Associated temperament: Difference between revisions
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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 == | ||
Revision as of 05:58, 1 August 2022
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 | 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 |
| 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 |