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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 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 ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 32: Line 31:
| [[36/35]]
| [[36/35]]
| [[12edo]]
| [[12edo]]
| [[Dominant]]
| [[Dominant (temperament)|Dominant]]
| 36/35, [[64/63]]
| 36/35, [[64/63]]
|-
|-
Line 58: Line 57:
| [[49edo]]
| [[49edo]]
| [[Superpyth]]
| [[Superpyth]]
| 64/63, 245/243
| 64/63, [[245/243]]
|-
|-
| 875/864
| 875/864
| [[41edo]]
| [[41edo]]
| [[Magic]]
| [[Magic]]
| [[225/224]], [[245/243]]
| [[225/224]], 245/243
|-
|-
| 875/864
| 875/864
Line 82: Line 81:
| 245/243
| 245/243
| [[283edo]]
| [[283edo]]
| [[Escapade family#Escaped|escaped]]
| [[Escaped]]
| 245/243, 65625/65536
| 245/243, 65625/65536
|-
|-
Line 92: Line 91:
| 1728/1715
| 1728/1715
| [[111edo]]
| [[111edo]]
| [[Hemifamity temperaments#Buzzard|buzzard]]
| [[Buzzard]]
| 1728/1715, 5120/5103
| 1728/1715, 5120/5103
|-
|-
Line 107: Line 106:
| 225/224
| 225/224
| [[197edo]]
| [[197edo]]
| [[catakleismic]]
| [[Catakleismic]]
| 225/224, 4375/4374
| 225/224, 4375/4374
|-
| 16875/16807
| [[224edo]]
| [[Octoid]]
| 4375/4374, 16875/16807
|-
| 4802000/4782969
| [[1131edo]]
| [[Amicable]]
| 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
|-
|-
| 6144/6125
| 6144/6125
| [[381edo]]
| [[381edo]]
| [[Porwell temperaments#Nessafof|nessafof]]
| [[Nessafof]]
| 6144/6125, 250047/250000
| 6144/6125, 250047/250000
|-
|-
| 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 151: Line 160:
|}
|}


=11 limit=
== 11-limit ==
 
{| class="wikitable"
{| class="wikitable"
|-
|-
| Comma
! Comma
| OPV
! OPV
| Temperament
! Temperament
| Basis
! Basis
|-
|-
| 33/32
| 33/32
| [[16edo]]
| [[16edo]]
| [[Pelogic family#Armodue|armodue]]
| [[Armodue]]
| 33/32, 36/35, 45/44
| 33/32, 36/35, 45/44
|-
|-
Line 187: 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 197: 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 212: 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 222: 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 237: 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 250: Line 263:
| 385/384, 1375/1372, 6250/6237
| 385/384, 1375/1372, 6250/6237
|-
|-
| 441/400
| 441/440
| [[320edo]]
| [[320edo]]
| [[Octowerck]]
| [[Octowerck]]
Line 257: 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
|-
|-
| 4000/3993
| 4000/3993
| [[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]]
| [[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
|}
|}


=13 limit=
== 13-limit ==
 
{| class="wikitable"
{| class="wikitable"
|-
|-
| Comma
! Comma
| OPV
! OPV
| Temperament
! Temperament
| Basis
! Basis
|-
|-
| 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:Rank-2]]
[[Category:Regular temperament theory]]
[[Category:Temperament]]
[[Category:Rank 2]]
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