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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]] 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 ==
Line 31: Line 31:
| [[36/35]]
| [[36/35]]
| [[12edo]]
| [[12edo]]
| [[Dominant]]
| [[Dominant (temperament)|Dominant]]
| 36/35, [[64/63]]
| 36/35, [[64/63]]
|-
|-
Line 71: Line 71:
| 3125/3087
| 3125/3087
| [[94edo]]
| [[94edo]]
| [[Garibaldi temperament|Garibaldi]]
| [[Garibaldi]]
| 225/224, 3125/3087
| 225/224, 3125/3087
|-
|-
Line 81: Line 81:
| 245/243
| 245/243
| [[283edo]]
| [[283edo]]
| [[Escapade family#Escaped|Escaped]]
| [[Escaped]]
| 245/243, 65625/65536
| 245/243, 65625/65536
|-
|-
Line 91: Line 91:
| 1728/1715
| 1728/1715
| [[111edo]]
| [[111edo]]
| [[Hemifamity temperaments#Buzzard|Buzzard]]
| [[Buzzard]]
| 1728/1715, 5120/5103
| 1728/1715, 5120/5103
|-
|-
Line 108: Line 108:
| [[Catakleismic]]
| [[Catakleismic]]
| 225/224, 4375/4374
| 225/224, 4375/4374
|-
| 16875/16807
| [[224edo]]
| [[Octoid]]
| 4375/4374, 16875/16807
|-
|-
| 4802000/4782969
| 4802000/4782969
|[[1131edo]]
| [[1131edo]]
|[[Amicable]]
| [[Amicable]]
| 2401/2400, 1600000/1594323
| 2401/2400, 1600000/1594323
|-
|-
| 3136/3125
| 3136/3125
| [[446edo]]
| [[446edo]]
| [[Hemimean clan#Sengagen|Sengagen]]
| [[Sengagen]]
| 3136/3125, 420175/419904
| 3136/3125, 420175/419904
|-
|-
| 5120/5103
| 5120/5103
| [[391edo]]
| [[391edo]]
| [[Hemifamity temperaments#Alphaquarter|Alphaquarter]]
| [[Alphaquarter]]
| 5120/5103, 29360128/29296875
| 5120/5103, 29360128/29296875
|-
|-
| 5120/5103
| 5120/5103
| [[391edo]]
| [[391edo]]
| [[Hemifamity temperaments#Septiquarter|Septiquarter]]
| [[Septiquarter]]
| 5120/5103, 420175/419904
| 5120/5103, 420175/419904
|-
|-
Line 136: Line 141:
| 65625/65536
| 65625/65536
| [[171edo]]
| [[171edo]]
| [[Breedsmic temperaments#Tertiaseptal|Tertiaseptal]]
| [[Tertiaseptal]]
| 2401/2400, 65625/65536
| 2401/2400, 65625/65536
|-
|-
| 703125/702464
| 703125/702464
| [[2185edo]]
| [[2185edo]]
| [[Ragismic microtemperaments#Enneadecal|Enneadecal]]
| [[Enneadecal]]
| 4375/4374, 703125/702464
| 4375/4374, 703125/702464
|-
|-
Line 165: Line 170:
| 33/32
| 33/32
| [[16edo]]
| [[16edo]]
| [[Pelogic family#Armodue|Armodue]]
| [[Armodue]]
| 33/32, 36/35, 45/44
| 33/32, 36/35, 45/44
|-
|-
Line 190: Line 195:
| 56/55
| 56/55
| [[36edo]]
| [[36edo]]
| [[Pythagorean family#Catler temperament|Catcall]]
| [[Catnip]]
| 56/55, 81/80, 128/125
| 56/55, 81/80, 128/125
|-
|-
Line 200: Line 205:
| 99/98
| 99/98
| [[127edo]]
| [[127edo]]
| [[Würschmidt family#Würschmidt|Würschmidt]]
| [[Würschmidt]]
| 99/98, 176/175, 243/242
| 99/98, 176/175, 243/242
|-
|-
Line 215: Line 220:
| 121/120
| 121/120
| [[99edo]]
| [[99edo]]
| [[Würschmidt family#Hemiwürschmidt|Hemiwur]]
| [[Hemiwur]]
| 121/120, 176/175, 1375/1372
| 121/120, 176/175, 1375/1372
|-
|-
Line 225: Line 230:
| 896/891
| 896/891
| [[208edo]]
| [[208edo]]
| [[Kleismic family#Metakleismic|Metakleismic]]
| [[Metakleismic]]
| 896/891, 2200/2187, 14700/14641
| 896/891, 2200/2187, 14700/14641
|-
|-
| 65536/65219
| 65536/65219
| [[282edo]]
| [[282edo]]
| [[Porwell temperaments#Septisuperfourth|Septisuperfourth]]
| [[Septisuperfourth]]
| 540/539, 4000/3993, 5632/5625
| 540/539, 4000/3993, 5632/5625
|-
| 14641/14580
| [[410edo]]
| [[Floral]]
| 2401/2400, 9801/9800, 14641/14580
|-
|-
| 243/242
| 243/242
Line 240: Line 250:
| 243/242
| 243/242
| [[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]]
| [[Ragismic microtemperaments#Semiparakleismic|Semiparakleismic]]
| [[Semiparakleismic]]
| 3025/3024, 3136/3125, 4375/4374
| 3025/3024, 3136/3125, 4375/4374
|-
|-
Line 260: Line 270:
| 540/539
| 540/539
| [[578edo]]
| [[578edo]]
| [[Schismatic family#Pogo|Pogo]]
| [[Pogo]]
| 540/539, 4000/3993, 32805/32768
| 540/539, 4000/3993, 32805/32768
|-
|-
Line 270: Line 280:
| 5632/5625
| 5632/5625
| [[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]]
| [[Ragismic microtemperaments#Supermajor-Semisupermajor|Semisupermajor]]
| [[Semisupermajor]]
| 3025/3024, 4375/4374, 35156250/35153041
| 3025/3024, 4375/4374, 35156250/35153041
|}
|}
Line 289: Line 299:
| 26/25
| 26/25
| [[12edo]]
| [[12edo]]
| [[Augmented family#August|Augustus]]
| [[Augustus]]
| 26/25, 36/35, 45/44, 56/55
| 26/25, 36/35, 45/44, 56/55
|-
|-
| 27/26
| 27/26
| [[35edo]]
| [[35edo]]
| [[Avicennmic temperaments#Secund|Secund]]
| [[Secund]]
| 27/26, 45/44, 99/98, 385/384
| 27/26, 45/44, 99/98, 385/384
|-
|-
| 27/26
| 27/26
| [[35edo]]
| [[35edo]]
| [[Greenwoodmic temperaments#Greenwood|Greenwood]]
| [[Greenwood]]
| 27/26, 45/44, 99/98, 640/637
| 27/26, 45/44, 99/98, 640/637
|-
|-
| 40/39
| 40/39
| [[15edo]]
| [[15edo]]
| [[Archytas clan#Blacksmith|Blacksmith]]
| [[Blacksmith]]
| 28/27, 40/39, 49/48, 55/54
| 28/27, 40/39, 49/48, 55/54
|-
|-
| 65/64
| 65/64
| [[29edo]]
| [[29edo]]
| [[Marvel temperaments#Negril|Negril]]
| [[Negril]]
| 49/48, 65/64, 91/90, 875/858
| 49/48, 65/64, 91/90, 875/858
|-
|-
| 65/64
| 65/64
| [[29edo]]
| [[29edo]]
| [[Porcupine family#Coendou|Coendou]]
| [[Coendou]]
| 55/54, 65/64, 100/99, 105/104
| 55/54, 65/64, 100/99, 105/104
|-
|-
| 66/65
| 66/65
| [[31edo]]
| [[31edo]]
| [[Semicomma family#Winston|Winston]]
| [[Winston]]
| 66/65, 99/98, 105/104, 121/120
| 66/65, 99/98, 105/104, 121/120
|-
|-
| 66/65
| 66/65
| [[31edo]]
| [[31edo]]
| [[Meantone family#Mohajira-13-limit|Mohajira]]
| [[Mohajira]]
| 66/65, 81/80, 105/104, 121/120
| 66/65, 81/80, 105/104, 121/120
|-
|-
| 66/65
| 66/65
| [[31edo]]
| [[31edo]]
| [[Meantone family#Squares-13-limit|Squares]]
| [[Squares]]
| 66/65, 81/80, 99/98, 121/120
| 66/65, 81/80, 99/98, 121/120
|-
|-
| 78/77
| 78/77
| [[43edo]]
| [[43edo]]
| [[Marvel temperaments#Amavil|Amavil]]
| [[Amavil]]
| 78/77, 99/98, 144/143, 176/175
| 78/77, 99/98, 144/143, 176/175
|-
|-
| 78/77
| 78/77
| [[43edo]]
| [[43edo]]
| [[Meantone family#Jerome|Jerome]]
| [[Jerome]]
| 78/77, 81/80, 99/98, 144/143
| 78/77, 81/80, 99/98, 144/143
|-
|-
| 91/90
| 91/90
| [[102edo]]
| [[102edo]]
| [[Diaschismic family#Echidnic|Echidnic]]
| [[Echidnic]]
| 91/90, 169/168, 385/384, 441/440
| 91/90, 169/168, 385/384, 441/440
|-
|-
| 105/104
| 105/104
| [[91edo]]
| [[91edo]]
| [[Marvel temperaments#Septimin|Septimin]]
| [[Septimin]]
| 105/104, 144/143, 196/195, 245/242
| 105/104, 144/143, 196/195, 245/242
|-
|-
| 275/273
| 275/273
| [[94edo]]
| [[94edo]]
| [[Schismatic family#Garibaldi|Garibaldi]]
| [[Garibaldi]]
| 225/224, 275/273, 325/324, 385/384
| 225/224, 275/273, 325/324, 385/384
|-
|-
| 144/143
| 144/143
| [[84edo]]
| [[84edo]]
| [[Marvel temperaments#Merman|Merman]]
| [[Merman]]
| 144/143, 225/224, 364/363, 441/440
| 144/143, 225/224, 364/363, 441/440
|-
|-
| 144/143
| 144/143
| [[84edo]]
| [[84edo]]
| [[Orwellismic temperaments|Secant]]
| [[Secant]]
| 144/143, 351/350, 364/363, 441/440
| 144/143, 351/350, 364/363, 441/440
|-
|-
| 169/168
| 169/168
| [[152edo]]
| [[152edo]]
| [[Ragismic microtemperaments#Octoid-Octopus|Octopus]]
| [[Octopus]]
| 169/168, 325/324, 364/363, 540/539
| 169/168, 325/324, 364/363, 540/539
|-
|-
| 196/195
| 196/195
| [[232edo]]
| [[232edo]]
| [[Hemifamity temperaments#Mystery|Mystery]]
| [[Mystery]]
| 196/195, 352/351, 364/363, 676/675
| 196/195, 352/351, 364/363, 676/675
|-
|-
| 640/637
| 640/637
| [[205edo]]
| [[205edo]]
| [[Hemifamity temperaments#Quanic|Quanic]]
| [[Quanic]]
| 352/351, 540/539, 729/728, 1331/1323
| 352/351, 540/539, 729/728, 1331/1323
|-
|-
| 1188/1183
| 1188/1183
| [[255edo]]
| [[255edo]]
| [[Mirkwai clan#Subsemifourth|Subsemifourth]]
| [[Subsemifourth]]
| 540/539, 847/845, 1375/1372, 1575/1573
| 540/539, 847/845, 1375/1372, 1575/1573
|-
| 1573/1568
| [[323edo]]
| [[Stockhausenic]]
| 676/675, 1001/1000, 1375/1372, 4096/4095
|-
|-
| 325/324
| 325/324
| [[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]]
| [[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]]
| [[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]]
| [[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]]
| [[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]]
| [[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]]
| [[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:Regular temperament theory]]
[[Category:Temperament]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Associated_temperament"