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Optimal patent val: Difference between revisions

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__FORCETOC__
The '''optimal patent val''' for a [[regular temperament]] is the unique [[patent val]] that [[support]]s the temperament with the lowest [[error]].  
Given any collection of ''p''-limit commas, there is a finite list of ''p''-limit [[patent val]]s tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[TE error]]; this is the (TE) '''optimal 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.


Given any temperament, which is characterized by the [[comma]]s it [[tempering out|tempers out]], there is a finite list of [[patent val]]s that temper out all the commas of the temperament in the same [[subgroup]]. 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 one which has the lowest [[TE error]]; this is the (TE) optimal patent val for the temperament. Note that other definitions of error lead to different results.
On this wiki, the optimal patent val for each temperament is given as the last patent val in the [[optimal ET sequence]], or stated explicitly in case it is not a member of the sequence.
== Instructions ==
By tempering a JI scale using the ''N''-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [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".
By tempering a JI scale using the ''N''-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [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''.
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''.


== Examples ==
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 [[wart notation]] are given; this adjusts the nth prime mapping to its second-best value by appending the ''n''-th lower-case letter in alphabetical order. Thus, "12f" adjusts a patent val for 12 in the 13-limit or above, for instance {{val| 12 19 28 34 42 44 }}, to {{val| 12 19 28 34 42 45 }} (which is actually a better mapping, and hence more useful for this purpose.)
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 [[wart notation]] are given; this adjusts the nth prime mapping to its second-best value by appending the ''n''-th lower-case letter in alphabetical order. Thus, "12f" adjusts a patent val for 12 in the 13-limit or above, for instance {{val| 12 19 28 34 42 44 }}, to {{val| 12 19 28 34 42 45 }} (which is actually a better mapping, and hence more useful for this purpose.)


= 5-limit rank two =
=== 5-limit rank two ===
Comma: ET w/ optimal patent val: 1000 * badness
Comma: ET w/ optimal patent val: 1000 * badness


Line 55: Line 60:
7629394531250/7625597484987: [[3501edo|3501et]] 17.191
7629394531250/7625597484987: [[3501edo|3501et]] 17.191


=7-limit rank two=
=== 7-limit rank two ===
Name: ET w/ optimal patent val: Val name: 1000*badness
Name: ET w/ optimal patent val: Val name: 1000*badness


Line 66: Line 71:
[[Archytas clan|Mother]]: [[5edo|5et]] 2&3 24.152
[[Archytas clan|Mother]]: [[5edo|5et]] 2&3 24.152


[[Sharptone]]: [[5edo|5et]] 5&7d 24.848
[[Meantone family|Sharptone]]: [[5edo|5et]] 5&7d 24.848


[[Father family#Baba|Baba]]: [[5edo|5et]] 1&5 44.321
[[Father family#Baba|Baba]]: [[5edo|5et]] 1&5 44.321
Line 98: Line 103:
[[Meantone family|Dominant]]: [[12edo|12et]] 5&7 20.690
[[Meantone family|Dominant]]: [[12edo|12et]] 5&7 20.690


[[Jubilismic clan|Diminished]]: [[12edo|12et]] 4&12 22.401
[[Diminished family #Septimal diminished|Diminished]]: [[12edo|12et]] 4&12 22.401


[[Augmented family|August]]: [[12edo|12et]] 9&12 26.459
[[Augmented family|August]]: [[12edo|12et]] 9&12 26.459
Line 150: Line 155:
[[Meantone family|Squares]]: [[31edo|31et]] 31&45 45.993
[[Meantone family|Squares]]: [[31edo|31et]] 31&45 45.993


[[Immunity family#Immunity-7-limit|Immunity]]: [[34edo|34et]] 5&29 77.631
[[Immunity family#Septimal immunity|Immunity]]: [[34edo|34et]] 5&29 77.631


[[Meantone family|Injera]]: [[38edo|38et]] 12&26 31.130
[[Meantone family|Injera]]: [[38edo|38et]] 12&26 31.130
Line 192: Line 197:
[[Marvel temperaments|Wizard]]: [[72edo|72et]] 22&50 40.846
[[Marvel temperaments|Wizard]]: [[72edo|72et]] 22&50 40.846


[[Unicorn family#Alicorn|Alicorn]] [[77edo|77et]] 19&58 40.913
[[Unicorn family|Unicorn]] [[77edo|77et]] 19&58 40.913


[[Diaschismic family|Bidia]]: [[80edo|80et]] 12&56 56.474
[[Diaschismic family|Bidia]]: [[80edo|80et]] 12&56 56.474
Line 242: Line 247:
[[Semicomma family|Orwell]]: [[137edo|137et]] 9&22 20.375
[[Semicomma family|Orwell]]: [[137edo|137et]] 9&22 20.375


[[Breedsmic temperaments|Tertiaseptal]]: [[171edo|171et]] 31&109 12.995
[[Breedsmic temperaments#Tertiaseptal|Tertiaseptal]]: [[171edo|171et]] 31&109 12.995


[[Breedsmic temperaments#Septidiasemi|Septidiasemi]]: [[171edo|171et]] 10&151 44.115
[[Breedsmic temperaments#Septidiasemi|Septidiasemi]]: [[171edo|171et]] 10&151 44.115
Line 298: Line 303:
[[Schismatic family|Bischismic]]: [[378edo|378et]] 12&118 54.744
[[Schismatic family|Bischismic]]: [[378edo|378et]] 12&118 54.744


[[Nessafof]]: [[381edo|381et]] 15&69 45.478
[[Porwell temperaments|Nessafof]]: [[381edo|381et]] 15&69 45.478


[[Hemifamity temperaments#Septiquarter|Septiquarter]]: [[391edo|391et]] 5&94 53.760
[[Hemifamity temperaments#Septiquarter|Septiquarter]]: [[391edo|391et]] 5&94 53.760
Line 308: Line 313:
[[Hemimean clan#Sengagen|Sengagen]]: [[446edo|446et]] 49&50 57.978
[[Hemimean clan#Sengagen|Sengagen]]: [[446edo|446et]] 49&50 57.978


[[Chromat]]: [[456edo|456et]] 60&99 57.499
[[Landscape microtemperaments|Chromat]]: [[456edo|456et]] 60&99 57.499


[[Kleismic family|Countercata]]: [[473edo|473et]] 34&53 52.128
[[Kleismic family|Countercata]]: [[473edo|473et]] 34&53 52.128
Line 328: Line 333:
[[Breedsmic temperaments#Emmthird|Emmthird]] [[1255edo|1255et]] 58&113 16.736
[[Breedsmic temperaments#Emmthird|Emmthird]] [[1255edo|1255et]] 58&113 16.736


[[Landscape microtemperaments#Septichrome|septichrome]] [[1308edo|1308et]] 60&111 16.814
[[Landscape microtemperaments#Septichrome|Septichrome]] [[1308edo|1308et]] 60&111 16.814


[[Schismatic family|Sesquiquartififths]]: [[1498edo|1498et]] 41&89 11.244
[[Schismatic family|Sesquiquartififths]]: [[1498edo|1498et]] 41&89 11.244


[[Brahmagupta]]: [[1547edo|1547et]] 7&217 29.122
[[Ragismic microtemperaments|Brahmagupta]]: [[1547edo|1547et]] 7&217 29.122


[[Breedsmic temperaments#Maviloid|Maviloid]]: [[1614edo|1614et]] 76&99 57.632
[[Breedsmic temperaments#Maviloid|Maviloid]]: [[1614edo|1614et]] 76&99 57.632
Line 362: Line 367:
[[Minortonic family#Domain|Domain]]: [[22038edo|22038et]] 171&1164 13.979
[[Minortonic family#Domain|Domain]]: [[22038edo|22038et]] 171&1164 13.979


= 7-limit rank three =
=== 7-limit rank three ===
Comma: ET w/ optimal patent val: 10^6 * badness
Comma: ET w/ optimal patent val: 10^6 * badness


Line 425: Line 430:
78125000/78121827: [[101654edo|101654et]] 20.457
78125000/78121827: [[101654edo|101654et]] 20.457


= 11-limit rank two =
=== 11-limit rank two ===
Name: ET w/ optimal patent val: Val name: 1000*badness
Name: ET w/ optimal patent val: Val name: 1000*badness


Line 436: Line 441:
[[Meantone family#Godzilla-Semafour|Semafour]]: [[5edo|5et]] 5&14c 28.510
[[Meantone family#Godzilla-Semafour|Semafour]]: [[5edo|5et]] 5&14c 28.510


[[Meantone family#Maqamic|Maqamic]]: [[7edo|7et]] 7&10c 40.240
[[Meantone family#Neutrominant|Neutrominant]]: [[7edo|7et]] 7&10c 40.240


[[Meantone family#Dominant-Arnold|Arnold]]: [[7edo|7et]] 5&7 26.141
[[Meantone family#Dominant-Arnold|Arnold]]: [[7edo|7et]] 5&7 26.141
Line 468: Line 473:
[[Dicot family#Decimal|Decibel]]: [[10edo|10et]] 4&6 32.385
[[Dicot family#Decimal|Decibel]]: [[10edo|10et]] 4&6 32.385


[[Jubilismic clan|Diminished]]: [[12edo|12et]] 4&12 22.132
[[Diminished family #Septimal diminished|Diminished]]: [[12edo|12et]] 4&12 22.132


[[Diaschismic family#Pajara-Pajaric|Pajaric]]: [[12edo|12et]] 2&10 23.798
[[Diaschismic family#Pajara-Pajaric|Pajaric]]: [[12edo|12et]] 2&10 23.798
Line 488: Line 493:
[[Mint temperaments#Ripple|Ripple]]: [[12edo|12et]] 1ce&12 38.811
[[Mint temperaments#Ripple|Ripple]]: [[12edo|12et]] 1ce&12 38.811


[[Jubilismic clan#Diminished-Hemidim|Hemidim]]: [[12edo|12et]] 12&20b 54.965
[Diminished family #Hemidim|Hemidim]]: [[12edo|12et]] 12&20b 54.965


[[Mint temperaments#Smate|Smate]]: [[14edo|14et]] 14&17c 42.518
[[Mint temperaments#Smate|Smate]]: [[14edo|14et]] 14&17c 42.518
Line 576: Line 581:
[[Augmented family#Hemiaug|Hemiaug]]: [[24edo|24et]] 24&27e 38.232
[[Augmented family#Hemiaug|Hemiaug]]: [[24edo|24et]] 24&27e 38.232


[[Dimipent family#Hemidim|Hemidim]]: [[24edo|24et]] 4e&24 56.576
[[Diminished family #Hemidim|Hemidim]]: [[24edo|24et]] 4e&24 56.576


[[Dimipent family#Semidim|Semidim]]: [[24edo|24et]] 24&32c 30.597
[[Diminished family #Octonion|Octonion]]: [[24edo|24et]] 24&32c 30.597


[[Meantone family#Injera-11-limit|Injera]]: [[26edo|26et]] 12&26 23.124
[[Meantone family#Injera-11-limit|Injera]]: [[26edo|26et]] 12&26 23.124
Line 596: Line 601:
[[Sensipent family#Sensi-Sensa|Sensa]]: [[27edo|27et]] 19e&27 36.835
[[Sensipent family#Sensi-Sensa|Sensa]]: [[27edo|27et]] 19e&27 36.835


[[Jubilismic clan#Diminished-Demolished|Demolished]]: [[28edo|28et]] 12&28 26.574
[[Diminished family #Demolished|Demolished]]: [[28edo|28et]] 12&28 26.574


[[Porcupine family|Nautilus]]: [[29edo|29et]] 15&29 26.023
[[Porcupine family|Nautilus]]: [[29edo|29et]] 15&29 26.023
Line 648: Line 653:
[[Meantone family#Liese-Elisa|Elisa]]: [[36edo|36e]] 19e&36e 41.592
[[Meantone family#Liese-Elisa|Elisa]]: [[36edo|36e]] 19e&36e 41.592


[[Pythagorean family#Catler temperament-Catcall|Catcall]]: [[36edo|36et]] 12&24 34.478
[[Compton family#Catnip|Catnip]]: [[36edo|36et]] 12&24 34.478


[[Augmented family#Niner|Niner]]: [[36edo|36et]] 9&36 34.861
[[Augmented family#Niner|Niner]]: [[36edo|36et]] 9&36 34.861
Line 828: Line 833:
[[Marvel temperaments|Septimin]]: [[91edo|91et]] 9&32 31.309
[[Marvel temperaments|Septimin]]: [[91edo|91et]] 9&32 31.309


[[31_comma temperaments#Prajapati|Prajapati]]: [[93edo|93et]] 31&93 42.959
[[31st-octave temperaments #Prajapati|Prajapati]]: [[93edo|93et]] 31&93 42.959


[[Schismatic family|Cassandra]]: [[94edo|94et]] 41&53 27.396
[[Schismatic family|Cassandra]]: [[94edo|94et]] 41&53 27.396
Line 964: Line 969:
[[Schismatic family|Bischismic]]: [[248edo|248et]] 12&118 28.160
[[Schismatic family|Bischismic]]: [[248edo|248et]] 12&118 28.160


[[31_comma temperaments#Birds|Birds]]: [[248edo|248et]] 31&217 39.921
[[31st-octave temperaments #Birds|Birds]]: [[248edo|248et]] 31&217 39.921


[[Quince clan#Essence|Essence]]: [[248edo|248et]] 58&190 46.447
[[Quince clan#Essence|Essence]]: [[248edo|248et]] 58&190 46.447
Line 1,066: Line 1,071:
[[Ragismic microtemperaments#Supermajor-Semisupermajor|Semisupermajor]]: [[2554edo|2554et]] 80&342 12.773
[[Ragismic microtemperaments#Supermajor-Semisupermajor|Semisupermajor]]: [[2554edo|2554et]] 80&342 12.773


= 11-limit rank three =
=== 11-limit rank three ===
Name: ET w/ optimal patent val: Val name: 10^5 * badness
Name: ET w/ optimal patent val: Val name: 10^5 * badness


Line 1,072: Line 1,077:


[[Marvel family #Potassium|Potassium]]: [[19edo|19et]] 2&9&10 46.3665
[[Marvel family #Potassium|Potassium]]: [[19edo|19et]] 2&9&10 46.3665
[[Jubilismic family #Festival|Festival]]: [[26edo|26et]] 2&10&26 68.8510


[[Mint family #Nickel|Nickel]]: [[28edo|28et]] 7&9&12 46.3463
[[Mint family #Nickel|Nickel]]: [[28edo|28et]] 7&9&12 46.3463
[[Jubilismic family #Festival|Festival]]: [[26edo|26et]] 2&10&26 68.8510


[[Starling family #Thrasher|Thrasher]]: [[34edo|34et]] 3&4&12 48.0188
[[Starling family #Thrasher|Thrasher]]: [[34edo|34et]] 3&4&12 48.0188
Line 1,137: Line 1,142:
[[Horwell family #Zelda|Zelda]]: [[118edo|118et]]: 9&22&87 64.1882
[[Horwell family #Zelda|Zelda]]: [[118edo|118et]]: 9&22&87 64.1882


[[Rastmic temperaments #Parahemif|Parahemif]]: 17&24&58 134.547
[[Rastmic temperaments #Parahemif|Parahemif]]: [[123edo|123et]] 17&24&58 134.547


[[Valinorismic temperaments #Varda|Varda]]: [[126edo|126et]] 12&22&46 72.8125
[[Valinorismic temperaments #Varda|Varda]]: [[126edo|126et]] 12&22&46 72.8125
Line 1,219: Line 1,224:
[[Kalismic temperaments #Odin|Odin]]: [[7464edo|7464et]] 12&42&72 11.6151
[[Kalismic temperaments #Odin|Odin]]: [[7464edo|7464et]] 12&42&72 11.6151


= 13-limit rank two =
=== 13-limit rank two ===
Name: ET w/ optimal patent val: Val name: 1000*badness
Name: ET w/ optimal patent val: Val name: 1000*badness


[[Jubilismic clan#Diminished-13-limit|Diminished]]: [[4edo|4et]] 4&8d 19.509
[[Diminished family #Septimal diminished|Diminished]]: [[4edo|4et]] 4&8d 19.509


[[Meantone family#Maqamic-13-limit|Maqamic]]: [[7edo|7et]] 7&10c 27.214
[[Meantone family#Neutrominant-13-limit|Neutrominant]]: [[7edo|7et]] 7&10c 27.214


[[Tetracot family#Modus|Modus]]: [[7edo|7et]] 7&27e 23.806
[[Tetracot family#Modus|Modus]]: [[7edo|7et]] 7&27e 23.806
Line 1,352: Line 1,357:
[[Archytas clan#Quasisuper-Quasisupra-13-limit|Quasisupra]]: [[22edo|22et]] 17c&22 30.219
[[Archytas clan#Quasisuper-Quasisupra-13-limit|Quasisupra]]: [[22edo|22et]] 17c&22 30.219


[[Pythagorean family#Catler temperament|Catcall]]: [[24edo|24et]] 12f&24 28.363
[[Compton family#Catnip|Catnip]]: [[24edo|24et]] 12f&24 28.363


[[Augmented family|Triforce]]: [[24edo|24et]] 9&15 20.248
[[Augmented family|Triforce]]: [[24edo|24et]] 9&15 20.248
Line 1,358: Line 1,363:
[[Augmented family#Hemiaug-13-limit|Hemiaug]]: [[24edo|24et]] 3de&24 30.159
[[Augmented family#Hemiaug-13-limit|Hemiaug]]: [[24edo|24et]] 3de&24 30.159


[[Dimipent family#Hemidim|Hemidim]]: [[24edo|24et]] 4ef&24 39.030
[[Diminished family #Hemidim|Hemidim]]: [[24edo|24et]] 4ef&24 39.030


[[Dimipent family#Semidim|Semidim]]: [[24edo|24et]] 24&32c 30.597
[[Diminished family #Octonion|Octonion]]: [[24edo|24et]] 24&32c 30.597


[[Meantone family#Mohamaq-13-limit|Mohamaq]]: [[24edo|24et]] 17c&24 28.738
[[Meantone family#Mohamaq-13-limit|Mohamaq]]: [[24edo|24et]] 17c&24 28.738
Line 1,428: Line 1,433:
[[Starling temperaments|Lupercalia]]: [[31edo|31et]] 15&31 21.328
[[Starling temperaments|Lupercalia]]: [[31edo|31et]] 15&31 21.328


[[Starling temperaments#Casablanca temperament-Marrakesh-Murakuc|Murakuc]]: [[31edo|31et]] 31&73f 41.394
[[Starling temperaments#Casablanca|Murakuc]]: [[31edo|31et]] 31&73f 41.394


[[Starling temperaments#Nusecond temperament-13-limit|Nusecond]]: [[31edo|31et]] 31&70f 23.323
[[Starling temperaments#Nusecond|Nusecond]]: [[31edo|31et]] 31&70f 23.323


[[Starling temperaments#Cypress-13-limit|Cypress]]: [[31edo|31et]] 11cdeef&31 37.849
[[Starling temperaments#Cypress-13-limit|Cypress]]: [[31edo|31et]] 11cdeef&31 37.849


[[Starling temperaments#x5-limit-Maneh|Maneh]]: [[31edo|31et]] 4ef&31 29.868
[[Starling temperaments#Maneh|Maneh]]: [[31edo|31et]] 4ef&31 29.868


[[31_comma temperaments#Prajapati|Prajapati]]: [[31edo|31et]] 31&93f 37.885
[[31st-octave temperaments #Prajapati|Prajapati]]: [[31edo|31et]] 31&93f 37.885


[[Würschmidt family#Würschmidt-13-limit|Würschmidt]]: [[31edo|31et]] 31&65d 23.593
[[Würschmidt family#Würschmidt-13-limit|Würschmidt]]: [[31edo|31et]] 31&65d 23.593
Line 1,588: Line 1,593:
[[Breedsmic temperaments#Gorgik|Gorgik]]: [[58edo|58et]] 21&37 32.205
[[Breedsmic temperaments#Gorgik|Gorgik]]: [[58edo|58et]] 21&37 32.205


[[31_comma temperaments#Gallium|Gallium]]: [[62edo|62et]] 31&62 25.484
[[31st-octave temperaments #Gallium|Gallium]]: [[62edo|62et]] 31&62 25.484


[[Starling temperaments#Valentine temperament-Semivalentine|Semivalentine]]: [[62edo|62et]] 16&30 32.749
[[Starling temperaments#Valentine temperament-Semivalentine|Semivalentine]]: [[62edo|62et]] 16&30 32.749
Line 1,628: Line 1,633:
[[Ragismic microtemperaments#Ennealimmal-Ennealiminal-13-limit|Ennealiminal]]: [[72edo|72et]]  27&72 30.325
[[Ragismic microtemperaments#Ennealimmal-Ennealiminal-13-limit|Ennealiminal]]: [[72edo|72et]]  27&72 30.325


[[Starling temperaments#Casablanca temperament-Marrakesh-13-limit|Marrakesh]]: [[73edo|73et]] 31&73 40.774
[[Starling temperaments#Casablanca|Marrakesh]]: [[73edo|73et]] 31&73 40.774


[[Meantone family#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone|Grosstone]]: [[74edo|74et]] 12&31 25.899
[[Meantone family#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone|Grosstone]]: [[74edo|74et]] 12&31 25.899
Line 1,640: Line 1,645:
[[Diaschismic family|Bidia]]: [[80edo|80et]] 12&68 41.137
[[Diaschismic family|Bidia]]: [[80edo|80et]] 12&68 41.137


[[Ragismic microtemperaments#Parakleismic-Paradigmic-Paradigmic-13-limit|Paradigmic]]: [[80edo|80et]] 19&80 35.781
[[Ragismic microtemperaments#Parakleismic|Paradigmic]]: [[80edo|80et]] 19&80 35.781


[[Orwellismic temperaments#Quartonic-13-limit|Quartonic]]: [[80edo|80et]] 53&80 23.875
[[Orwellismic temperaments#Quartonic-13-limit|Quartonic]]: [[80edo|80et]] 53&80 23.875
Line 1,664: Line 1,669:
[[Marvel temperaments#Septimin-13-limit|Septimin]]: [[91edo|91et]] 9&41 23.117
[[Marvel temperaments#Septimin-13-limit|Septimin]]: [[91edo|91et]] 9&41 23.117


[[31_comma temperaments#Prajapati|Kumhar]]: [[93edo|93et]] 31&93 48.582
[[31st-octave temperaments #Prajapati|Kumhar]]: [[93edo|93et]] 31&93 48.582


[[Marvel temperaments|Slender]]: [[94edo|94et]] 31&63 25.913
[[Marvel temperaments|Slender]]: [[94edo|94et]] 31&63 25.913


[[Schismatic family#Sanjaab-13-limit|Sanjaab]]: [[94edo|94et]] 29&94 33.849
[[Schismatic family#Sanjaab-13-limit|Sanjaab]]: [[94edo|94et]] 29&94 33.849
[[Hemifamity temperaments#Supers|Supers]]: [[94edo|94et]] 58&94 21.644


[[Marvel temperaments#Submajor-Interpental-13-limit|Interpental]]: [[96edo|96et]] 43&53 29.680
[[Marvel temperaments#Submajor-Interpental-13-limit|Interpental]]: [[96edo|96et]] 43&53 29.680
[[Hemifamity temperaments#Supers|Supers]]: [[94edo|94et]] 58&94 21.644


[[Meantone family#Mothra|Mosura]]: [[98edo|98et]] 31&67 36.857
[[Meantone family#Mothra|Mosura]]: [[98edo|98et]] 31&67 36.857
Line 1,714: Line 1,719:
[[Breedsmic temperaments|Harry]]: [[130edo|130et]] 58&72 13.046
[[Breedsmic temperaments|Harry]]: [[130edo|130et]] 58&72 13.046


[[11-limit_comma temperaments#Bosonic|Bosonic]]: [[130edo|130et]] 26&130 32.946
[[26th-octave temperaments#Bosonic|Bosonic]]: [[130edo|130et]] 26&130 32.946


[[Sensipent family#Bison-13-limit|Bison]]: [[130edo|130et]] 46&84 23.504
[[Sensipent family#Bison-13-limit|Bison]]: [[130edo|130et]] 46&84 23.504
Line 1,802: Line 1,807:
[[Porwell temperaments#Septisuperfourth|Septisuperfourth]]: [[282edo|282et]] 130&282 22.887
[[Porwell temperaments#Septisuperfourth|Septisuperfourth]]: [[282edo|282et]] 130&282 22.887


[[Sensamagic clan|Sensa]]: [[283edo|283et]] 22&65 31.366
[[Escapade family#Escaped|Escaped]]: [[283edo|283et]] 22&65 31.366


[[Schismatic family|Sextilififths]]: [[289edo|289et]] 29&130 25.276
[[Schismatic family#Sextilifourths|Sextilifourths]]: [[289edo|289et]] 29&130 25.276


[[Ragismic microtemperaments#Quincy|Quincy]]: [[289edo|289et]] 72&145 23.862
[[Ragismic microtemperaments#Quincy|Quincy]]: [[289edo|289et]] 72&145 23.862
Line 1,810: Line 1,815:
[[Würschmidt family#Hemiwürschmidt-11-limit-13-limit|Hemiwürschmidt]]: [[291edo|291et]] 31&130 23.074
[[Würschmidt family#Hemiwürschmidt-11-limit-13-limit|Hemiwürschmidt]]: [[291edo|291et]] 31&130 23.074


[[Porwell temperaments|Hemischis]]: [[313edo|313et]] 53&130 20.816
[[Schismatic family#Hemischis|Hemischis]]: [[313edo|313et]] 53&130 20.816


[[Werckismic temperaments#Octowerck|Octowerck]]: [[320edo|320et]] 72&248 27.632
[[Werckismic temperaments#Octowerck|Octowerck]]: [[320edo|320et]] 72&248 27.632


[[Pental family|Decal]]: [[320edo|320et]] 130&190 23.948
[[Quintile family#Decile|Decile]]: [[320edo|320et]] 130&190 23.948


[[Kleismic family#Novemkleismic|Novemkleismic]]: [[333edo|333et]] 72&261 39.072
[[Kleismic family#Novemkleismic|Novemkleismic]]: [[333edo|333et]] 72&261 39.072
Line 1,828: Line 1,833:
[[Ragismic microtemperaments|Semiennealimmal]]: [[441edo|441et]] 72&441 26.122
[[Ragismic microtemperaments|Semiennealimmal]]: [[441edo|441et]] 72&441 26.122


[[31_comma temperaments#Birds|Birds]]: [[465edo|465et]] 31&217 35.680
[[31st-octave temperaments #Birds|Birds]]: [[465edo|465et]] 31&217 35.680


[[Horwell temperaments#Emkay|Emkay]]: [[535edo|535et]] 87&137 17.853
[[Horwell temperaments#Emkay|Emkay]]: [[535edo|535et]] 87&137 17.853


[[11-limit_comma temperaments#Fermionic|Fermionic]]: [[546edo|546et]] 130&286 43.581
[[26th-octave temperaments#Fermionic|Fermionic]]: [[546edo|546et]] 130&286 43.581


[[Schismatic family#Pogo-13-limit|Pogo]]: [[578edo|578et]] 36&94 17.514
[[Schismatic family#Pogo-13-limit|Pogo]]: [[578edo|578et]] 36&94 17.514
Line 1,862: Line 1,867:
[[Vishnuzmic family#Vishnu-Ananta|Ananta]]: [[1468edo|1468et]] 118&152 23.678
[[Vishnuzmic family#Vishnu-Ananta|Ananta]]: [[1468edo|1468et]] 118&152 23.678


= 13-limit rank three =
=== 13-limit rank three ===
Name: ET w/ optimal patent val: Val name: 10^5 * badness
Name: ET w/ optimal patent val: Val name: 10^5 * badness


Line 2,016: Line 2,021:


[[Breed family #Freya|Freya]]: [[2615edo|2615et]] 10&31&229 85.515
[[Breed family #Freya|Freya]]: [[2615edo|2615et]] 10&31&229 85.515
== See also ==
* [[Associated temperament]]


{{todo
{{todo
| improve layout
| improve layout
| add introduction
| introduce lemma
| increase applicability
| increase applicability
| text = add structure by switching to tables
| text = add structure by switching to tables
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[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament]]
[[Category:Val]]
[[Category:Val]]
[[Category:List]]
[[Category:Lists]]
Retrieved from "https://en.xen.wiki/w/Optimal_patent_val"