Associated temperament
In regular temperament theory, an associated temperament to a 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 rank-1 associated temperament. For rank-2 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 (OPV), the rank-2 temperament, and a comma basis for the rank-2 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 | Catnip | 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 |
| 14641/14580 | 410edo | Floral | 2401/2400, 9801/9800, 14641/14580 |
| 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 |
| 1573/1568 | 323edo | Stockhausenic | 676/675, 1001/1000, 1375/1372, 4096/4095 |
| 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 |