Starling temperaments: Difference between revisions

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* [[Diaschismic family #Diaschismic|diaschismic]] ({126/125, 2048/2025}, diaschismic family)
* [[Diaschismic family #Diaschismic|diaschismic]] ({126/125, 2048/2025}, diaschismic family)
* [[Tetracot family #Wollemia|wollemia]] ({126/125, 2240/2187}, tetracot family)
* [[Tetracot family #Wollemia|wollemia]] ({126/125, 2240/2187}, tetracot family)
* [[Unicorn family|unicorn]] ({126/125, 10976/10935}, unicorn family)
* [[Cloudy clan #Coblack|coblack]] ({126/125, 16807/16384}, cloudy clan)
* [[Cloudy clan #Coblack|coblack]] ({126/125, 16807/16384}, cloudy clan)
* [[Schismatic family #Grackle|grackle]] ({126/125, 32805/32768}, schismatic family)
* [[Schismatic family #Grackle|grackle]] ({126/125, 32805/32768}, schismatic family)
Line 25: Line 26:


In addition to 126/125, myna tempers out [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the 27&amp;31 temperament. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
In addition to 126/125, myna tempers out [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the 27&amp;31 temperament. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
== 5-limit (mynic) ==


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 38: Line 37:
{{Val list|legend=1| 27, 31, 58, 89, 325cc }}
{{Val list|legend=1| 27, 31, 58, 89, 325cc }}


[[Badness]]: 0.2500
[[Badness]]: 0.249965


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 61: Line 59:
{{Val list|legend=1| 27, 31, 58, 89 }}
{{Val list|legend=1| 27, 31, 58, 89 }}


[[Badness]]: 0.0270
[[Badness]]: 0.027044


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 73: Line 70:
POTE generator: ~6/5 = 310.144
POTE generator: ~6/5 = 310.144


{{Val list|legend=1| 27e, 31, 58, 89 }}
Vals: {{Val list| 27e, 31, 58, 89 }}


Badness: 0.0168
Badness: 0.016842


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 87: Line 83:
POTE generator: ~6/5 = 310.276
POTE generator: ~6/5 = 310.276


{{Val list|legend=1| 27e, 31, 58 }}
Vals: {{Val list| 27e, 31, 58 }}


Badness: 0.0171
Badness: 0.017125


=== Minah ===
=== Minah ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 101: Line 96:
POTE generator: ~6/5 = 310.381
POTE generator: ~6/5 = 310.381


{{Val list|legend=1| 27e, 31f, 58f }}
Vals: {{Val list| 27e, 31f, 58f }}


Badness: 0.0276
Badness: 0.027568


=== Maneh ===
=== Maneh ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 115: Line 109:
POTE generator: ~6/5 = 309.804
POTE generator: ~6/5 = 309.804


{{Val list|legend=1| 27eff, 31 }}
Vals: {{Val list| 27eff, 31 }}


Badness: 0.0299
Badness: 0.029868


== Myno ==
== Myno ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 129: Line 122:
POTE generator: ~6/5 = 309.737
POTE generator: ~6/5 = 309.737


{{Val list|legend=1| 27, 31 }}
Vals: {{Val list| 27, 31 }}


Badness: 0.0334
Badness: 0.033434


== Coleto ==
== Coleto ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 143: Line 135:
POTE generator: ~6/5 = 310.853
POTE generator: ~6/5 = 310.853


{{Val list|legend=1| 4, …, 23bc, 27e }}
Vals: {{Val list| 4, 23bc, 27e }}


Badness: 0.0487
Badness: 0.048687


= Valentine =
= Valentine =
Line 154: Line 146:


Valentine is very closely related to [[Carlos Alpha]], the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
Valentine is very closely related to [[Carlos Alpha]], the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
== 5-limit ==


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 167: Line 157:
{{Val list|legend=1| 15, 31, 46, 77, 123 }}
{{Val list|legend=1| 15, 31, 46, 77, 123 }}


[[Badness]]: 0.1228
[[Badness]]: 0.122765


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 193: Line 182:
{{Val list|legend=1| 15, 31, 46, 77, 185, 262cd }}
{{Val list|legend=1| 15, 31, 46, 77, 185, 262cd }}


[[Badness]]: 0.0311
[[Badness]]: 0.031056


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 214: Line 202:
Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.


{{Val list|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}
Vals: {{Val list| 15, 31, 46, 77, 262cdee, 339cdeee }}


Badness: 0.0167
Badness: 0.016687


=== Dwynwen ===
=== Dwynwen ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 228: Line 215:
POTE generator: ~21/20 = 78.219
POTE generator: ~21/20 = 78.219


{{Val list|legend=1| 15, 31f, 46 }}
Vals: {{Val list| 15, 31f, 46 }}


Badness: 0.0235
Badness: 0.023461


=== Lupercalia ===
=== Lupercalia ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 242: Line 228:
POTE generator: ~21/20 = 77.709
POTE generator: ~21/20 = 77.709


{{Val list|legend=1| 15, 31, 77ff, 108eff, 139efff }}
Vals: {{Val list| 15, 31, 77ff, 108eff, 139efff }}


Badness: 0.0213
Badness: 0.021328


=== Valentino ===
=== Valentino ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 256: Line 241:
POTE generator: ~21/20 = 77.958
POTE generator: ~21/20 = 77.958


{{Val list|legend=1| 15f, 31, 46, 77, 431ccdeeeef }}
Vals: {{Val list| 15f, 31, 46, 77 }}


Badness: 0.0207
Badness: 0.020665


=== Semivalentine ===
=== Semivalentine ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 270: Line 254:
POTE generator: ~21/20 = 77.839
POTE generator: ~21/20 = 77.839


{{Val list|legend=1| 16, 30, 46, 62, 108ef }}
Vals: {{Val list| 16, 30, 46, 62, 108ef }}
 
Badness: 0.0327
 
= Alicorn =
{{see also| Unicorn family #Alicorn }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 126/125, 10976/10935
 
[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -8 -13 -23 }}]
 
{{Multival|legend=1| 8 13 23 2 14 17 }}


[[POTE generator]]: ~28/27 = 62.278
Badness: 0.032749
 
{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
 
[[Badness]]: 0.040913
 
== 11-limit ==
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 540/539, 896/891
 
Mapping: [{{val| 1 2 3 4 3 }}, {{val| 0 -8 -13 -23 9 }}]
 
POTE generator: ~28/27 = 62.101
 
Vals: {{Val list| 19, 39d, 58 }}
 
Badness: 0.039156
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 196/195, 676/675
 
Mapping: [{{val| 1 2 3 4 3 5 }}, {{val| 0 -8 -13 -23 9 -25 }}]
 
POTE generator: ~28/27 = 62.119
 
Vals: {{Val list| 19, 39df, 58 }}
 
Badness: 0.023667
 
== Camahueto ==
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 10976/10935
 
Mapping: [{{val| 1 2 3 4 2 }}, {{val| 0 -8 -13 -23 28 }}]
 
POTE generator: ~28/27 = 62.431
 
Vals: {{Val list| 19, 58e, 77, 96d, 173d }}
 
Badness: 0.065940
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 385/384, 676/675
 
Mapping: [{{val| 1 2 3 4 2 5 }}, {{val| 0 -8 -13 -23 28 -25 }}]
 
POTE generator: ~28/27 = 62.434
 
Vals: {{Val list| 19, 58e, 77, 96d, 173d }}
 
Badness: 0.036155
 
== Qilin ==
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 10976/10935
 
Mapping: [{{val| 1 2 3 4 6 }}, {{val| 0 -8 -13 -23 -49 }}]
 
POTE generator: ~28/27 = 62.196
 
Vals: {{Val list| 58, 77, 135c, 193c, 328cc }}
 
Badness: 0.041426
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195, 2200/2197
 
Mapping: [{{val| 1 2 3 4 6 5 }}, {{val| 0 -8 -13 -23 -49 -25 }}]
 
POTE generator: ~28/27 = 62.197
 
Vals: {{Val list| 58, 77, 135c, 193cf, 328ccff }}
 
Badness: 0.022842
 
== Monocerus ==
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 5488/5445
 
Mapping: [{{val| 2 4 6 8 9 }}, {{val| 0 -8 -13 -23 -20 }}]
 
POTE generator: ~28/27 = 62.292
 
Vals: {{Val list| 58, 96d, 154, 212ce, 366cce }}
 
Badness: 0.052757
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 364/363, 676/675
 
Mapping: [{{val| 2 4 6 8 9 10 }}, {{val| 0 -8 -13 -23 -20 -25 }}]
 
POTE generator: ~28/27 = 62.301
 
Vals: {{Val list| 58, 96d, 154, 366ccef }}
 
Badness: 0.028795


= Casablanca =
= Casablanca =
Line 412: Line 275:
{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


[[Badness]]: 0.1012
[[Badness]]: 0.101191


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 424: Line 286:
POTE generator: ~16/11 = 657.923
POTE generator: ~16/11 = 657.923


{{Val list|legend=1| 11b, 20b, 31 }}
Vals: {{Val list| 11b, 20b, 31 }}


Badness: 0.0623
Badness: 0.067291


== Marrakesh ==
== Marrakesh ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 438: Line 299:
POTE generator: ~22/15 = 657.791
POTE generator: ~22/15 = 657.791


{{Val list|legend=1| 31, 73, 104c, 135c }}
Vals: {{Val list| 31, 73, 104c, 135c }}


Badness: 0.0405
Badness: 0.040539


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 452: Line 312:
POTE generator: ~22/15 = 657.756
POTE generator: ~22/15 = 657.756


{{Val list|legend=1| 31, 73, 104c, 135c, 239ccf }}
Vals: {{Val list| 31, 73, 104c, 135c, 239ccf }}


Badness: 0.0408
Badness: 0.040774


=== Murakuc ===
=== Murakuc ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 466: Line 325:
POTE generator: ~22/15 = 657.700
POTE generator: ~22/15 = 657.700


{{Val list|legend=1| 31, 104cf, 135cf, 166c }}
Vals: {{Val list| 31, 104cff, 135cff }}


Badness: 0.0414
Badness: 0.041395


= Nusecond =
= Nusecond =
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.
== 5-limit ==


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 488: Line 345:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 514: Line 370:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 537: Line 392:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 551: Line 405:


= Thuja =
= Thuja =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 567: Line 420:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 581: Line 433:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 595: Line 446:


== 29-limit ==
== 29-limit ==
The ''raison d'etre'' of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.
The ''raison d'etre'' of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.


Line 607: Line 457:


= Cypress =
= Cypress =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 622: Line 470:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 638: Line 485:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 652: Line 498:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 666: Line 511:


= Bisemidim =
= Bisemidim =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 682: Line 526:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 696: Line 539:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 710: Line 552:


= Vines =
= Vines =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 724: Line 565:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 738: Line 578:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 752: Line 591:


= Kumonga =
= Kumonga =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 767: Line 604:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 783: Line 619:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 797: Line 632:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 826: Line 660:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 840: Line 673:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 855: Line 687:
= Oolong =
= Oolong =
{{main|Oolong}}
{{main|Oolong}}
== 5-limit ==


Subgroup: 2.3.5
Subgroup: 2.3.5


[[Comma list]]: [11 18 -17>
[[Comma list]]: {{monzo|11 18 -17}}


[[Mapping]]: [{{val| 1 6 7 }}, {{val| 0 -17 -18 }}]
[[Mapping]]: [{{val| 1 6 7 }}, {{val| 0 -17 -18 }}]
Line 871: Line 701:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 885: Line 714:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 899: Line 727:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Revision as of 01:04, 19 May 2021

This page discusses miscellaneous rank-2 temperaments tempering out 126/125, the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:

  • pater ({16/15, 126/125}, father family)
  • flat ({21/20, 25/24}, dicot family)
  • opossum ({28/27, 126/125}, trienstonic clan)
  • diminished ({36/35, 50/49}, dimipent family / jubilismic clan)
  • keemun ({49/48, 126/125}, kleismic family / slendro clan)
  • augene ({64/63, 126/125}, augmented family / archytas clan)
  • septimal meantone ({81/80, 126/125}, meantone family)
  • mavila ({126/125, 135/128}, pelogic family)
  • sensi ({126/125, 245/243}, sensipent family / sensamagic clan)
  • gilead ({126/125, 343/324}, shibboleth family)
  • muggles ({126/125, 525/512}, magic family)
  • diaschismic ({126/125, 2048/2025}, diaschismic family)
  • wollemia ({126/125, 2240/2187}, tetracot family)
  • unicorn ({126/125, 10976/10935}, unicorn family)
  • coblack ({126/125, 16807/16384}, cloudy clan)
  • grackle ({126/125, 32805/32768}, schismatic family)
  • worschmidt ({126/125, 33075/32768}, würschmidt family)

Since (6/5)3 = 126/125 × 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess the starling tetrad, the 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.

Myna

Main article: Myna

In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&31 temperament. It has 6/5 as a generator, and 58edo can be used as a tuning, with 89edo being a better one, and fans of round amounts in cents may like 120edo. It is also possible to tune myna with pure fifths by taking 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

Subgroup: 2.3.5

Comma list: 10077696/9765625

Mapping: [⟨1 9 9], ⟨0 -10 -9]]

POTE generator: ~6/5 = 310.140

Template:Val list

Badness: 0.249965

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 1728/1715

Mapping: [⟨1 9 9 8], ⟨0 -10 -9 -7]]

Mapping generators: ~2, ~5/3

Wedgie: ⟨⟨ 10 9 7 -9 -17 -9 ]]

POTE generator: ~6/5 = 310.146

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [9/10 9/10 0 0⟩, [17/10 7/10 0 0⟩]
Eigenmonzos: 2, 3

Template:Val list

Badness: 0.027044

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: [⟨1 9 9 8 22], ⟨0 -10 -9 -7 -25]]

POTE generator: ~6/5 = 310.144

Vals: Template:Val list

Badness: 0.016842

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: [⟨1 9 9 8 22 0], ⟨0 -10 -9 -7 -25 5]]

POTE generator: ~6/5 = 310.276

Vals: Template:Val list

Badness: 0.017125

Minah

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: [⟨1 9 9 8 22 20], ⟨0 -10 -9 -7 -25 -22]]

POTE generator: ~6/5 = 310.381

Vals: Template:Val list

Badness: 0.027568

Maneh

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 540/539

Mapping: [⟨1 9 9 8 22 23], ⟨0 -10 -9 -7 -25 -26]]

POTE generator: ~6/5 = 309.804

Vals: Template:Val list

Badness: 0.029868

Myno

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: [⟨1 9 9 8 -1], ⟨0 -10 -9 -7 6]]

POTE generator: ~6/5 = 309.737

Vals: Template:Val list

Badness: 0.033434

Coleto

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: [⟨1 9 9 8 2], ⟨0 -10 -9 -7 2]]

POTE generator: ~6/5 = 310.853

Vals: Template:Val list

Badness: 0.048687

Valentine

Main article: Valentine
See also: Gamelismic clan § Valentine

Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3×7/5. In this respect it resembles miracle, with a generator of 3×5/7, and casablanca, with a generator of 5×7/3. These three generators are the simplest in terms of the relationship of tetrads in the lattice of 7-limit tetrads. Valentine can also be described as the 31&46 temperament, and 77edo, 108edo or 185edo make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)1/9 as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as ⟨⟨ 9 5 -3 7 … ]], tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)1/10.

Valentine is very closely related to Carlos Alpha, the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in Beauty in the Beast suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.

Subgroup: 2.3.5

Comma list: 1990656/1953125

Mapping: [⟨1 1 2], ⟨0 9 5]]

POTE generator: ~25/24 = 78.039

Template:Val list

Badness: 0.122765

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 1029/1024

Mapping: [⟨1 1 2 3], ⟨0 9 5 -3]]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 77.864

Minimax tuning:

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [17/6 5/12 0 -5/12⟩, [5/2 -1/4 0 1/4⟩]
Eigenmonzos: 2, 7/6
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [47/21 10/21 0 -5/21⟩, [20/7 -2/7 0 1/7⟩]
Eigenmonzos: 2, 9/7

Algebraic generator: smaller root of x2 - 89x + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

Template:Val list

Badness: 0.031056

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 176/175

Mapping: [⟨1 1 2 3 3], ⟨0 9 5 -3 7]]

Mapping generators: ~2, ~21/20

POTE generator: ~21/20 = 77.881

Minimax tuning:

  • 11-odd-limit
[[1 0 0 0 0⟩, [1 0 0 -9/10 9/10⟩, [2 0 0 -1/2 1/2⟩, [3 0 0 3/10 -3/10⟩, [3 0 0 -7/10 7/10⟩]
Eigenmonzos: 2, 11/7

Algebraic generator: positive root of 4x3 + 15x2 - 21, or else Gontrand2, the smallest positive root of 4x7 - 8x6 + 5.

Vals: Template:Val list

Badness: 0.016687

Dwynwen

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 176/175

Mapping: [⟨1 1 2 3 3 2], ⟨0 9 5 -3 7 26]]

POTE generator: ~21/20 = 78.219

Vals: Template:Val list

Badness: 0.023461

Lupercalia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 126/125

Mapping: [⟨1 1 2 3 3 3], ⟨0 9 5 -3 7 11]]

POTE generator: ~21/20 = 77.709

Vals: Template:Val list

Badness: 0.021328

Valentino

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 196/195

Mapping: [⟨1 1 2 3 3 5], ⟨0 9 5 -3 7 -20]]

POTE generator: ~21/20 = 77.958

Vals: Template:Val list

Badness: 0.020665

Semivalentine

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 169/168, 176/175

Mapping: [⟨2 2 4 6 6 7], ⟨0 9 5 -3 7 3]]

POTE generator: ~21/20 = 77.839

Vals: Template:Val list

Badness: 0.032749

Casablanca

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described as 31&73. 74\135 or 91\166 supply good tunings for the generator, and 20 and 31 note MOS are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a hexany and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.

Subgroup: 2.3.5.7

Comma list: 126/125, 589824/588245

Mapping: [⟨1 12 10 5], ⟨0 -19 -14 -4]]

Wedgie: ⟨⟨ 19 14 4 -22 -47 -30 ]]

POTE generator: ~35/24 = 657.818

Template:Val list

Badness: 0.101191

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: [⟨1 12 10 5 4], ⟨0 -19 -14 -4 -1]]

POTE generator: ~16/11 = 657.923

Vals: Template:Val list

Badness: 0.067291

Marrakesh

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: [⟨1 12 10 5 21], ⟨0 -19 -14 -4 -32]]

POTE generator: ~22/15 = 657.791

Vals: Template:Val list

Badness: 0.040539

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: [⟨1 12 10 5 21 -10], ⟨0 -19 -14 -4 -32 25]]

POTE generator: ~22/15 = 657.756

Vals: Template:Val list

Badness: 0.040774

Murakuc

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: [⟨1 12 10 5 21 7], ⟨0 -19 -14 -4 -32 -6]]

POTE generator: ~22/15 = 657.700

Vals: Template:Val list

Badness: 0.041395

Nusecond

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. 31edo can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.

Subgroup: 2.3.5

Comma list: 51018336/48828125

Mapping: [⟨1 3 4], ⟨0 -11 -13]]

POTE generator: ~3125/2916 = 154.523

Template:Val list

Badness: 0.4665

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 2430/2401

Mapping: [⟨1 3 4 5], ⟨0 -11 -13 -17]]

Mapping generators: ~2, ~49/45

Wedgie: ⟨⟨ 11 13 17 -5 -4 3 ]]

POTE generator: ~49/45 = 154.579

Minimax tuning:

[[1 0 0 0⟩, [-5/13 0 11/13 0⟩, [0 0 1 0⟩, [-3/13 0 17/13 0⟩]
Eigenmonzos: 2, 5
[[1 0 0 0⟩, [0 1 0 0⟩, [5/11 13/11 0 0⟩, [4/11 17/11 0 0⟩]
Eigenmonzos: 2, 3

Template:Val list

Badness: 0.0504

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: [⟨1 3 4 5 5], ⟨0 -11 -13 -17 -12]]

Mapping generators: ~2, ~11/10

POTE generator: ~11/10 = 154.645

Minimax tuning:

  • 11-odd-limit
[[1 0 0 0 0⟩, [19/10 11/5 0 0 -11/10⟩, [27/10 13/5 0 0 -13/10⟩, [33/10 17/5 0 0 -17/10⟩, [19/5 12/5 0 0 -6/5⟩]
Eigenmonzos: 2, 11/9

Algebraic generator: positive root of 15x2 - 10x - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

Template:Val list

Badness: 0.0256

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: [⟨1 3 4 5 5 5], ⟨0 -11 -13 -17 -12 -10]]

POTE generator: ~11/10 = 154.478

Template:Val list

Badness: 0.0233

Thuja

Subgroup: 2.3.5.7

Comma list: 126/125, 65536/64827

Mapping: [⟨1 8 5 -2], ⟨0 -12 -5 9]]

Wedgie: ⟨⟨ 12 5 -9 -20 -48 -35 ]]

POTE generator: ~175/128 = 558.605

Template:Val list

Badness: 0.0884

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 1344/1331

Mapping: [⟨1 8 5 -2 4], ⟨0 -12 -5 9 -1]]

POTE generator: ~11/8 = 558.620

Template:Val list

Badness: 0.0331

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: [⟨1 8 5 -2 4 16], ⟨0 -12 -5 9 -1 -23]]

POTE generator: ~11/8 = 558.589

Template:Val list

Badness: 0.0228

29-limit

The raison d'etre of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.

Subgroup: 2.3.5.7.11.13.17.19.23.29

Mapping: [⟨1 -4 0 7 3 -7 12 1 5 3], ⟨0 12 5 -9 1 23 -17 7 -1 4]]

POTE generator: ~11/8 = 558.520

Template:Val list

Cypress

Subgroup: 2.3.5

Comma list: 258280326/244140625

Mapping: [⟨1 7 10], ⟨0 -12 -17]]

POTE generator: ~4374/3125 = 541.726

Template:Val list

Badness: 0.8166

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 19683/19208

Mapping: [⟨1 7 10 15], ⟨0 -12 -17 -27]]

Wedgie: ⟨⟨ 12 17 27 -1 9 15 ]]

POTE generator: ~135/98 = 541.828

Template:Val list

Badness: 0.0998

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: [⟨1 7 10 15 17], ⟨0 -12 -17 -27 -30]]

POTE generator: ~15/11 = 541.772

Template:Val list

Badness: 0.0427

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98. 126/125, 243/242

Mapping: [⟨1 7 10 15 17 15], ⟨0 -12 -17 -27 -30 -25]]

POTE generator: ~15/11 = 541.778

Template:Val list

Badness: 0.0378

Bisemidim

Subgroup: 2.3.5.7

Comma list: 126/125, 118098/117649

Mapping: [⟨2 1 2 2], ⟨0 9 11 15]]

Wedgie: ⟨⟨ 18 22 30 -7 -3 8 ]]

POTE generator: ~35/27 = 455.445

Template:Val list

Badness: 0.0978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: [⟨2 1 2 2 5], ⟨0 9 11 15 8]]

POTE generator: ~35/27 = 455.373

Template:Val list

Badness: 0.0412

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: [⟨2 1 2 2 5 5], ⟨0 9 11 15 8 10]]

POTE generator: ~35/27 = 455.347

Template:Val list

Badness: 0.0239

Vines

Subgroup: 2.3.5.7

Comma list: 126/125, 84035/82944

Mapping: [⟨2 7 8 8], ⟨0 -8 -7 -5]]

POTE generator: ~6/5 = 312.602

Template:Val list

Badness: 0.0780

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: [⟨2 7 8 8 5], ⟨0 -8 -7 -5 4]]

POTE generator: ~6/5 = 312.601

Template:Val list

Badness: 0.0445

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: [⟨2 7 8 8 5 5], ⟨0 -8 -7 -5 4 5]]

POTE generator: ~6/5 = 312.564

Template:Val list

Badness: 0.0297

Kumonga

Subgroup: 2.3.5

Comma list: 1289945088/1220703125

Mapping: [⟨1 4 4], ⟨0 -13 -9]]

POTE generator: ~144/125 = 222.912

Template:Val list

Badness: 0.7296

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 12288/12005

Mapping: [⟨1 4 4 3], ⟨0 -13 -9 -1]]

Wedgie: ⟨⟨ 13 9 1 -16 -35 -23 ]]

POTE generator: ~8/7 = 222.797

Template:Val list

Badness: 0.0875

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: [⟨1 4 4 3 7], ⟨0 -13 -9 -1 -19]]

POTE generator: ~8/7 = 222.898

Template:Val list

Badness: 0.0433

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: [⟨1 4 4 3 7 5], ⟨0 -13 -9 -1 -19 -7]]

POTE generator: ~8/7 = 222.961

Template:Val list

Badness: 0.0289

Amigo

See also: Sensamagic clan § Magus

Subgroup: 2.3.5.7

Comma list: 126/125, 2097152/2083725

Mapping: [⟨1 9 3 -10], ⟨0 -11 -1 19]]

POTE generator: ~5/4 = 391.094

Template:Val list

Badness: 0.1109

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: [⟨1 9 3 -10 -8], ⟨0 -11 -1 19 17]]

POTE generator: ~5/4 = 391.075

Template:Val list

Badness: 0.0434

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: [⟨1 9 3 -10 -8 1], ⟨0 -11 -1 19 17 4]]

POTE generator: ~5/4 = 391.072

Template:Val list

Badness: 0.0307

Oolong

Main article: Oolong

Subgroup: 2.3.5

Comma list: [11 18 -17⟩

Mapping: [⟨1 6 7], ⟨0 -17 -18]]

POTE generator: ~6/5 = 311.6942

Template:Val list

Badness: 0.9428

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 117649/116640

Mapping: [⟨1 6 7 8], ⟨0 -17 -18 -20]]

POTE generator: ~6/5 = 311.6793

Template:Val list

Badness: 0.0735

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: [⟨1 6 7 8 18], ⟨0 -17 -18 -20 -56]]

POTE generator: ~6/5 = 311.5873

Template:Val list

Badness: 0.0569

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: [⟨1 6 7 8 18 5], ⟨0 -17 -18 -20 -56 -5]]

POTE generator: ~6/5 = 311.5908

Template:Val list

Badness: 0.0356

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