Associated temperament: Difference between revisions

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By an '''associated temperament''' to a ''p''-limit [[comma]] is meant a [[harmonic 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.
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 ==
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| [[28/27]]
| [[28/27]]
| [[15edo]]
| [[15edo]]
| [[Blacksmith]]
| [[Blackwood]]
| 28/27, [[49/48]]
| 28/27, [[49/48]]
|-
|-
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| [[36/35]]
| [[36/35]]
| [[12edo]]
| [[12edo]]
| [[Diminished]]
| [[Diminished (temperament)|Diminished]]
| 36/35, [[50/49]]
| 36/35, [[50/49]]
|-
|-
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| [[36/35]]
| [[36/35]]
| [[12edo]]
| [[12edo]]
| [[Dominant]]
| [[Dominant (temperament)|Dominant]]
| 36/35, [[64/63]]
| 36/35, [[64/63]]
|-
|-
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| 33/32
| 33/32
| [[16edo]]
| [[16edo]]
| [[Armodue]]
| [[Armodue (temperament)|Armodue]]
| 33/32, 36/35, 45/44
| 33/32, 36/35, 45/44
|-
|-
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| 56/55
| 56/55
| [[36edo]]
| [[36edo]]
| [[Catcall]]
| [[Catnip]]
| 56/55, 81/80, 128/125
| 56/55, 81/80, 128/125
|-
|-
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| 40/39
| 40/39
| [[15edo]]
| [[15edo]]
| [[Blacksmith]]
| [[Blackwood]]
| 28/27, 40/39, 49/48, 55/54
| 28/27, 40/39, 49/48, 55/54
|-
|-

Latest revision as of 14:12, 6 September 2025

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 Blackwood 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 Blackwood 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
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