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