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Associated temperament: Difference between revisions

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By an '''associated temperament''' to a ''p''-limit [[comma]] is meant a [[prime limit|''p''-limit]] [[regular temperament|temperament]] [[temper out|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 [[ET|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 115: Line 115:
|-
|-
| 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 141: 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 170: 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 195: 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 205: 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 220: 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 230: 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 245: 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 265: 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 275: 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 294: 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"