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. | 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. | ||
== 7-limit == | == 7-limit == | ||
Revision as of 23:05, 8 November 2021
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 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.
7-limit
| Comma | OPV | Temperament | Basis |
|---|---|---|---|
| 28/27 | 15edo | Blacksmith | 28/27, 49/48 |
| 1029/1000 | 55edo | Liese | 81/80, 686/675 |
| 36/35 | 12edo | Diminished | 36/35, 50/49 |
| 36/35 | 12edo | August | 36/35, 128/125 |
| 36/35 | 12edo | Dominant | 36/35, 64/63 |
| 525/512 | 45edo | Flattone | 81/80, 525/512 |
| 49/48 | 19edo | Keemun | 49/48, 126/125 |
| 49/48 | 19edo | Godzilla | 49/48, 81/80 |
| 50/49 | 48edo | Doublewide | 50/49, 875/864 |
| 64/63 | 49edo | Superpyth | 64/63, 245/243 |
| 875/864 | 41edo | Magic | 225/224, 245/243 |
| 875/864 | 41edo | Superkleismic | 875/864, 1029/1024 |
| 3125/3087 | 94edo | Garibaldi | 225/224, 3125/3087 |
| 2430/2401 | 137edo | Orwell | 225/224, 1728/1715 |
| 245/243 | 283edo | Escaped | 245/243, 65625/65536 |
| 126/125 | 185edo | Valentine | 126/125, 1029/1024 |
| 1728/1715 | 111edo | Buzzard | 1728/1715, 5120/5103 |
| 1728/1715 | 111edo | Semisept | 1728/1715, 3136/3125 |
| 1029/1024 | 190edo | Unidec | 1029/1024, 4375/4374 |
| 225/224 | 197edo | Catakleismic | 225/224, 4375/4374 |
| 16875/16807 | 224edo | Octoid | 4375/4374, 16875/16807 |
| 4802000/4782969 | 1131edo | Amicable | 2401/2400, 1600000/1594323 |
| 3136/3125 | 446edo | Sengagen | 3136/3125, 420175/419904 |
| 5120/5103 | 391edo | Alphaquarter | 5120/5103, 29360128/29296875 |
| 5120/5103 | 391edo | Septiquarter | 5120/5103, 420175/419904 |
| 6144/6125 | 381edo | Nessafof | 6144/6125, 250047/250000 |
| 65625/65536 | 171edo | Tertiaseptal | 2401/2400, 65625/65536 |
| 703125/702464 | 2185edo | Enneadecal | 4375/4374, 703125/702464 |
| 4375/4374 | 8419edo | Semidimi | 4375/4374, 3955078125/3954653486 |
| 250047/250000 | 12555edo | 250047/250000, 281484423828125/281474976710656 |
11-limit
| Comma | OPV | Temperament | Basis |
|---|---|---|---|
| 33/32 | 16edo | Armodue | 33/32, 36/35, 45/44 |
| 77/75 | 39edo | Triforce | 49/48, 56/55, 77/75 |
| 352/343 | 22edo | Hedgehog | 50/49, 55/54, 99/98 |
| 45/44 | 45edo | Flattone | 45/44, 81/80, 385/384 |
| 55/54 | 51edo | Porky | 55/54, 100/99, 225/224 |
| 56/55 | 36edo | Catcall | 56/55, 81/80, 128/125 |
| 245/242 | 91edo | Septimin | 225/224, 245/242, 385/384 |
| 99/98 | 127edo | Würschmidt | 99/98, 176/175, 243/242 |
| 100/99 | 104edo | Magic | 100/99, 225/224, 245/243 |
| 121/120 | 99edo | Hitchcock | 121/120, 176/175, 2200/2187 |
| 121/120 | 99edo | Hemiwur | 121/120, 176/175, 1375/1372 |
| 176/175 | 111edo | Semisept | 176/175, 540/539, 1331/1323 |
| 896/891 | 208edo | Metakleismic | 896/891, 2200/2187, 14700/14641 |
| 65536/65219 | 282edo | Septisuperfourth | 540/539, 4000/3993, 5632/5625 |
| 243/242 | 202edo | Harry | 243/242, 441/440, 4000/3993 |
| 243/242 | 202edo | Tertiaseptal | 243/242, 441/440, 65625/65536 |
| 3388/3375 | 316edo | Semiparakleismic | 3025/3024, 3136/3125, 4375/4374 |
| 385/384 | 284edo | Quadritikleismic | 385/384, 1375/1372, 6250/6237 |
| 441/440 | 320edo | Octowerck | 441/440, 8019/8000, 41503/41472 |
| 540/539 | 578edo | Pogo | 540/539, 4000/3993, 32805/32768 |
| 4000/3993 | 665edo | Brahmagupta | 4000/3993, 4375/4374, 131072/130977 |
| 5632/5625 | 1092edo | Sextile | 5632/5625, 9801/9800, 151263/151250 |
| 3025/3024 | 2554edo | Semisupermajor | 3025/3024, 4375/4374, 35156250/35153041 |
13-limit
| Comma | OPV | Temperament | Basis |
|---|---|---|---|
| 26/25 | 12edo | Augustus | 26/25, 36/35, 45/44, 56/55 |
| 27/26 | 35edo | Secund | 27/26, 45/44, 99/98, 385/384 |
| 27/26 | 35edo | Greenwood | 27/26, 45/44, 99/98, 640/637 |
| 40/39 | 15edo | Blacksmith | 28/27, 40/39, 49/48, 55/54 |
| 65/64 | 29edo | Negril | 49/48, 65/64, 91/90, 875/858 |
| 65/64 | 29edo | Coendou | 55/54, 65/64, 100/99, 105/104 |
| 66/65 | 31edo | Winston | 66/65, 99/98, 105/104, 121/120 |
| 66/65 | 31edo | Mohajira | 66/65, 81/80, 105/104, 121/120 |
| 66/65 | 31edo | Squares | 66/65, 81/80, 99/98, 121/120 |
| 78/77 | 43edo | Amavil | 78/77, 99/98, 144/143, 176/175 |
| 78/77 | 43edo | Jerome | 78/77, 81/80, 99/98, 144/143 |
| 91/90 | 102edo | Echidnic | 91/90, 169/168, 385/384, 441/440 |
| 105/104 | 91edo | Septimin | 105/104, 144/143, 196/195, 245/242 |
| 275/273 | 94edo | Garibaldi | 225/224, 275/273, 325/324, 385/384 |
| 144/143 | 84edo | Merman | 144/143, 225/224, 364/363, 441/440 |
| 144/143 | 84edo | Secant | 144/143, 351/350, 364/363, 441/440 |
| 169/168 | 152edo | Octopus | 169/168, 325/324, 364/363, 540/539 |
| 196/195 | 232edo | Mystery | 196/195, 352/351, 364/363, 676/675 |
| 640/637 | 205edo | Quanic | 352/351, 540/539, 729/728, 1331/1323 |
| 1188/1183 | 255edo | Subsemifourth | 540/539, 847/845, 1375/1372, 1575/1573 |
| 325/324 | 333edo | Novemkleismic | 325/324, 625/624, 1375/1372, 4000/3993 |
| 351/350 | 546edo | Fermionic | 351/350, 540/539, 40656/40625, 142884/142805 |
| 352/351 | 198edo | Semihemi | 352/351, 676/675, 847/845, 1716/1715 |
| 352/351 | 198edo | Hemimist | 352/351, 676/675, 847/845, 3025/3024 |
| 847/845 | 388edo | Neusec | 847/845, 1001/1000, 3025/3024, 4375/4374 |
| 676/675 | 940edo | Decoid | 676/675, 1001/1000, 1716/1715, 4225/4224 |
| 2200/2197 | 836edo | Quasithird | 2200/2197, 3025/3024, 4375/4374, 468512/468195 |