvel temperaments|Wizard]]: [[72edo]] 22&50 40.846
[[Unicorn family#Alicorn|Alicorn]] [[77edo]] 19&58 40.913
[[Diaschismic family|Bidia]]: [[80edo]] 12&56 56.474
[[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
[[Schismatic family#Guiron|Guiron]]: [[118edo]] 36&41 47.544
[[Gamelismic clan|Hemithirds]]: [[118edo]] 25&31 44.284
[[Kleismic family#Hemikleismic|Hemikleismic]]: [[121edo]] 15&38 52.054
[[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
[[Schismatic family|Tsaharuk]]: [[171edo]] 17&77 30.697
[[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
[[Porwell temperaments|Hemischis]]: [[313edo]] 24&53 45.817
[[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
[[Porwell temperaments|Nessafof]]: [[381edo]] 15&69 45.478
[[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
[[Vulture family#Vulture|Vulture]]: [[1133edo]] 53&164 36.985
[[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
[[Minortonic family#Domain|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
[[Meantone family#Maqamic|Maqamic]]: [[7edo]] 7&10c 40.240
[[Porcupine family#Hystrix|Hystrix]]: [[7edo]] 7&15d 44.944
[[Marvel temperaments|Negric]]: [[9edo]] 1&9 30.617
[[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.132
[[Trienstonic clan|Opossum]]: [[15edo]] 7d&8d 22.325
[[Augmented family#x-Augene-11-limit|Augene]]: [[15edo]] 3&12 19.613
[[Marvel temperaments|Negri]]: [[19edo]] 9&10 26.190
[[Meantone family#Septimal%20meantone-Meanenneadecal|Meanenneadecal]]: [[19edo]] 12&19 21.182
[[Diaschismic family|Pajara]]: [[22edo]] 12&22 20.343
[[Magic family#Magic-Telepathy|Telepathy]]: [[22edo]] 19e&22 27.109
[[Porcupine family#Hedgehog|Hedgehog]]: [[22edo]] 14c&22 43.983
[[Porcupine family|Nautilus]]: [[29edo]] 15&29 26.023
[[Marvel temperaments|Negril]]: [[29edo]] 19&29 38.679
[[Meantone family|Mohajira]]: [[31edo]] 7&24 26.064
[[Meantone family|Squares]]: [[31edo]] 31&45e 21.636
[[Marvel temperaments|Tritonic]]: [[31edo]] 2&29 23.659
[[Starling temperaments|Casablanca]]: [[31edo]] 31&42 67.291
[[Meantone family#Cuboctahedra|Cuboctahedra]]: [[31edo]] 31&45 56.826
[[Kleismic family|Keemun]]: [[34edo]] 4&15 27.410
[[Meantone family#Liese-Liesel|Liesel]]: [[36edo]] 19&36 40.721
[[Meantone family#Liese-Elisa|Elisa]]: [[36edo|36e]] 19e&36e 41.592
[[Augmented family|Triforce]]: [[39edo]] 6&9 20.248
[[Schismatic family|Cassandra]]: [[41edo]] 12&29 23.556
[[Gamelismic clan|Superkleismic]]: [[41edo]] 15&26 25.659
[[Tetracot family#Monkey|Monkey]]: [[41edo]] 7&34 38.836
[[Tetracot family#Bunya|Bunya]]: [[41edo]] 7d&41 31.332
[[Marvel temperaments#Amavil|Amavil]]: [[43edo]] 9&43 42.649
[[Starling temperaments|Sensor]]: [[46edo]] 19&27 37.942
[[Starling temperaments|Sensus]]: [[46edo]] 46&73 29.486
[[Hemifamity temperaments|Leapday]]: [[46edo]] 29&46 38.624
[[Porwell temperaments|Twothirdtonic]]: [[46edo]] 9&37 40.768
[[Jubilismic clan|Doublewide]]: [[48edo]] 4&18 32.058
[[Archytas clan|Superpyth]]: [[49edo]] 22&49 24.976
[[Porcupine family#Porky|Porky]]: [[51edo]] 22&51 27.268
[[Semicomma family|Orwell]]: [[53edo]] 9&22 15.231
[[Kleismic family#Cataclysmic]]: [[53edo]] 53&140d 39.954
[[Diaschismic family|Diaschismic]]: [[58edo]] 12&46 25.034
[[Breedsmic temperaments|Hemififths]]: [[58edo]] 41&58 23.498
[[Starling temperaments#Thuja-11-limit|Thuja]]: [[58edo]] 15&43 33.078
[[Unicorn family#Alicorn|Alicorn]] [[58edo]] 19&58 39.156
[[Porcupine family|Porcupine]]: [[59edo]] 7&15 21.562
[[Porcupine family#Ammonite|Ammonite]]: [[66edo]] 29&37 45.694
[[Kleismic family#Hemikleismic|Hemikleismic]]: [[68edo]] 15&38 38.023
[[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#Casablanca temperament-Marrakesh|Marrakesh]]: [[73edo]] 31&73 40.539
[[Starling temperaments|Valentine]]: [[77edo]] 15&31 16.687
[[Unicorn family#Alicorn|Camahueto]] [[77edo]] 19&77 65.940
[[Diaschismic family|Keen]]: [[78edo]] 22&56 45.270
[[Diaschismic family|Echidna]]: [[80edo]] 22&58 25.987
[[Diaschismic family|Bidia]]: [[80edo]] 12&68 40.191
[[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
[[31 comma temperaments#Prajapati|Prajapati]]: [[93edo]] 31&93 42.959
[[Schismatic family|Garibaldi]]: [[94edo]] 41&53 27.396
[[Ragismic microtemperaments|Hitchcock]]: [[99edo]] 7&39 35.187
[[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|Tritoni]]: [[122edo]] 31&91 45.456
[[Marvel temperaments|Slender]]: [[125edo]] 31&63 25.342
[[Meantone family#Mothra-Mosura|Mosura]]: [[129edo]] 31&129 31.334
[[Wuerschmidt family|Hemiwuerschmidt]]: [[130edo]] 31&130 21.069
[[Schismatic family#Sesquiquartififths|Sesquart]]: [[130edo]] 41&89 29.306
[[Semicomma family#Borwell|Borwell]]: [[137edo]] 31&106 38.377
[[Hemifamity temperaments|Mystery]]: [[145edo]] 29&58 34.291
[[Diaschismic family|Echidnic]]: [[148edo]] 10&46 45.127
[[Sensipent family|Heinz]]: [[149edo]] 46&103 42.412
[[Mirkwai clan|Grendel]]: [[152edo]] 31&121 19.845
[[Mirkwai clan|Kwai]]: [[152edo]] 41&111 26.219
[[Hemifamity temperaments#Supers|Supers]]: [[152edo]] 58&94 28.240
[[Porwell temperaments#Whoops|Whoops]]: [[152edo]] 15&137 43.743
[[Unicorn family#Alicorn|Monocerus]]: [[154edo]] 58&154 52.757
[[Schismatic family|Guiron]]: [[159edo]] 41&77 26.648
[[Marvel temperaments|Wizard]]: [[166edo]] 22&50 18.539
[[Ragismic microtemperaments#Parakleismic-Parkleismic|Parkleismic]]: [[179edo]] 80&179 55.844
[[Gamelismic clan|Unidec]]: [[190edo]] 26&46 15.479
[[Breedsmic temperaments|Harry]]: [[202edo]] 58&72 15.867
[[Breedsmic temperaments|Tertiaseptal]]: [[202edo]] 31&171 35.576
[[Marvel temperaments#Marvo|Marvo]]: [[209edo]] 72&137 26.882
[[Ragismic microtemperaments|Octoid]]: [[224edo]] 72&152 14.097
[[Kleismic family|Countercata]]: [[227edo]] 34&53 39.770
[[Kleismic family|Tritikleismic]]: [[231edo]] 15&57 19.333
[[Ragismic microtemperaments#Ennealimmal|Ennealimmic]]: [[243edo]] 72&171 20.347
[[Schismatic family|Bischismic]]: [[248edo]] 12&118 28.160
[[31 comma temperaments#Birds|Birds]]: [[248edo]] 31&217 39.921
[[Mirkwai clan|Mirkat]]: [[255edo]] 72&183 22.126
[[Kleismic family|Sqrtphi]]: [[265edo]] 49&72 25.515
[[Breedsmic temperaments|Quasiorwell]]: [[270edo]] 31&208 17.540
[[Marvel temperaments|Marvo]]: [[281edo]] 72&281 31.685
[[Sensamagic clan|Sensa]]: [[283edo]] 22&65 35.844
[[Kleismic family|Quadritikleismic]]: [[284edo]] 4&68 23.406
[[Ragismic microtemperaments#Quincy|Quincy]]: [[289edo]] 72&217 30.875
[[Breedsmic temperaments#Neominor|Neominor]]: [[305edo]] 72&161 27.958
[[Porwell temperaments|Hemischis]]: [[313edo]] 53&130 36.289
[[Werckismic temperaments#Octowerck|Octowerck]]: [[320edo]] 72&176 30.159
[[Ragismic microtemperaments|Hemiamity]]: [[350edo]] 46&106 31.307
[[Ragismic microtemperaments|Semiennealimmal]]: [[441edo]] 72&441 34.196
[[Mirkwai clan#Eris|Eris]]: [[703edo]]: [[703edo]] 31&193 27.621
[[Ragismic microtemperaments|Abigail]]: [[764edo]] 46&132 12.860
[[Breedsmic temperaments|Decoid]]: [[1210edo]] 130&270 18.735
[[Ragismic microtemperaments|Deca]]: [[1270edo]] 80&190 24.329
[[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283

=11-limit rank three= 
Name: Optimal patent val: Val name: 10^5 * badness
[[Archytas family|Vulcan]]: [[15edo]] 2&3&10 57.7113
[[Marvel family|Potassium]]: [[19edo]] 2&9&10 46.3665
[[Quartonic family|Nickel]]: [[28edo]] 7&9&12 46.3463
[[Jubilismic family|Festival]]: [[26edo]] 2&10&26 68.8510
[[Starling family|Thrasher]]: [[34edo]] 3&4&12 48.0188
[[Semaphore family|Selenium]]: [[34edo]] 4&5&10 66.5377
[[Didymus rank three family|Calliope]]: [[45edo]] 7&12&19 52.9778
[[Jubilismic family|Jubilee]]: [[48edo]] 4&12&22 55.9842
[[Archytas family|Ares]]: [[49edo]] 3&7&12 46.8874
[[Porcupine rank three family|Sonic]]: [[51edo]] 7&15&51 52.3158
[[Nuwell family|Big Brother]]: [[53edo]] 9&22&70 50.6556
[[Marvel family|Prodigy]]: [[72edo]] 2&10&29 33.4745
[[Didymus rank three family|Erato]]: [[81edo]] 12&19&31 55.7521
[[Sensamagic family#Sensawer|Sensawer]]: [[87edo]] 41&46&73 79.632
[[Didymus rank three family|Euterpe]]: [[88edo]] 5&12&26 53.5809
[[Starling family|Thrush]]: [[89edo]] 3&12&31 35.2591
[[Starling family|Aplonis]]: [[89edo]] 19&31&58 64.7511
[[Valinorismic temperaments#Mandos|Mandos]]: [[89edo]] 7&24&58 75.1090
[[Porwell family|Zeus]]: [[99edo]] 6&7&9 39.9882
[[Marvel family|Apollo]]: [[104edo]] 12&19&22 52.9766
[[Orwellismic family|Guanyin]]: [[111edo]] 9&22&58 41.1252
[[Valinorismic temperaments#Manwe|Manwe]]: [[111edo]] 15&31&80 67.9150
[[Valinorismic temperaments#Varda|Varda]]: [[126edo]] 12&22&46 72.8125
[[Marvel family|Minerva]]: [[127edo]] 1&9&12 38.1064
[[Didymus rank three family|Clio]]: [[129edo]] 7&12&24 73.7590
[[Hemifamity family|Pele]]: [[145edo]] 5&12&29 64.8199
[[Hemifamity family|Laka]]: [[152edo]] 41&53&58 82.5317
[[Marvel family|Marvel]]: [[166edo]] 9&10&22 25.5033
[[Gamelismic family|Portent]]: [[190edo]] 5&10&26 23.4092
[[Breed family|Jove]]: [[202edo]] 10&31&58 24.1055
[[Hemimean family#Belobog|Belobog]]: [[248edo]] 12&31&87 60.9402
[[Marvel family|Spectacle]]: [[281edo]] 10&31&281 49.8669
[[Porwell family|Jupiter]]: [[282edo]] 9&22&108 56.2368
[[Sensamagic family#Sensamagic-Unidecimal sensamagic|Sensamagic]]: [[283edo]] 5&17&19 72.231
[[Breed family|Agni]]: [[284edo]] 4&6&27 49.3790
[[Kleismic rank three family#Enlil|Enlil]]: [[284edo]] 4&15&38 79.8062
[[Trimyna family#Trimyna-11-limit|Trimyna]]: [[294edo]] 2&29&58 81.4047
[[Hemimean family#Triglav|Triglav]]: [[316edo]] 31&80&87 81.896
[[Werckismic temperaments|Varuna]]: [[320edo]] 12&26&46 41.8714
[[Werckismic temperaments|History]]: [[491edo]] 15&29&43 60.3636
[[Swetismic temperaments|Hades]]: [[578edo]] 22&50&58 36.1398
[[Cataharry family|Albus]]: [[640edo]] 19&53&58 62.1100
[[Mirkwai family|Indra]]: [[703edo]] 10&31&152 26.6020
[[Lehmerismic temperaments#Hanuman|Hanuman]]: [[998edo]] 15&57&152 49.965
[[Kalismic temperaments|Loki]]: [[1092edo]] 12&22&86 49.2550
[[Lehmerismic temperaments|Ganesha]]: [[1502edo]] 31&87&152 39.0462
[[Breed family|Freya]]: [[1566edo]] 10&31&167 16.9702
[[Lehmerismic temperaments|Tyr]]: [[1566edo]] 15&57&183 25.8374
[[Breed family|Baldur]]: [[1696edo]] 4&58&72 16.6115
[[Lehmerismic temperaments|Skadi]]: [[2014edo]] 31&97&121 34.8944
[[Ragisma family|Thor]]: [[2554edo]] 26&46&152 8.8834
[[Breed family|Ennealimmic]]: [[2619edo]] 27&45&171 27.4685
[[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 11.6151

=13-limit rank two= 
Name: Optimal patent val: Val name: 1000*badness
[[Meantone family#Maqamic-13-limit|Maqamic]]: [[7edo]] 7&10c 27.214
[[Quartonic temperaments|Progression]]: [[9edo]] 8d&9 18.158
[[Marvel temperaments|Negric]]: [[9edo]] 1&9 20.205
[[Trienstonic clan|Opossum]]: [[15edo]] 7d&8d 19.389
[[Marvel temperaments|Negri]]: [[19edo]] 9&10 17.833
[[Starling temperaments|Sensis]]: [[19edo]] 8d&19 20.017
[[Starling temperaments|Keemun]]: [[19edo]] 4&19 29.749
[[Meantone family|Meanenneadecal]]: [[19edo]] 12f&19 21.182
[[Augmented family|August]]: [[21edo]] 9&21 18.448
[[Magic family#Magic-Telepathy-13-limit telepathy|Telepathy]]: [[22edo]] 19e&22 25.522
[[Magic family#Magic-13-limit-Sorcery|Sorcery]]: [[22edo]] 19&22 25.829
[[Augmented family|Triforce]]: [[24edo]] 9&15 20.248
[[Avicennmic temperaments|Roman]]: [[29edo]] 3&23 30.043
[[Porcupine family|Nautilus]]: [[29edo]] 15&29 22.285
[[Marvel temperaments|Negril]]: [[29edo]] 19&29 24.383
[[Meantone family|Meantone]]: [[31edo]] 12f&31 18.048
[[Meantone family|Squares]]: [[31edo]] 31&45e 25.514
[[Semicomma family|Winston]]: [[31edo]] 9&31 19.931
[[Starling temperaments|Lupercalia]]: [[31edo]] 15&31 21.328
[[Marvel temperaments#Tritonic|Tritonic]]: [[31edo]] 29&31 22.993
[[Starling temperaments#Casablanca temperament-Marrakesh-Murakuc|Murakuc]]: [[31edo]] 31&73f 41.394
[[31 comma temperaments#Prajapati|Prajapati]]: [[31edo]] 31&93f 37.885
[[Gamelismic clan|Miraculous]]: [[41edo]] 10&31 18.669
[[Tetracot family|Octacot]]: [[41edo]] 27e&41 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
[[Tetracot family#Monkey|Monkey]]: [[41edo]] 7&34 28.410
[[Tetracot family#Bunya|Bunya]]: [[41edo]] 7d&41 24.886
[[Meantone family#Jerome|Jerome]]: [[43edo]] 26&43 29.285
[[Marvel temperaments#Amavil|Amavil]]: [[43edo]] 9&43 25.791
[[Starling temperaments|Sensor]]: [[46edo]] 19&27 25.474
[[Starling temperaments|Sensus]]: [[46edo]] 46&73 20.789
[[Starling temperaments|Dwynwen]]: [[46edo]] 15&46 23.461
[[Hemifamity temperaments|Leapday]]: [[46edo]] 29&46 24.732
[[Porwell temperaments|Twothirdtonic]]: [[46edo]] 9&37 25.941
[[Kleismic family#Cataclysmic]]: [[53edo]] 53&140d 22.554
[[Diaschismic family|Diaschismic]]: [[58edo]] 46&58 18.926
[[Starling temperaments|Myna]]: [[58edo]] 31&58 17.125
[[Breedsmic temperaments|Hemififths]]: [[58edo]] 41&58 19.090
[[Starling temperaments#Thuja-13-limit|Thuja]]: [[58edo]] 15&43 22.838
[[Unicorn family#Alicorn|Alicorn]]: [[58edo]] 19&58 23.667
[[31 comma temperaments#Gallium]]: [[62edo]] 31&62 25.484
[[Porcupine family|Ammonite]]: [[66edo]] 29&37 27.168
[[Gamelismic clan|Manna]]: [[72edo]] 41&72 17.012
[[Gamelismic clan|Hendec]]: [[72edo]] 26&46 17.707
[[Kleismic family|Catakleismic]]: [[72edo]] 19&53 16.883
[[Marvel temperaments|Lizard]]: [[72edo]] 22&50 21.781
[[Breedsmic temperaments#Neominor|Neominor]]: [[72edo]] 72&161f 26.942
[[Pythagorean family#Compton temperament|Compton]]: [[72edo]] 72&228f 21.852
[[Pythagorean family#Compton temperament|Comptone]]: [[72edo]] 12&72 25.144
[[Starling temperaments#Casablanca temperament-Marrakesh-13-limit|Marrakesh]]: [[73edo]] 31&73 40.774
[[Meantone family|Grosstone]]: [[74edo]] 12&31 25.899
[[Starling temperaments|Valentino]]: [[77edo]] 31&46 20.665
[[Unicorn family#Alicorn|Camahueto]] [[77edo]] 19&77 36.155
[[Diaschismic family|Echidna]]: [[80edo]] 22&58 23.679
[[Diaschismic family|Bidia]]: [[80edo]] 12&68 41.137
[[Meantone family|Meanpop]]: [[81edo]] 19&31 20.883
[[Gamelismic clan|Rodan]]: [[87edo]] 41&46 18.448
[[Meantone family#Mothra|Mothra]]: [[88edo]] 5&26 23.954
[[31 comma temperaments#Prajapati|Kumhar]]: [[93edo]] 31&93 48.582
[[Marvel temperaments|Slender]]: [[94edo]] 31&63 25.913
[[Meantone family#Mothra|Mosura]]: [[98edo]] 31&67 36.857
[[Diaschismic family|Echidnic]]: [[102edo]] 10&46 28.874
[[Gamelismic clan|Benediction]]: [[103edo]] 31&72 15.715
[[Magic family#Magic-13-limit-Necromancy|Necromancy]]: [[104edo]] 22&41 25.275
[[Hemifamity temperaments|Buzzard]]: [[111edo]] 53&58 18.842
[[Diaschismic family|Shrutar]]: [[114edo]] 46&68 28.057
[[Schismatic family|Guiron]]: [[118edo]] 41&77 28.444
[[Mirkwai clan|Grendel]]: [[121edo]] 31&121 24.839
[[Breedsmic temperaments|Harry]]: [[130edo]] 58&72 13.046
[[Kleismic family|Countercata]]: [[140edo]] 34&53 20.156
[[Pythagorean family#Omicronbeta temperament|Omicronbeta]]: [[144edo]] 72&144 29.956
[[Sensipent family|Heinz]]: [[149edo]] 46&103 25.779
[[Hemifamity temperaments#Supers|Supers]]: [[152edo]] 58&94 21.644
[[Unicorn family#Alicorn|Monocerus]]: [[154edo]] 58&154 28.795
[[Kleismic family|Tritikleismic]]: [[159edo]] 15&72 15.652
[[Marvel temperaments|Gizzard]]: [[166edo]] 72&166 20.252
[[Ragismic microtemperaments#Parakleismic-Parkleismic-13-limit|Parkleismic]]: [[179edo]] 80&179 36.559
[[Mirkwai clan|Mirkat]]: [[183edo]] 72&111 18.632
[[Gamelismic clan|Ekadash]]: [[190edo]] 72&118 20.381
[[Kleismic family|Sqrtphi]]: [[193edo]] 72&121 20.040
[[Misty family#Hemimist|Hemimist]]: [[198edo]] 87&111 22.923
[[Hemifamity temperaments|Quanic]]: [[205edo]] 94&111 32.481
[[Kleismic family|Metakleismic]]: [[208edo]] 87&121 24.371
[[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#Ennealimmal|Ennealimmic]]: [[243edo]] 72&171 20.250
[[Ragismic microtemperaments|Hemiennealimmal]]: [[270edo]] 72&198 12.505
[[Sensamagic clan|Sensa]]: [[283edo]] 22&65 31.366
[[Schismatic family|Sextilififths]]: [[289edo]] 29&130 25.276
[[Ragismic microtemperaments#Quincy|Quincy]]: [[289edo]] 72&145 23.862
[[Porwell temperaments|Hemischis]]: [[313edo]] 53&130 20.816
[[Werckismic temperaments#Octowerck|Octowerck]]: [[320edo]] 72&248 27.632
[[Schismatic family|Bischismic]]: [[378edo]] 12&118 28.722
[[Ragismic microtemperaments|Semiennealimmal]]: [[441edo]] 72&441 26.122
[[31 comma temperaments#Birds|Birds]]: [[465edo]] 31&217 35.680
[[Breedsmic temperaments|Decoid]]: [[940edo]] 130&270 13.475
[[Ragismic microtemperaments#Ennealimmal|Semihemiennealimmal]] [[954edo]] 126&144 13.104
[[Ragismic microtemperaments|Deca]]: [[1000edo]] 80&190 16.810
[[Breedsmic temperaments|Quasiorwell]]: [[1111edo]] 31&239 17.921
[[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]] 4&12f&19 90.475
[[Marvel family|Potassium]]: [[19edo]] 9&10&11 73.346
[[Quartonic family|Nickel]]: [[28edo]] 7&9&21 64.705
[[Semaphore family#Negric|Negric]]: [[29edo]] 1&9&10 80.756
[[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#Prodigy|Prodigy]]: [[41edo]] 10&29&31 73.639
[[Marvel family|Athene]]: [[53edo]] 12f&22&53 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#Prodigy|Prodigious]]: [[72edo]] 2&29&41 89.974
[[Sensamagic family#Sensawer-13-limit|Sensawer]]: [[87edo]] 41&46&73 92.807
[[Marvel family|Tripod]]: [[91edo]] 9&10&31 74.546
[[Marvel family|Isis]]: [[94edo]] 10&22&31 86.583
[[Marvel family|Marvel]]: [[103edo]] 19&22&31 68.977
[[Marvel family#Prodigy|Prodigal]]: [[103edo]] 31&72&84 88.910
[[Marvel family#Mirage|Mirage]]: [[103edo]] 10&31&62 73.850
[[Swetismic temperaments#Hades-Cerberus|Cerberus]]: [[130edo]] 22&50&58 76.512
[[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|Hecate]]: [[166edo]] 19&41&53 72.113
[[Marvel family|Marvell]]: [[166edo]] 9&31&63 86.160
[[Valinorismic temperaments#Ulmo|Ulmo]]: [[179edo]] 7&39&80 93.162
[[Mirkwai family#Indra-Vibhu|Vibhu]]: [[183edo]] 31&72&111 82.598
[[Gamelismic family|Portentous]]: [[190edo]] 15&31&56 66.119
[[Mirkwai family#Indra-Vish|Vish]]: [[193edo]] 41&72&121 92.976
[[Swetismic temperaments#Minos-13-limit|Minos]]: [[193edo]] 9&54&58 72.381
[[Hemifamity family#Namaka|Namaka]]: [[198edo]] 29&53&58 78.124
[[Hemifamity family|Laka]]: [[205edo]] 41&53&58 82.186
[[Breed family|Agni]]: [[212edo]] 6&31&72 92.251
[[Mirkwai family#Indra-Yajna|Yajna]]: [[224edo]] 31&72&121 69.986
[[Hemifamity family|Pele]]: [[232edo]] 29&41&46 70.328
[[Trimyna family#Trimyna-13-limit|Trimyna]]: [[232edo]] 29&31&56 81.405
[[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
[[Kleismic rank three family#Enlil|Enlil]]: [[299edo]] 15&19&53 68.005
[[Cataharry family|Madagascar]]: [[313edo]] 19&53&58 56.006
[[Hemifamity family#Akea|Akea]]: [[321edo]] 7&34&46 82.191
[[Breed family|Jovis]]: [[363edo]] 31&58&72 54.221
[[Lehmerismic temperaments#Hanuman|Hanuman]]: [[535edo]] 15&72&137 65.278
[[Swetismic temperaments#Hades|Hades]]: [[578edo]] 36&58&72 
[[Breed family|Vili]]: [[643edo]] 37&93&103 73.821
[[Mirkwai family#Indra-Shibi|Shibi]]: [[703edo]] 41&72&111 72.094
[[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

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<html><head><title>Optimal patent val</title></head><body><!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#x5-limit rank two">5-limit rank two</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#x7-limit rank two">7-limit rank two</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#x7-limit rank three">7-limit rank three</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#x11-limit rank two">11-limit rank two</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#x11-limit rank three">11-limit rank three</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#x13-limit rank two">13-limit rank two</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#x13-limit rank three">13-limit rank three</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
<!-- ws:end:WikiTextTocRule:22 --><br />
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 ≤ 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 vals 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, 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, &quot;12f&quot; adjusts a patent val for 12 in the 13-limit or above, for instance &lt;12 19 28 34 42 44|, to &lt;12 19 28 34 42 45| (which is actually a better mapping, and hence more useful for this purpose.)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x5-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit rank two</h1>
 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;h1&gt; --><h1 id="toc1"><a name="x7-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit rank two</h1>
 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> 3d&amp;5 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> 5&amp;7d 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> 3d&amp;7d 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> 7d&amp;8d 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="/Trienstonic%20clan">Uncle</a>: <a class="wiki_link" href="/18edo">18edo</a> 5&amp;18 72.653<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="/Porcupine%20family">Nautilus</a>: <a class="wiki_link" href="/29edo">29edo</a> 15&amp;29 57.420<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">Squares</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;45 45.993<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">Triforce</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="/Porcupine%20family#Porky">Porky</a>: <a class="wiki_link" href="/51edo">51edo</a> 22&amp;51 54.389<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="/Diaschismic%20family">Diaschismic</a>: <a class="wiki_link" href="/58edo">58edo</a> 12&amp;34 37.914<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="/Unicorn%20family#Alicorn">Alicorn</a> <a class="wiki_link" href="/77edo">77edo</a> 19&amp;58 40.913<br />
<a class="wiki_link" href="/Diaschismic%20family">Bidia</a>: <a class="wiki_link" href="/80edo">80edo</a> 12&amp;56 56.474<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="/Schismatic%20family#Guiron">Guiron</a>: <a class="wiki_link" href="/118edo">118edo</a> 36&amp;41 47.544<br />
<a class="wiki_link" href="/Gamelismic%20clan">Hemithirds</a>: <a class="wiki_link" href="/118edo">118edo</a> 25&amp;31 44.284<br />
<a class="wiki_link" href="/Kleismic%20family#Hemikleismic">Hemikleismic</a>: <a class="wiki_link" href="/121edo">121edo</a> 15&amp;38 52.054<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="/Schismatic%20family">Tsaharuk</a>: <a class="wiki_link" href="/171edo">171edo</a> 17&amp;77 30.697<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="/Porwell%20temperaments">Hemischis</a>: <a class="wiki_link" href="/313edo">313edo</a> 24&amp;53 45.817<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="/Porwell%20temperaments">Nessafof</a>: <a class="wiki_link" href="/381edo">381edo</a> 15&amp;69 45.478<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="/Vulture%20family#Vulture">Vulture</a>: <a class="wiki_link" href="/1133edo">1133edo</a> 53&amp;164 36.985<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 />
<a class="wiki_link" href="/Minortonic%20family#Domain">Domain</a>: <a class="wiki_link" href="/22038edo">22038edo</a> 171&amp;1164 13.979<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x7-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit rank three</h1>
 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 />
5120/5103: <a class="wiki_link" href="/391edo">391edo</a> 153.052<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;h1&gt; --><h1 id="toc3"><a name="x11-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit rank two</h1>
 Name: Optimal patent val: Val name: 1000*badness<br />
<a class="wiki_link" href="/Meantone%20family#Maqamic">Maqamic</a>: <a class="wiki_link" href="/7edo">7edo</a> 7&amp;10c 40.240<br />
<a class="wiki_link" href="/Porcupine%20family#Hystrix">Hystrix</a>: <a class="wiki_link" href="/7edo">7edo</a> 7&amp;15d 44.944<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> 7d&amp;8d 22.325<br />
<a class="wiki_link" href="/Augmented%20family#x-Augene-11-limit">Augene</a>: <a class="wiki_link" href="/15edo">15edo</a> 3&amp;12 19.613<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="/Meantone%20family#Septimal%20meantone-Meanenneadecal">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="/Magic%20family#Magic-Telepathy">Telepathy</a>: <a class="wiki_link" href="/22edo">22edo</a> 19e&amp;22 27.109<br />
<a class="wiki_link" href="/Porcupine%20family#Hedgehog">Hedgehog</a>: <a class="wiki_link" href="/22edo">22edo</a> 14c&amp;22 43.983<br />
<a class="wiki_link" href="/Porcupine%20family">Nautilus</a>: <a class="wiki_link" href="/29edo">29edo</a> 15&amp;29 26.023<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negril</a>: <a class="wiki_link" href="/29edo">29edo</a> 19&amp;29 38.679<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;45e 21.636<br />
<a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/31edo">31edo</a> 2&amp;29 23.659<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="/Meantone%20family#Cuboctahedra">Cuboctahedra</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;45 56.826<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="/Meantone%20family#Liese-Liesel">Liesel</a>: <a class="wiki_link" href="/36edo">36edo</a> 19&amp;36 40.721<br />
<a class="wiki_link" href="/Meantone%20family#Liese-Elisa">Elisa</a>: <a class="wiki_link" href="/36edo">36e</a> 19e&amp;36e 41.592<br />
<a class="wiki_link" href="/Augmented%20family">Triforce</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="/Tetracot%20family#Monkey">Monkey</a>: <a class="wiki_link" href="/41edo">41edo</a> 7&amp;34 38.836<br />
<a class="wiki_link" href="/Tetracot%20family#Bunya">Bunya</a>: <a class="wiki_link" href="/41edo">41edo</a> 7d&amp;41 31.332<br />
<a class="wiki_link" href="/Marvel%20temperaments#Amavil">Amavil</a>: <a class="wiki_link" href="/43edo">43edo</a> 9&amp;43 42.649<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="/Porwell%20temperaments">Twothirdtonic</a>: <a class="wiki_link" href="/46edo">46edo</a> 9&amp;37 40.768<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="/Archytas%20clan">Superpyth</a>: <a class="wiki_link" href="/49edo">49edo</a> 22&amp;49 24.976<br />
<a class="wiki_link" href="/Porcupine%20family#Porky">Porky</a>: <a class="wiki_link" href="/51edo">51edo</a> 22&amp;51 27.268<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="/Kleismic%20family#Cataclysmic">Kleismic family</a>: <a class="wiki_link" href="/53edo">53edo</a> 53&amp;140d 39.954<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="/Starling%20temperaments#Thuja-11-limit">Thuja</a>: <a class="wiki_link" href="/58edo">58edo</a> 15&amp;43 33.078<br />
<a class="wiki_link" href="/Unicorn%20family#Alicorn">Alicorn</a> <a class="wiki_link" href="/58edo">58edo</a> 19&amp;58 39.156<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="/Porcupine%20family#Ammonite">Ammonite</a>: <a class="wiki_link" href="/66edo">66edo</a> 29&amp;37 45.694<br />
<a class="wiki_link" href="/Kleismic%20family#Hemikleismic">Hemikleismic</a>: <a class="wiki_link" href="/68edo">68edo</a> 15&amp;38 38.023<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#Casablanca temperament-Marrakesh">Marrakesh</a>: <a class="wiki_link" href="/73edo">73edo</a> 31&amp;73 40.539<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="/Unicorn%20family#Alicorn">Camahueto</a> <a class="wiki_link" href="/77edo">77edo</a> 19&amp;77 65.940<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="/Diaschismic%20family">Bidia</a>: <a class="wiki_link" href="/80edo">80edo</a> 12&amp;68 40.191<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="/31%20comma%20temperaments#Prajapati">Prajapati</a>: <a class="wiki_link" href="/93edo">93edo</a> 31&amp;93 42.959<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="/Ragismic%20microtemperaments">Hitchcock</a>: <a class="wiki_link" href="/99edo">99edo</a> 7&amp;39 35.187<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">Tritoni</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="/Meantone%20family#Mothra-Mosura">Mosura</a>: <a class="wiki_link" href="/129edo">129edo</a> 31&amp;129 31.334<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="/Schismatic%20family#Sesquiquartififths">Sesquart</a>: <a class="wiki_link" href="/130edo">130edo</a> 41&amp;89 29.306<br />
<a class="wiki_link" href="/Semicomma%20family#Borwell">Borwell</a>: <a class="wiki_link" href="/137edo">137edo</a> 31&amp;106 38.377<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="/Sensipent%20family">Heinz</a>: <a class="wiki_link" href="/149edo">149edo</a> 46&amp;103 42.412<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="/Hemifamity%20temperaments#Supers">Supers</a>: <a class="wiki_link" href="/152edo">152edo</a> 58&amp;94 28.240<br />
<a class="wiki_link" href="/Porwell%20temperaments#Whoops">Whoops</a>: <a class="wiki_link" href="/152edo">152edo</a> 15&amp;137 43.743<br />
<a class="wiki_link" href="/Unicorn%20family#Alicorn">Monocerus</a>: <a class="wiki_link" href="/154edo">154edo</a> 58&amp;154 52.757<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="/Ragismic%20microtemperaments#Parakleismic-Parkleismic">Parkleismic</a>: <a class="wiki_link" href="/179edo">179edo</a> 80&amp;179 55.844<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="/Marvel%20temperaments#Marvo">Marvo</a>: <a class="wiki_link" href="/209edo">209edo</a> 72&amp;137 26.882<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="/Ragismic%20microtemperaments#Ennealimmal">Ennealimmic</a>: <a class="wiki_link" href="/243edo">243edo</a> 72&amp;171 20.347<br />
<a class="wiki_link" href="/Schismatic%20family">Bischismic</a>: <a class="wiki_link" href="/248edo">248edo</a> 12&amp;118 28.160<br />
<a class="wiki_link" href="/31%20comma%20temperaments#Birds">Birds</a>: <a class="wiki_link" href="/248edo">248edo</a> 31&amp;217 39.921<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="/Kleismic%20family">Sqrtphi</a>: <a class="wiki_link" href="/265edo">265edo</a> 49&amp;72 25.515<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="/Sensamagic%20clan">Sensa</a>: <a class="wiki_link" href="/283edo">283edo</a> 22&amp;65 35.844<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#Quincy">Quincy</a>: <a class="wiki_link" href="/289edo">289edo</a> 72&amp;217 30.875<br />
<a class="wiki_link" href="/Breedsmic%20temperaments#Neominor">Neominor</a>: <a class="wiki_link" href="/305edo">305edo</a> 72&amp;161 27.958<br />
<a class="wiki_link" href="/Porwell%20temperaments">Hemischis</a>: <a class="wiki_link" href="/313edo">313edo</a> 53&amp;130 36.289<br />
<a class="wiki_link" href="/Werckismic%20temperaments#Octowerck">Octowerck</a>: <a class="wiki_link" href="/320edo">320edo</a> 72&amp;176 30.159<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">Semiennealimmal</a>: <a class="wiki_link" href="/441edo">441edo</a> 72&amp;441 34.196<br />
<a class="wiki_link" href="/Mirkwai%20clan#Eris">Eris</a>: <a class="wiki_link" href="/703edo">703edo</a>: <a class="wiki_link" href="/703edo">703edo</a> 31&amp;193 27.621<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">Deca</a>: <a class="wiki_link" href="/1270edo">1270edo</a> 80&amp;190 24.329<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;h1&gt; --><h1 id="toc4"><a name="x11-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit rank three</h1>
 Name: Optimal patent val: Val name: 10^5 * badness<br />
<a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&amp;3&amp;10 57.7113<br />
<a class="wiki_link" href="/Marvel%20family">Potassium</a>: <a class="wiki_link" href="/19edo">19edo</a> 2&amp;9&amp;10 46.3665<br />
<a class="wiki_link" href="/Quartonic%20family">Nickel</a>: <a class="wiki_link" href="/28edo">28edo</a> 7&amp;9&amp;12 46.3463<br />
<a class="wiki_link" href="/Jubilismic%20family">Festival</a>: <a class="wiki_link" href="/26edo">26edo</a> 2&amp;10&amp;26 68.8510<br />
<a class="wiki_link" href="/Starling%20family">Thrasher</a>: <a class="wiki_link" href="/34edo">34edo</a> 3&amp;4&amp;12 48.0188<br />
<a class="wiki_link" href="/Semaphore%20family">Selenium</a>: <a class="wiki_link" href="/34edo">34edo</a> 4&amp;5&amp;10 66.5377<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 52.9778<br />
<a class="wiki_link" href="/Jubilismic%20family">Jubilee</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&amp;12&amp;22 55.9842<br />
<a class="wiki_link" href="/Archytas%20family">Ares</a>: <a class="wiki_link" href="/49edo">49edo</a> 3&amp;7&amp;12 46.8874<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 52.3158<br />
<a class="wiki_link" href="/Nuwell%20family">Big Brother</a>: <a class="wiki_link" href="/53edo">53edo</a> 9&amp;22&amp;70 50.6556<br />
<a class="wiki_link" href="/Marvel%20family">Prodigy</a>: <a class="wiki_link" href="/72edo">72edo</a> 2&amp;10&amp;29 33.4745<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 55.7521<br />
<a class="wiki_link" href="/Sensamagic%20family#Sensawer">Sensawer</a>: <a class="wiki_link" href="/87edo">87edo</a> 41&amp;46&amp;73 79.632<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 53.5809<br />
<a class="wiki_link" href="/Starling%20family">Thrush</a>: <a class="wiki_link" href="/89edo">89edo</a> 3&amp;12&amp;31 35.2591<br />
<a class="wiki_link" href="/Starling%20family">Aplonis</a>: <a class="wiki_link" href="/89edo">89edo</a> 19&amp;31&amp;58 64.7511<br />
<a class="wiki_link" href="/Valinorismic%20temperaments#Mandos">Mandos</a>: <a class="wiki_link" href="/89edo">89edo</a> 7&amp;24&amp;58 75.1090<br />
<a class="wiki_link" href="/Porwell%20family">Zeus</a>: <a class="wiki_link" href="/99edo">99edo</a> 6&amp;7&amp;9 39.9882<br />
<a class="wiki_link" href="/Marvel%20family">Apollo</a>: <a class="wiki_link" href="/104edo">104edo</a> 12&amp;19&amp;22 52.9766<br />
<a class="wiki_link" href="/Orwellismic%20family">Guanyin</a>: <a class="wiki_link" href="/111edo">111edo</a> 9&amp;22&amp;58 41.1252<br />
<a class="wiki_link" href="/Valinorismic%20temperaments#Manwe">Manwe</a>: <a class="wiki_link" href="/111edo">111edo</a> 15&amp;31&amp;80 67.9150<br />
<a class="wiki_link" href="/Valinorismic%20temperaments#Varda">Varda</a>: <a class="wiki_link" href="/126edo">126edo</a> 12&amp;22&amp;46 72.8125<br />
<a class="wiki_link" href="/Marvel%20family">Minerva</a>: <a class="wiki_link" href="/127edo">127edo</a> 1&amp;9&amp;12 38.1064<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 73.7590<br />
<a class="wiki_link" href="/Hemifamity%20family">Pele</a>: <a class="wiki_link" href="/145edo">145edo</a> 5&amp;12&amp;29 64.8199<br />
<a class="wiki_link" href="/Hemifamity%20family">Laka</a>: <a class="wiki_link" href="/152edo">152edo</a> 41&amp;53&amp;58 82.5317<br />
<a class="wiki_link" href="/Marvel%20family">Marvel</a>: <a class="wiki_link" href="/166edo">166edo</a> 9&amp;10&amp;22 25.5033<br />
<a class="wiki_link" href="/Gamelismic%20family">Portent</a>: <a class="wiki_link" href="/190edo">190edo</a> 5&amp;10&amp;26 23.4092<br />
<a class="wiki_link" href="/Breed%20family">Jove</a>: <a class="wiki_link" href="/202edo">202edo</a> 10&amp;31&amp;58 24.1055<br />
<a class="wiki_link" href="/Hemimean%20family#Belobog">Belobog</a>: <a class="wiki_link" href="/248edo">248edo</a> 12&amp;31&amp;87 60.9402<br />
<a class="wiki_link" href="/Marvel%20family">Spectacle</a>: <a class="wiki_link" href="/281edo">281edo</a> 10&amp;31&amp;281 49.8669<br />
<a class="wiki_link" href="/Porwell%20family">Jupiter</a>: <a class="wiki_link" href="/282edo">282edo</a> 9&amp;22&amp;108 56.2368<br />
<a class="wiki_link" href="/Sensamagic%20family#Sensamagic-Unidecimal sensamagic">Sensamagic</a>: <a class="wiki_link" href="/283edo">283edo</a> 5&amp;17&amp;19 72.231<br />
<a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/284edo">284edo</a> 4&amp;6&amp;27 49.3790<br />
<a class="wiki_link" href="/Kleismic%20rank%20three%20family#Enlil">Enlil</a>: <a class="wiki_link" href="/284edo">284edo</a> 4&amp;15&amp;38 79.8062<br />
<a class="wiki_link" href="/Trimyna%20family#Trimyna-11-limit">Trimyna</a>: <a class="wiki_link" href="/294edo">294edo</a> 2&amp;29&amp;58 81.4047<br />
<a class="wiki_link" href="/Hemimean%20family#Triglav">Triglav</a>: <a class="wiki_link" href="/316edo">316edo</a> 31&amp;80&amp;87 81.896<br />
<a class="wiki_link" href="/Werckismic%20temperaments">Varuna</a>: <a class="wiki_link" href="/320edo">320edo</a> 12&amp;26&amp;46 41.8714<br />
<a class="wiki_link" href="/Werckismic%20temperaments">History</a>: <a class="wiki_link" href="/491edo">491edo</a> 15&amp;29&amp;43 60.3636<br />
<a class="wiki_link" href="/Swetismic%20temperaments">Hades</a>: <a class="wiki_link" href="/578edo">578edo</a> 22&amp;50&amp;58 36.1398<br />
<a class="wiki_link" href="/Cataharry%20family">Albus</a>: <a class="wiki_link" href="/640edo">640edo</a> 19&amp;53&amp;58 62.1100<br />
<a class="wiki_link" href="/Mirkwai%20family">Indra</a>: <a class="wiki_link" href="/703edo">703edo</a> 10&amp;31&amp;152 26.6020<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments#Hanuman">Hanuman</a>: <a class="wiki_link" href="/998edo">998edo</a> 15&amp;57&amp;152 49.965<br />
<a class="wiki_link" href="/Kalismic%20temperaments">Loki</a>: <a class="wiki_link" href="/1092edo">1092edo</a> 12&amp;22&amp;86 49.2550<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments">Ganesha</a>: <a class="wiki_link" href="/1502edo">1502edo</a> 31&amp;87&amp;152 39.0462<br />
<a class="wiki_link" href="/Breed%20family">Freya</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 10&amp;31&amp;167 16.9702<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments">Tyr</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 15&amp;57&amp;183 25.8374<br />
<a class="wiki_link" href="/Breed%20family">Baldur</a>: <a class="wiki_link" href="/1696edo">1696edo</a> 4&amp;58&amp;72 16.6115<br />
<a class="wiki_link" href="/Lehmerismic%20temperaments">Skadi</a>: <a class="wiki_link" href="/2014edo">2014edo</a> 31&amp;97&amp;121 34.8944<br />
<a class="wiki_link" href="/Ragisma%20family">Thor</a>: <a class="wiki_link" href="/2554edo">2554edo</a> 26&amp;46&amp;152 8.8834<br />
<a class="wiki_link" href="/Breed%20family">Ennealimmic</a>: <a class="wiki_link" href="/2619edo">2619edo</a> 27&amp;45&amp;171 27.4685<br />
<a class="wiki_link" href="/Kalismic%20temperaments">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</a> 12&amp;42&amp;72 11.6151<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="x13-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit rank two</h1>
 Name: Optimal patent val: Val name: 1000*badness<br />
<a class="wiki_link" href="/Meantone%20family#Maqamic-13-limit">Maqamic</a>: <a class="wiki_link" href="/7edo">7edo</a> 7&amp;10c 27.214<br />
<a class="wiki_link" href="/Quartonic%20temperaments">Progression</a>: <a class="wiki_link" href="/9edo">9edo</a> 8d&amp;9 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> 7d&amp;8d 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> 8d&amp;19 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> 12f&amp;19 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="/Magic%20family#Magic-Telepathy-13-limit telepathy">Telepathy</a>: <a class="wiki_link" href="/22edo">22edo</a> 19e&amp;22 25.522<br />
<a class="wiki_link" href="/Magic%20family#Magic-13-limit-Sorcery">Sorcery</a>: <a class="wiki_link" href="/22edo">22edo</a> 19&amp;22 25.829<br />
<a class="wiki_link" href="/Augmented%20family">Triforce</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="/Porcupine%20family">Nautilus</a>: <a class="wiki_link" href="/29edo">29edo</a> 15&amp;29 22.285<br />
<a class="wiki_link" href="/Marvel%20temperaments">Negril</a>: <a class="wiki_link" href="/29edo">29edo</a> 19&amp;29 24.383<br />
<a class="wiki_link" href="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/31edo">31edo</a> 12f&amp;31 18.048<br />
<a class="wiki_link" href="/Meantone%20family">Squares</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;45e 25.514<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="/Marvel%20temperaments#Tritonic">Tritonic</a>: <a class="wiki_link" href="/31edo">31edo</a> 29&amp;31 22.993<br />
<a class="wiki_link" href="/Starling%20temperaments#Casablanca temperament-Marrakesh-Murakuc">Murakuc</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;73f 41.394<br />
<a class="wiki_link" href="/31%20comma%20temperaments#Prajapati">Prajapati</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&amp;93f 37.885<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> 27e&amp;41 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="/Tetracot%20family#Monkey">Monkey</a>: <a class="wiki_link" href="/41edo">41edo</a> 7&amp;34 28.410<br />
<a class="wiki_link" href="/Tetracot%20family#Bunya">Bunya</a>: <a class="wiki_link" href="/41edo">41edo</a> 7d&amp;41 24.886<br />
<a class="wiki_link" href="/Meantone%20family#Jerome">Jerome</a>: <a class="wiki_link" href="/43edo">43edo</a> 26&amp;43 29.285<br />
<a class="wiki_link" href="/Marvel%20temperaments#Amavil">Amavil</a>: <a class="wiki_link" href="/43edo">43edo</a> 9&amp;43 25.791<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">Dwynwen</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="/Porwell%20temperaments">Twothirdtonic</a>: <a class="wiki_link" href="/46edo">46edo</a> 9&amp;37 25.941<br />
<a class="wiki_link" href="/Kleismic%20family#Cataclysmic">Kleismic family</a>: <a class="wiki_link" href="/53edo">53edo</a> 53&amp;140d 22.554<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="/Starling%20temperaments#Thuja-13-limit">Thuja</a>: <a class="wiki_link" href="/58edo">58edo</a> 15&amp;43 22.838<br />
<a class="wiki_link" href="/Unicorn%20family#Alicorn">Alicorn</a>: <a class="wiki_link" href="/58edo">58edo</a> 19&amp;58 23.667<br />
<a class="wiki_link" href="/31%20comma%20temperaments#Gallium">31 comma temperaments</a>: <a class="wiki_link" href="/62edo">62edo</a> 31&amp;62 25.484<br />
<a class="wiki_link" href="/Porcupine%20family">Ammonite</a>: <a class="wiki_link" href="/66edo">66edo</a> 29&amp;37 27.168<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="/Marvel%20temperaments">Lizard</a>: <a class="wiki_link" href="/72edo">72edo</a> 22&amp;50 21.781<br />
<a class="wiki_link" href="/Breedsmic%20temperaments#Neominor">Neominor</a>: <a class="wiki_link" href="/72edo">72edo</a> 72&amp;161f 26.942<br />
<a class="wiki_link" href="/Pythagorean%20family#Compton temperament">Compton</a>: <a class="wiki_link" href="/72edo">72edo</a> 72&amp;228f 21.852<br />
<a class="wiki_link" href="/Pythagorean%20family#Compton temperament">Comptone</a>: <a class="wiki_link" href="/72edo">72edo</a> 12&amp;72 25.144<br />
<a class="wiki_link" href="/Starling%20temperaments#Casablanca temperament-Marrakesh-13-limit">Marrakesh</a>: <a class="wiki_link" href="/73edo">73edo</a> 31&amp;73 40.774<br />
<a class="wiki_link" href="/Meantone%20family">Grosstone</a>: <a class="wiki_link" href="/74edo">74edo</a> 12&amp;31 25.899<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="/Unicorn%20family#Alicorn">Camahueto</a> <a class="wiki_link" href="/77edo">77edo</a> 19&amp;77 36.155<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="/Diaschismic%20family">Bidia</a>: <a class="wiki_link" href="/80edo">80edo</a> 12&amp;68 41.137<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="/Meantone%20family#Mothra">Mothra</a>: <a class="wiki_link" href="/88edo">88edo</a> 5&amp;26 23.954<br />
<a class="wiki_link" href="/31%20comma%20temperaments#Prajapati">Kumhar</a>: <a class="wiki_link" href="/93edo">93edo</a> 31&amp;93 48.582<br />
<a class="wiki_link" href="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/94edo">94edo</a> 31&amp;63 25.913<br />
<a class="wiki_link" href="/Meantone%20family#Mothra">Mosura</a>: <a class="wiki_link" href="/98edo">98edo</a> 31&amp;67 36.857<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="/Magic%20family#Magic-13-limit-Necromancy">Necromancy</a>: <a class="wiki_link" href="/104edo">104edo</a> 22&amp;41 25.275<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="/Diaschismic%20family">Shrutar</a>: <a class="wiki_link" href="/114edo">114edo</a> 46&amp;68 28.057<br />
<a class="wiki_link" href="/Schismatic%20family">Guiron</a>: <a class="wiki_link" href="/118edo">118edo</a> 41&amp;77 28.444<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="/Pythagorean%20family#Omicronbeta temperament">Omicronbeta</a>: <a class="wiki_link" href="/144edo">144edo</a> 72&amp;144 29.956<br />
<a class="wiki_link" href="/Sensipent%20family">Heinz</a>: <a class="wiki_link" href="/149edo">149edo</a> 46&amp;103 25.779<br />
<a class="wiki_link" href="/Hemifamity%20temperaments#Supers">Supers</a>: <a class="wiki_link" href="/152edo">152edo</a> 58&amp;94 21.644<br />
<a class="wiki_link" href="/Unicorn%20family#Alicorn">Monocerus</a>: <a class="wiki_link" href="/154edo">154edo</a> 58&amp;154 28.795<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="/Marvel%20temperaments">Gizzard</a>: <a class="wiki_link" href="/166edo">166edo</a> 72&amp;166 20.252<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments#Parakleismic-Parkleismic-13-limit">Parkleismic</a>: <a class="wiki_link" href="/179edo">179edo</a> 80&amp;179 36.559<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">Sqrtphi</a>: <a class="wiki_link" href="/193edo">193edo</a> 72&amp;121 20.040<br />
<a class="wiki_link" href="/Misty%20family#Hemimist">Hemimist</a>: <a class="wiki_link" href="/198edo">198edo</a> 87&amp;111 22.923<br />
<a class="wiki_link" href="/Hemifamity%20temperaments">Quanic</a>: <a class="wiki_link" href="/205edo">205edo</a> 94&amp;111 32.481<br />
<a class="wiki_link" href="/Kleismic%20family">Metakleismic</a>: <a class="wiki_link" href="/208edo">208edo</a> 87&amp;121 24.371<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#Ennealimmal">Ennealimmic</a>: <a class="wiki_link" href="/243edo">243edo</a> 72&amp;171 20.250<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="/Sensamagic%20clan">Sensa</a>: <a class="wiki_link" href="/283edo">283edo</a> 22&amp;65 31.366<br />
<a class="wiki_link" href="/Schismatic%20family">Sextilififths</a>: <a class="wiki_link" href="/289edo">289edo</a> 29&amp;130 25.276<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments#Quincy">Quincy</a>: <a class="wiki_link" href="/289edo">289edo</a> 72&amp;145 23.862<br />
<a class="wiki_link" href="/Porwell%20temperaments">Hemischis</a>: <a class="wiki_link" href="/313edo">313edo</a> 53&amp;130 20.816<br />
<a class="wiki_link" href="/Werckismic%20temperaments#Octowerck">Octowerck</a>: <a class="wiki_link" href="/320edo">320edo</a> 72&amp;248 27.632<br />
<a class="wiki_link" href="/Schismatic%20family">Bischismic</a>: <a class="wiki_link" href="/378edo">378edo</a> 12&amp;118 28.722<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Semiennealimmal</a>: <a class="wiki_link" href="/441edo">441edo</a> 72&amp;441 26.122<br />
<a class="wiki_link" href="/31%20comma%20temperaments#Birds">Birds</a>: <a class="wiki_link" href="/465edo">465edo</a> 31&amp;217 35.680<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#Ennealimmal">Semihemiennealimmal</a> <a class="wiki_link" href="/954edo">954edo</a> 126&amp;144 13.104<br />
<a class="wiki_link" href="/Ragismic%20microtemperaments">Deca</a>: <a class="wiki_link" href="/1000edo">1000edo</a> 80&amp;190 16.810<br />
<a class="wiki_link" href="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/1111edo">1111edo</a> 31&amp;239 17.921<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;h1&gt; --><h1 id="toc6"><a name="x13-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit rank three</h1>
 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> 4&amp;12f&amp;19 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="/Semaphore%20family#Negric">Negric</a>: <a class="wiki_link" href="/29edo">29edo</a> 1&amp;9&amp;10 80.756<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#Prodigy">Prodigy</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&amp;29&amp;31 73.639<br />
<a class="wiki_link" href="/Marvel%20family">Athene</a>: <a class="wiki_link" href="/53edo">53edo</a> 12f&amp;22&amp;53 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#Prodigy">Prodigious</a>: <a class="wiki_link" href="/72edo">72edo</a> 2&amp;29&amp;41 89.974<br />
<a class="wiki_link" href="/Sensamagic%20family#Sensawer-13-limit">Sensawer</a>: <a class="wiki_link" href="/87edo">87edo</a> 41&amp;46&amp;73 92.807<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="/Marvel%20family">Isis</a>: <a class="wiki_link" href="/94edo">94edo</a> 10&amp;22&amp;31 86.583<br />
<a class="wiki_link" href="/Marvel%20family">Marvel</a>: <a class="wiki_link" href="/103edo">103edo</a> 19&amp;22&amp;31 68.977<br />
<a class="wiki_link" href="/Marvel%20family#Prodigy">Prodigal</a>: <a class="wiki_link" href="/103edo">103edo</a> 31&amp;72&amp;84 88.910<br />
<a class="wiki_link" href="/Marvel%20family#Mirage">Mirage</a>: <a class="wiki_link" href="/103edo">103edo</a> 10&amp;31&amp;62 73.850<br />
<a class="wiki_link" href="/Swetismic%20temperaments#Hades-Cerberus">Cerberus</a>: <a class="wiki_link" href="/130edo">130edo</a> 22&amp;50&amp;58 76.512<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">Hecate</a>: <a class="wiki_link" href="/166edo">166edo</a> 19&amp;41&amp;53 72.113<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="/Valinorismic%20temperaments#Ulmo">Ulmo</a>: <a class="wiki_link" href="/179edo">179edo</a> 7&amp;39&amp;80 93.162<br />
<a class="wiki_link" href="/Mirkwai%20family#Indra-Vibhu">Vibhu</a>: <a class="wiki_link" href="/183edo">183edo</a> 31&amp;72&amp;111 82.598<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="/Mirkwai%20family#Indra-Vish">Vish</a>: <a class="wiki_link" href="/193edo">193edo</a> 41&amp;72&amp;121 92.976<br />
<a class="wiki_link" href="/Swetismic%20temperaments#Minos-13-limit">Minos</a>: <a class="wiki_link" href="/193edo">193edo</a> 9&amp;54&amp;58 72.381<br />
<a class="wiki_link" href="/Hemifamity%20family#Namaka">Namaka</a>: <a class="wiki_link" href="/198edo">198edo</a> 29&amp;53&amp;58 78.124<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="/Mirkwai%20family#Indra-Yajna">Yajna</a>: <a class="wiki_link" href="/224edo">224edo</a> 31&amp;72&amp;121 69.986<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="/Trimyna%20family#Trimyna-13-limit">Trimyna</a>: <a class="wiki_link" href="/232edo">232edo</a> 29&amp;31&amp;56 81.405<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="/Kleismic%20rank%20three%20family#Enlil">Enlil</a>: <a class="wiki_link" href="/299edo">299edo</a> 15&amp;19&amp;53 68.005<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="/Hemifamity%20family#Akea">Akea</a>: <a class="wiki_link" href="/321edo">321edo</a> 7&amp;34&amp;46 82.191<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="/Lehmerismic%20temperaments#Hanuman">Hanuman</a>: <a class="wiki_link" href="/535edo">535edo</a> 15&amp;72&amp;137 65.278<br />
<a class="wiki_link" href="/Swetismic%20temperaments#Hades">Hades</a>: <a class="wiki_link" href="/578edo">578edo</a> 36&amp;58&amp;72 <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="/Mirkwai%20family#Indra-Shibi">Shibi</a>: <a class="wiki_link" href="/703edo">703edo</a> 41&amp;72&amp;111 72.094<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>
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