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By an ''associated temperament'' to a p-limit comma is meant a p-limit temperament tempering out that comma which shares the same [[Optimal_patent_val|optimal patent val]] as the [[Rank_and_codimension|codimension one]] temperament tempering out that comma. By definition, the optimal patent val defines the unique rank one associated temperament. For rank two temperaments, it is possible for more than one temperament to be associated, and some of these are listed below. The column headings are the comma being associated, the optimal patent val, the rank two temperament, and a comma basis for the rank two temperament.
In [[regular temperament theory]], an '''associated temperament''' to a [[harmonic limit|''p''-limit]] [[comma]] is a ''p''-limit [[regular temperament]] [[tempering out]] that comma which shares the same [[optimal patent val]] as the [[codimension]]-1 temperament tempering out that comma. By definition, the optimal patent val defines the unique [[ET|rank-1]] associated temperament. For [[rank-2 temperament]]s, it is possible for more than one temperament to be associated, and some of these are listed below. The column headings are the comma being associated, the optimal patent val (OPV), the rank-2 temperament, and a [[comma basis]] for the rank-2 temperament.
 
=7 limit=


== 7-limit ==
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Comma
! Comma
| | OPV
! OPV
| | Temperament
! Temperament
| | Basis
! Basis
|-
| [[28/27]]
| [[15edo]]
| [[Blacksmith]]
| 28/27, [[49/48]]
|-
| 1029/1000
| [[55edo]]
| [[Liese]]
| [[81/80]], 686/675
|-
|-
| | 28/27
| [[36/35]]
| | [[15edo|15edo]]
| [[12edo]]
| | [[blacksmith|blacksmith]]
| [[Diminished]]
| | 28/27, 49/48
| 36/35, [[50/49]]
|-
|-
| | 1029/1000
| [[36/35]]
| | [[55edo|55edo]]
| [[12edo]]
| | [[Liese|liese]]
| [[August]]
| | 81/80, 686/675
| 36/35, [[128/125]]
|-
|-
| | 36/35
| [[36/35]]
| | [[12edo|12edo]]
| [[12edo]]
| | [[Diminished|diminished]]
| [[Dominant (temperament)|Dominant]]
| | 36/35, 50/49
| 36/35, [[64/63]]
|-
|-
| | 36/35
| [[525/512]]
| | [[12edo|12edo]]
| [[45edo]]
| | [[august|august]]
| [[Flattone]]
| | 36/35, 128/125
| 81/80, 525/512
|-
|-
| | 36/35
| [[49/48]]
| | [[12edo|12edo]]
| [[19edo]]
| | [[dominant|dominant]]
| [[Keemun]]
| | 36/35, 64/63
| 49/48, [[126/125]]
|-
|-
| | 525/512
| [[49/48]]
| | [[45edo|45edo]]
| [[19edo]]
| | [[Flattone|flattone]]
| [[Godzilla]]
| | 81/80, 525/512
| 49/48, [[81/80]]
|-
|-
| | 49/48
| [[50/49]]
| | [[19edo|19edo]]
| [[48edo]]
| | [[Keemun|keemun]]
| [[Doublewide]]
| | 49/48, 126/125
| 50/49, 875/864
|-
|-
| | 49/48
| [[64/63]]
| | [[19edo|19edo]]
| [[49edo]]
| | [[Godzilla|godzilla]]
| [[Superpyth]]
| | 49/48, 81/80
| 64/63, [[245/243]]
|-
|-
| | 50/49
| 875/864
| | [[48edo|48edo]]
| [[41edo]]
| | [[Doublewide|doublewide]]
| [[Magic]]
| | 50/49, 875/864
| [[225/224]], 245/243
|-
|-
| | 64/63
| 875/864
| | [[49edo|49edo]]
| [[41edo]]
| | [[Superpyth|superpyth]]
| [[Superkleismic]]
| | 64/63, 245/243
| 875/864, [[1029/1024]]
|-
|-
| | 875/864
| 3125/3087
| | [[41edo|41edo]]
| [[94edo]]
| | [[Magic|magic]]
| [[Garibaldi]]
| | 225/224, 245/243
| 225/224, 3125/3087
|-
|-
| | 875/864
| 2430/2401
| | [[41edo|41edo]]
| [[137edo]]
| | [[Superkleismic|superkleismic]]
| [[Orwell]]
| | 875/864, 1029/1024
| [[225/224]], 1728/1715
|-
|-
| | 3125/3087
| 245/243
| | [[94edo|94edo]]
| [[283edo]]
| | [[Garibaldi|garibaldi]]
| [[Escaped]]
| | 225/224, 3125/3087
| 245/243, 65625/65536
|-
|-
| | 2430/2401
| 126/125
| | [[137edo|137edo]]
| [[185edo]]
| | [[Orwell|orwell]]
| [[Valentine]]
| | 225/224, 1728/1715
| 126/125, 1029/1024
|-
|-
| | 245/243
| 1728/1715
| | [[283edo|283edo]]
| [[111edo]]
| | [[Escapade_family#Escaped|escaped]]
| [[Buzzard]]
| | 245/243, 65625/65536
| 1728/1715, 5120/5103
|-
|-
| | 126/125
| 1728/1715
| | [[185edo|185edo]]
| [[111edo]]
| | [[Valentine|valentine]]
| [[Semisept]]
| | 126/125, 1029/1024
| 1728/1715, 3136/3125
|-
|-
| | 1728/1715
| 1029/1024
| | [[111edo|111edo]]
| [[190edo]]
| | [[Hemifamity_temperaments#Buzzard|buzzard]]
| [[Unidec]]
| | 1728/1715, 5120/5103
| 1029/1024, 4375/4374
|-
|-
| | 1728/1715
| 225/224
| | [[111edo|111edo]]
| [[197edo]]
| | [[Semisept|semisept]]
| [[Catakleismic]]
| | 1728/1715, 3136/3125
| 225/224, 4375/4374
|-
|-
| | 1029/1024
| 16875/16807
| | [[190edo|190edo]]
| [[224edo]]
| | [[Unidec|unidec]]
| [[Octoid]]
| | 1029/1024, 4375/4374
| 4375/4374, 16875/16807
|-
|-
| | 225/224
| 4802000/4782969
| | [[197edo|197edo]]
| [[1131edo]]
| | [[catakleismic|catakleismic]]
| [[Amicable]]
| | 225/224, 4375/4374
| 2401/2400, 1600000/1594323
|-
|-
| | 3136/3125
| 3136/3125
| | [[446edo|446edo]]
| [[446edo]]
| | [[Hemimean_clan#Sengagen|sengagen]]
| [[Sengagen]]
| | 3136/3125, 420175/419904
| 3136/3125, 420175/419904
|-
|-
| | 5120/5103
| 5120/5103
| | [[391edo|391edo]]
| [[391edo]]
| | [[Hemifamity_temperaments#Alphaquarter|alphaquarter]]
| [[Alphaquarter]]
| | 5120/5103, 29360128/29296875
| 5120/5103, 29360128/29296875
|-
|-
| | 5120/5103
| 5120/5103
| | [[391edo|391edo]]
| [[391edo]]
| | [[Hemifamity_temperaments#Septiquarter|septiquarter]]
| [[Septiquarter]]
| | 5120/5103, 420175/419904
| 5120/5103, 420175/419904
|-
|-
| | 6144/6125
| 6144/6125
| | [[381edo|381edo]]
| [[381edo]]
| | [[Porwell_temperaments#Nessafof|nessafof]]
| [[Nessafof]]
| | 6144/6125, 250047/250000
| 6144/6125, 250047/250000
|-
|-
| | 65625/65536
| 65625/65536
| | [[171edo|171edo]]
| [[171edo]]
| | [[Breedsmic_temperaments#Tertiaseptal|tertiaseptal]]
| [[Tertiaseptal]]
| | 2401/2400, 65625/65536
| 2401/2400, 65625/65536
|-
|-
| | 703125/702464
| 703125/702464
| | [[2185edo|2185edo]]
| [[2185edo]]
| | [[Ragismic_microtemperaments#Enneadecal|enneadecal]]
| [[Enneadecal]]
| | 4375/4374, 703125/702464
| 4375/4374, 703125/702464
|-
|-
| | 4375/4374
| 4375/4374
| | [[8419edo|8419edo]]
| [[8419edo]]
| | [[Ragismic_microtemperaments#Semidimi|semidimi]]
| [[Semidimi]]
| | 4375/4374, 3955078125/3954653486
| 4375/4374, 3955078125/3954653486
|-
|-
| | 250047/250000
| 250047/250000
| | [[12555edo|12555edo]]
| [[12555edo]]
| |  
|  
| | 250047/250000, 281484423828125/281474976710656
| 250047/250000, 281484423828125/281474976710656
|}
|}


=11 limit=
== 11-limit ==
 
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Comma
! Comma
| | OPV
! OPV
| | Temperament
! Temperament
| | Basis
! Basis
|-
| 33/32
| [[16edo]]
| [[Armodue]]
| 33/32, 36/35, 45/44
|-
|-
| | 33/32
| 77/75
| | [[16edo|16edo]]
| [[39edo]]
| | [[Pelogic_family#Armodue|armodue]]
| [[Triforce]]
| | 33/32, 36/35, 45/44
| 49/48, 56/55, 77/75
|-
|-
| | 77/75
| 352/343
| | [[39edo|39edo]]
| [[22edo]]
| | [[Triforce|triforce]]
| [[Hedgehog]]
| | 49/48, 56/55, 77/75
| 50/49, 55/54, 99/98
|-
|-
| | 352/343
| 45/44
| | [[22edo|22edo]]
| [[45edo]]
| | [[Hedgehog|hedgehog]]
| [[Flattone]]
| | 50/49, 55/54, 99/98
| 45/44, 81/80, 385/384
|-
|-
| | 45/44
| 55/54
| | [[45edo|45edo]]
| [[51edo]]
| | [[Flattone|flattone]]
| [[Porky]]
| | 45/44, 81/80, 385/384
| 55/54, 100/99, 225/224
|-
|-
| | 55/54
| 56/55
| | [[51edo|51edo]]
| [[36edo]]
| | [[Porky|porky]]
| [[Catnip]]
| | 55/54, 100/99, 225/224
| 56/55, 81/80, 128/125
|-
|-
| | 56/55
| 245/242
| | [[36edo|36edo]]
| [[91edo]]
| | [[Pythagorean_family#Catler temperament|catcall]]
| [[Septimin]]
| | 56/55, 81/80, 128/125
| 225/224, 245/242, 385/384
|-
|-
| | 245/242
| 99/98
| | [[91edo|91edo]]
| [[127edo]]
| | [[Septimin|septimin]]
| [[Würschmidt]]
| | 225/224, 245/242, 385/384
| 99/98, 176/175, 243/242
|-
|-
| | 99/98
| 100/99
| | [[127edo|127edo]]
| [[104edo]]
| | [[Würschmidt_family#Würschmidt|würschmidt]]
| [[Magic]]
| | 99/98, 176/175, 243/242
| 100/99, 225/224, 245/243
|-
|-
| | 100/99
| 121/120
| | [[104edo|104edo]]
| [[99edo]]
| | [[Magic|magic]]
| [[Hitchcock]]
| | 100/99, 225/224, 245/243
| 121/120, 176/175, 2200/2187
|-
|-
| | 121/120
| 121/120
| | [[99edo|99edo]]
| [[99edo]]
| | [[Hitchcock|hitchcock]]
| [[Hemiwur]]
| | 121/120, 176/175, 2200/2187
| 121/120, 176/175, 1375/1372
|-
|-
| | 121/120
| 176/175
| | [[99edo|99edo]]
| [[111edo]]
| | [[Würschmidt_family#Hemiwürschmidt|hemiwur]]
| [[Semisept]]
| | 121/120, 176/175, 1375/1372
| 176/175, 540/539, 1331/1323
|-
|-
| | 176/175
| 896/891
| | [[111edo|111edo]]
| [[208edo]]
| | [[Semisept|semisept]]
| [[Metakleismic]]
| | 176/175, 540/539, 1331/1323
| 896/891, 2200/2187, 14700/14641
|-
|-
| | 896/891
| 65536/65219
| | [[208edo|208edo]]
| [[282edo]]
| | [[Kleismic_family#Metakleismic|metakleismic]]
| [[Septisuperfourth]]
| | 896/891, 2200/2187, 14700/14641
| 540/539, 4000/3993, 5632/5625
|-
|-
| | 65536/65219
| 14641/14580
| | [[282edo|282edo]]
| [[410edo]]
| | [[Porwell_temperaments#Septisuperfourth|septisuperfourth]]
| [[Floral]]
| | 540/539, 4000/3993, 5632/5625
| 2401/2400, 9801/9800, 14641/14580
|-
|-
| | 243/242
| 243/242
| | [[202edo|202edo]]
| [[202edo]]
| | [[Harry|harry]]
| [[Harry]]
| | 243/242, 441/440, 4000/3993
| 243/242, 441/440, 4000/3993
|-
|-
| | 243/242
| 243/242
| | [[202edo|202edo]]
| [[202edo]]
| | [[Breedsmic_temperaments#Tertiaseptal|tertiaseptal]]
| [[Tertiaseptal]]
| | 243/242, 441/440, 65625/65536
| 243/242, 441/440, 65625/65536
|-
|-
| | 3388/3375
| 3388/3375
| | [[316edo|316edo]]
| [[316edo]]
| | [[Ragismic_microtemperaments#Semiparakleismic|semiparakleismic]]
| [[Semiparakleismic]]
| | 3025/3024, 3136/3125, 4375/4374
| 3025/3024, 3136/3125, 4375/4374
|-
|-
| | 385/384
| 385/384
| | [[284edo|284edo]]
| [[284edo]]
| | [[Quadritikleismic|quadritikleismic]]
| [[Quadritikleismic]]
| | 385/384, 1375/1372, 6250/6237
| 385/384, 1375/1372, 6250/6237
|-
|-
| | 441/400
| 441/440
| | [[320edo|320edo]]
| [[320edo]]
| | [[Octowerck|octowerck]]
| [[Octowerck]]
| | 441/440, 8019/8000, 41503/41472
| 441/440, 8019/8000, 41503/41472
|-
|-
| | 540/539
| 540/539
| | [[578edo|578edo]]
| [[578edo]]
| | [[Schismatic_family#Pogo|pogo]]
| [[Pogo]]
| | 540/539, 4000/3993, 32805/32768
| 540/539, 4000/3993, 32805/32768
|-
|-
| | 4000/3993
| 4000/3993
| | [[665edo|665edo]]
| [[665edo]]
| | [[Ragismic_microtemperaments#Brahmagupta|brahmagupta]]
| [[Brahmagupta]]
| | 4000/3993, 4375/4374, 131072/130977
| 4000/3993, 4375/4374, 131072/130977
|-
|-
| | 5632/5625
| 5632/5625
| | [[1092edo|1092edo]]
| [[1092edo]]
| | [[Landscape_microtemperaments#Sextile|sextile]]
| [[Sextile]]
| | 5632/5625, 9801/9800, 151263/151250
| 5632/5625, 9801/9800, 151263/151250
|-
|-
| | 3025/3024
| 3025/3024
| | [[2554edo|2554edo]]
| [[2554edo]]
| | [[Ragismic_microtemperaments#Supermajor-Semisupermajor|semisupermajor]]
| [[Semisupermajor]]
| | 3025/3024, 4375/4374, 35156250/35153041
| 3025/3024, 4375/4374, 35156250/35153041
|}
|}


=13 limit=
== 13-limit ==
 
{| class="wikitable"
{| class="wikitable"
|-
|-
| | Comma
! Comma
| | OPV
! OPV
| | Temperament
! Temperament
| | Basis
! Basis
|-
| 26/25
| [[12edo]]
| [[Augustus]]
| 26/25, 36/35, 45/44, 56/55
|-
|-
| | 26/25
| 27/26
| | [[12edo|12edo]]
| [[35edo]]
| | [[Augmented_family#August|augustus]]
| [[Secund]]
| | 26/25, 36/35, 45/44, 56/55
| 27/26, 45/44, 99/98, 385/384
|-
|-
| | 27/26
| 27/26
| | [[35edo|35edo]]
| [[35edo]]
| | [[Avicennmic_temperaments#Secund|secund]]
| [[Greenwood]]
| | 27/26, 45/44, 99/98, 385/384
| 27/26, 45/44, 99/98, 640/637
|-
|-
| | 27/26
| 40/39
| | [[35edo|35edo]]
| [[15edo]]
| | [[Greenwoodmic_temperaments#Greenwood|greenwood]]
| [[Blacksmith]]
| | 27/26, 45/44, 99/98, 640/637
| 28/27, 40/39, 49/48, 55/54
|-
|-
| | 40/39
| 65/64
| | [[15edo|15edo]]
| [[29edo]]
| | [[Archytas_clan#Blacksmith|blacksmith]]
| [[Negril]]
| | 28/27, 40/39, 49/48, 55/54
| 49/48, 65/64, 91/90, 875/858
|-
|-
| | 65/64
| 65/64
| | [[29edo|29edo]]
| [[29edo]]
| | [[Marvel_temperaments#Negril|negril]]
| [[Coendou]]
| | 49/48, 65/64, 91/90, 875/858
| 55/54, 65/64, 100/99, 105/104
|-
|-
| | 65/64
| 66/65
| | [[29edo|29edo]]
| [[31edo]]
| | [[Porcupine_family#Coendou|coendou]]
| [[Winston]]
| | 55/54, 65/64, 100/99, 105/104
| 66/65, 99/98, 105/104, 121/120
|-
|-
| | 66/65
| 66/65
| | [[31edo|31edo]]
| [[31edo]]
| | [[Semicomma_family#Winston|winston]]
| [[Mohajira]]
| | 66/65, 99/98, 105/104, 121/120
| 66/65, 81/80, 105/104, 121/120
|-
|-
| | 66/65
| 66/65
| | [[31edo|31edo]]
| [[31edo]]
| | [[Meantone_family#Mohajira-13-limit|mohajira]]
| [[Squares]]
| | 66/65, 81/80, 105/104, 121/120
| 66/65, 81/80, 99/98, 121/120
|-
|-
| | 66/65
| 78/77
| | [[31edo|31edo]]
| [[43edo]]
| | [[Meantone_family#Squares-13-limit|squares]]
| [[Amavil]]
| | 66/65, 81/80, 99/98, 121/120
| 78/77, 99/98, 144/143, 176/175
|-
|-
| | 78/77
| 78/77
| | [[43edo|43edo]]
| [[43edo]]
| | [[Marvel_temperaments#Amavil|amavil]]
| [[Jerome]]
| | 78/77, 99/98, 144/143, 176/175
| 78/77, 81/80, 99/98, 144/143
|-
|-
| | 78/77
| 91/90
| | [[43edo|43edo]]
| [[102edo]]
| | [[Meantone_family#Jerome|jerome]]
| [[Echidnic]]
| | 78/77, 81/80, 99/98, 144/143
| 91/90, 169/168, 385/384, 441/440
|-
|-
| | 91/90
| 105/104
| | [[102edo|102edo]]
| [[91edo]]
| | [[Diaschismic_family#Echidnic|echidnic]]
| [[Septimin]]
| | 91/90, 169/168, 385/384, 441/440
| 105/104, 144/143, 196/195, 245/242
|-
|-
| | 105/104
| 275/273
| | [[91edo|91edo]]
| [[94edo]]
| | [[Marvel_temperaments#Septimin|septimin]]
| [[Garibaldi]]
| | 105/104, 144/143, 196/195, 245/242
| 225/224, 275/273, 325/324, 385/384
|-
|-
| | 275/273
| 144/143
| | [[94edo|94edo]]
| [[84edo]]
| | [[Schismatic_family#Garibaldi|garibaldi]]
| [[Merman]]
| | 225/224, 275/273, 325/324, 385/384
| 144/143, 225/224, 364/363, 441/440
|-
|-
| | 144/143
| 144/143
| | [[84edo|84edo]]
| [[84edo]]
| | [[Marvel_temperaments#Merman|merman]]
| [[Secant]]
| | 144/143, 225/224, 364/363, 441/440
| 144/143, 351/350, 364/363, 441/440
|-
|-
| | 144/143
| 169/168
| | [[84edo|84edo]]
| [[152edo]]
| | [[Orwellismic_temperaments|secant]]
| [[Octopus]]
| | 144/143, 351/350, 364/363, 441/440
| 169/168, 325/324, 364/363, 540/539
|-
|-
| | 169/168
| 196/195
| | [[152edo|152edo]]
| [[232edo]]
| | [[Ragismic_microtemperaments#Octoid-Octopus|octopus]]
| [[Mystery]]
| | 169/168, 325/324, 364/363, 540/539
| 196/195, 352/351, 364/363, 676/675
|-
|-
| | 196/195
| 640/637
| | [[232edo|232edo]]
| [[205edo]]
| | [[Hemifamity_temperaments#Mystery|mystery]]
| [[Quanic]]
| | 196/195, 352/351, 364/363, 676/675
| 352/351, 540/539, 729/728, 1331/1323
|-
|-
| | 640/637
| 1188/1183
| | [[205edo|205edo]]
| [[255edo]]
| | [[Hemifamity_temperaments#Quanic|quanic]]
| [[Subsemifourth]]
| | 352/351, 540/539, 729/728, 1331/1323
| 540/539, 847/845, 1375/1372, 1575/1573
|-
|-
| | 1188/1183
| 1573/1568
| | [[255edo|255edo]]
| [[323edo]]
| | [[Mirkwai_clan#Subsemifourth|subsemifourth]]
| [[Stockhausenic]]
| | 540/539, 847/845, 1375/1372, 1575/1573
| 676/675, 1001/1000, 1375/1372, 4096/4095
|-
|-
| | 325/324
| 325/324
| | [[333edo|333edo]]
| [[333edo]]
| | [[Kleismic_family#Novemkleismic|novemkleismic]]
| [[Novemkleismic]]
| | 325/324, 625/624, 1375/1372, 4000/3993
| 325/324, 625/624, 1375/1372, 4000/3993
|-
|-
| | 351/350
| 351/350
| | [[546edo|546edo]]
| [[546edo]]
| | [[11-limit_comma_temperaments#Fermionic|fermionic]]
| [[Fermionic]]
| | 351/350, 540/539, 40656/40625, 142884/142805
| 351/350, 540/539, 40656/40625, 142884/142805
|-
|-
| | 352/351
| 352/351
| | [[198edo|198edo]]
| [[198edo]]
| | [[Breedsmic_temperaments#Semihemi|semihemi]]
| [[Semihemi]]
| | 352/351, 676/675, 847/845, 1716/1715
| 352/351, 676/675, 847/845, 1716/1715
|-
|-
| | 352/351
| 352/351
| | [[198edo|198edo]]
| [[198edo]]
| | [[Misty_family#Hemimist|hemimist]]
| [[Hemimist]]
| | 352/351, 676/675, 847/845, 3025/3024
| 352/351, 676/675, 847/845, 3025/3024
|-
|-
| | 847/845
| 847/845
| | [[388edo|388edo]]
| [[388edo]]
| | [[Ragismic_microtemperaments#Neusec-13-limit|neusec]]
| [[Neusec]]
| | 847/845, 1001/1000, 3025/3024, 4375/4374
| 847/845, 1001/1000, 3025/3024, 4375/4374
|-
|-
| | 676/675
| 676/675
| | [[940edo|940edo]]
| [[940edo]]
| | [[Breedsmic_temperaments#Decoid|decoid]]
| [[Decoid]]
| | 676/675, 1001/1000, 1716/1715, 4225/4224
| 676/675, 1001/1000, 1716/1715, 4225/4224
|-
|-
| | 2200/2197
| 2200/2197
| | [[836edo|836edo]]
| [[836edo]]
| | [[Ragismic_microtemperaments#Quasithird-13-limit|quasithird]]
| [[Quasithird]]
| | 2200/2197, 3025/3024, 4375/4374, 468512/468195
| 2200/2197, 3025/3024, 4375/4374, 468512/468195
|}
|}
[[Category:Regular temperament theory]]
[[Category:Rank 2]]
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