Kleismic family: Difference between revisions

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Line 177: Line 177:


= Catalan =
= Catalan =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 190: Line 189:
{{Val list|legend=1| 15, 34d, 49, 132bcd }}
{{Val list|legend=1| 15, 34d, 49, 132bcd }}


[[Badness]]: 0.0949
[[Badness]]: 0.094872


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


Line 202: Line 200:
POTE generator: ~6/5 = 318.282
POTE generator: ~6/5 = 318.282


{{Val list|legend=1| 15, 34d, 49 }}
Vals: {{Val list| 15, 34d, 49 }}


Badness: 0.0369
Badness: 0.036894


= Catakleismic =
= Catakleismic =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 225: Line 222:
{{Val list|legend=1| 19, 34d, 53, 72, 197, 269c }}
{{Val list|legend=1| 19, 34d, 53, 72, 197, 269c }}


[[Badness]]: 0.0215
[[Badness]]: 0.021501


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


Line 242: Line 238:
* strict range: [315.789, 316.981]
* strict range: [315.789, 316.981]


{{Val list|legend=1| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}
Vals: {{Val list| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}


Badness: 0.0218
Badness: 0.021849


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


Line 261: Line 256:
* strict range: [315.789, 316.981]
* strict range: [315.789, 316.981]


{{Val list|legend=1| 19, 34de, 53, 72, 125f, 197ef, 269cef }}
Vals: {{Val list| 19, 34de, 53, 72, 125f, 197ef, 269cef }}


Badness: 0.0169
Badness: 0.016883


== Cataclysmic ==
== Cataclysmic ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 275: Line 269:
POTE generator: ~6/5 = 317.042
POTE generator: ~6/5 = 317.042


{{Val list|legend=1| 19e, 34d, 53, 193dde, …, 352… }}
Vals: {{Val list| 19e, 34d, 53 }}


Badness: 0.0400
Badness: 0.039954


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


Line 289: Line 282:
POTE generator: ~6/5 = 317.036
POTE generator: ~6/5 = 317.036


{{Val list|legend=1| 19e, 34d, 53, 246ddee, …, 405… }}
Vals: {{Val list| 19e, 34d, 53 }}


Badness: 0.0226
Badness: 0.022555


== Catalytic ==
== Catalytic ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 303: Line 295:
POTE generator: ~6/5 = 316.653
POTE generator: ~6/5 = 316.653


{{Val list|legend=1| 19e, 53e, 72 }}
Vals: {{Val list| 19e, 53e, 72 }}


Badness: 0.0304
Badness: 0.030422


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


Line 317: Line 308:
POTE generator: ~6/5 = 316.639
POTE generator: ~6/5 = 316.639


{{Val list|legend=1| 19e, 53e, 72 }}
Vals: {{Val list| 19e, 53e, 72 }}


Badness: 0.0223
Badness: 0.022337


== Cataleptic ==
== Cataleptic ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 331: Line 321:
POTE generator: ~6/5 = 317.083
POTE generator: ~6/5 = 317.083


{{Val list|legend=1| 19, 34d, 53e }}
Vals: {{Val list| 19, 34d, 53e }}


Badness: 0.0443
Badness: 0.044335


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


Line 345: Line 334:
POTE generator: ~6/5 = 317.118
POTE generator: ~6/5 = 317.118


{{Val list|legend=1| 19, 34d, 53e, 87dee }}
Vals: {{Val list| 19, 34d, 53e, 87dee }}


Badness: 0.0273
Badness: 0.027343


== Bikleismic ==
== Bikleismic ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 359: Line 347:
POTE generator: ~6/5 = 316.721
POTE generator: ~6/5 = 316.721


{{Val list|legend=1| 34d, 72, 322c, …, 610bcc }}
Vals: {{Val list| 34d, 72, 322c, …, 610bcc }}


Badness: 0.0293
Badness: 0.029319


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


Line 373: Line 360:
POTE generator: ~6/5 = 316.726
POTE generator: ~6/5 = 316.726


{{Val list|legend=1| 34d, 72, 322cfff, …, 538… }}
Vals: {{Val list| 34d, 72 }}


Badness: 0.0218
Badness: 0.021814


= Countercata =
= Countercata =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 391: Line 377:
{{Val list|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
{{Val list|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}


[[Badness]]: 0.0521
[[Badness]]: 0.052129


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


Line 403: Line 388:
POTE generator: ~6/5 = 317.162
POTE generator: ~6/5 = 317.162


{{Val list|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
Vals: {{Val list| 34, 53, 87, 140, 227, 367e, 507e }}


Badness: 0.0398
Badness: 0.039770


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


Line 417: Line 401:
POTE generator: ~6/5 = 317.162
POTE generator: ~6/5 = 317.162


{{Val list|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
Vals: {{Val list| 34, 53, 87, 140, 227, 367e, 507e }}


Badness: 0.0202
Badness: 0.020156


= Metakleismic =
= Metakleismic =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 435: Line 418:
{{Val list|legend=1| 34d, 87, 121, 208 }}
{{Val list|legend=1| 34d, 87, 121, 208 }}


[[Badness]]: 0.1635
[[Badness]]: 0.163519


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


Line 447: Line 429:
POTE generator: ~6/5 = 317.311
POTE generator: ~6/5 = 317.311


{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
Vals: {{Val list| 34d, 53d, 87, 121, 208 }}


Badness: 0.0486
Badness: 0.048570


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


Line 461: Line 442:
POTE generator: ~6/5 = 317.311
POTE generator: ~6/5 = 317.311


{{Val list|legend=1| 34d, 53d, 87, 121, 208 }}
Vals: {{Val list| 34d, 53d, 87, 121, 208 }}


Badness: 0.0244
Badness: 0.024371


= Hemikleismic =
= Hemikleismic =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 472: Line 452:


[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 12 10 -9 }}]
[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 12 10 -9 }}]
{{Multival|legend=1| 12 10 -9 -12 -48 -49 }}


[[POTE generator]]: ~35/32 = 158.649
[[POTE generator]]: ~35/32 = 158.649
Line 477: Line 459:
{{Val list|legend=1| 15, 38, 53, 121 }}
{{Val list|legend=1| 15, 38, 53, 121 }}


[[Badness]]: 0.0521
[[Badness]]: 0.052054


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


Line 489: Line 470:
POTE generator: ~11/10 = 158.677
POTE generator: ~11/10 = 158.677


{{Val list|legend=1| 15, 38, 53, 68, 121e }}
Vals: {{Val list| 15, 38, 53, 68, 121e }}


Badness: 0.0380
Badness: 0.038023


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


Line 503: Line 483:
Mapping: [{{val| 1 0 1 4 2 0 }}, {{val| 0 12 10 -9 11 28 }}]
Mapping: [{{val| 1 0 1 4 2 0 }}, {{val| 0 12 10 -9 11 28 }}]


{{Val list|legend=1| 15, 38f, 53, 121e }}
Vals: {{Val list| 15, 38f, 53, 121e }}


Badness: 0.0260
Badness: 0.026005


= Clyde =
= Clyde =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 516: Line 495:


Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7
{{Multival|legend=1| 12 10 25 -12 6 30 }}


[[POTE generator]]: ~9/7 = 441.335
[[POTE generator]]: ~9/7 = 441.335
Line 528: Line 509:
{{Val list|legend=1| 19, 49, 68, 87, 155 }}
{{Val list|legend=1| 19, 49, 68, 87, 155 }}


[[Badness]]: 0.0473
[[Badness]]: 0.047261


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


Line 540: Line 520:
POTE generator: ~9/7 = 441.355
POTE generator: ~9/7 = 441.355


{{Val list|legend=1| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}
Vals: {{Val list| 19, 49e, 68, 87, 329bd, 419bd, 503bd, 590bd }}


Badness: 0.0474
Badness: 0.047417


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


Line 554: Line 533:
Mapping: [{{val| 1 6 6 12 -5 14 }}, {{val| 0 -12 -10 -25 23 -28 }}]
Mapping: [{{val| 1 6 6 12 -5 14 }}, {{val| 0 -12 -10 -25 23 -28 }}]


{{Val list|legend=1| 19, 49ef, 68, 87, 503bdf, 590bdf }}
{{Val list| 19, 49ef, 68, 87, 503bdf, 590bdf }}


Badness: 0.0268
Badness: 0.026842


= Tritikleismic =
= Tritikleismic =
Line 636: Line 615:


= Quadritikleismic =
= Quadritikleismic =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 649: Line 627:
{{Val list|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
{{Val list|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}


[[Badness]]: 0.0392
[[Badness]]: 0.039231


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


Line 661: Line 638:
POTE generator: ~6/5 = 316.925
POTE generator: ~6/5 = 316.925


{{Val list|legend=1| 68, 72, 140, 212, 284, 496, 780 }}
Vals: {{Val list| 68, 72, 140, 212, 284, 496, 780 }}


Badness: 0.0234
Badness: 0.023406


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


Line 675: Line 651:
POTE generator: ~6/5 = 316.989
POTE generator: ~6/5 = 316.989


{{Val list|legend=1| 68, 72, 140, 212 }}
Vals: {{Val list| 68, 72, 140, 212 }}


Badness: 0.0187
Badness: 0.018731


= Kleiboh =
= Kleiboh =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 686: Line 661:


[[Mapping]]: [{{val| 1 6 6 6 }}, {{val| 0 -18 -15 -13 }}]
[[Mapping]]: [{{val| 1 6 6 6 }}, {{val| 0 -18 -15 -13 }}]
{{Multival|legend=1| 18 15 13 -18 -30 -12 }}


[[POTE generator]]: ~25/21 = 294.303
[[POTE generator]]: ~25/21 = 294.303
{{Multival|legend=1| 18 15 13 -18 -30 -12 }}


{{Val list|legend=1| 49, 53, 314d }}
{{Val list|legend=1| 49, 53, 314d }}


[[Badness]]: 0.0765
[[Badness]]: 0.076460


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


Line 705: Line 679:
POTE generator: ~25/21 = 294.181
POTE generator: ~25/21 = 294.181


{{Val list|legend=1| 49, 53, 102d, 155d }}
Vals: {{Val list| 49, 53, 102d, 155d }}


Badness: 0.0528
Badness: 0.052805


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


Line 719: Line 692:
POTE generator: ~13/11 = 294.187
POTE generator: ~13/11 = 294.187


{{Val list|legend=1| 49f, 53, 102df, 155d }}
Vals: {{Val list| 49f, 53, 102df, 155d }}
 
Badness: 0.031074
 
= Marfifths =
The ''marfifths'' temperament (19&140, named by [[User:Xenllium|Xenllium]]) tempers out the [[hemimage comma]], 10976/10935. It splits the interval of major tenth (~10/3) into three marvelous fifth ([[112/75]]) intervals, and uses it for a generator.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 10976/10935, 15625/15552
 
[[Mapping]]: [{{val| 1 -6 -4 -17 }}, {{val| 0 18 15 47 }}]
 
{{Multival|legend=1| 18 15 47 -18 24 67 }}
 
[[POTE generator]]: ~75/56 = 505.705
 
{{Val list|legend=1| 19, 121, 140, 579, 719, 859bcd, 999bcd, 1858bbccdd }}
 
[[Badness]]: 0.063448
 
== 11-limit ==
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 6250/6237, 10976/10935
 
Mapping: [{{val| 1 -6 -4 -17 22 }}, {{val| 0 18 15 47 -44 }}]
 
POTE generator: ~75/56 = 505.684
 
Vals: {{Val list| 19, 140, 159, 299 }}
 
Badness: 0.058902
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 625/624, 10976/10935
 
Mapping: [{{val| 1 -6 -4 -17 22 -14 }}, {{val| 0 18 15 47 -44 42 }}]
 
POTE generator: ~75/56 = 505.686
 
Vals: {{Val list|19, 140, 159, 299 }}
 
Badness: 0.030082
 
== Diatessic ==
The ''marfifths'' temperament (121&140, named by [[User:Xenllium|Xenllium]]) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
 
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 2200/2187, 5632/5625
 
Mapping: [{{val| 1 -6 -4 -17 -37 }}, {{val| 0 18 15 47 96 }}]
 
POTE generator: ~75/56 = 505.740
 
Vals: {{Val list|19e, 121, 140, 261, 401, 662bd }}
 
Badness: 0.061172
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 625/624, 1375/1372
 
Mapping: [{{val| 1 -6 -4 -17 -37 -14 }}, {{val| 0 18 15 47 96 42 }}]
 
POTE generator: ~75/56 = 505.740
 
Vals: {{Val list| 19e, 121, 140, 261, 401, 662bd }}
 
Badness: 0.028671
 
== Marf ==
The ''marf'' temperament (19&121, named by [[User:Xenllium|Xenllium]]) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
 
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 15625/15552
 
Mapping: [{{val| 1 -6 -4 -17 14 }}, {{val| 0 18 15 47 -25 }}]
 
POTE generator: ~75/56 = 505.769
 
Vals: {{Val list| 19, 102d, 121 }}
 
Badness: 0.075112
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 540/539, 625/624, 896/891
 
Mapping: [{{val| 1 -6 -4 -17 14 -14 }}, {{val| 0 18 15 47 -25 42 }}]
 
POTE generator: ~75/56 = 505.771
 
Vals: {{Val list| 19, 121 }}


Badness: 0.0311
Badness: 0.038317


= Novemkleismic =
= Novemkleismic =

Revision as of 13:04, 30 April 2021

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is [-6 -5 6⟩, and flipping that yields ⟨⟨ 6 5 -6 ]] for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)5 = 5/2 × 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called Hanson, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

Hanson

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping: [⟨1 0 1], ⟨0 6 5]]

POTE generator: ~6/5 = 317.007

Tuning ranges:

  • valid range: [300.000, 327.273] (1\4 to 3\11)
  • nice range: [315.641, 317.263]
  • strict range: [315.641, 317.263]

Template:Val list

Badness: 0.013234

Music
Analysis and diagrams

Seven limit extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.

Keemun

Main article: Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping: [⟨1 0 1 2], ⟨0 6 5 3]]

Wedgie: ⟨⟨ 6 5 3 -6 -12 -7 ]]

POTE generator: ~6/5 = 316.473

Tuning ranges:

  • valid range: [300.000, 327.273] (1\4 to 3\11)
  • nice range: [308.744, 322.942]
  • strict range: [308.744, 322.942]

Template:Val list

Badness: 0.027408

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 100/99

Mapping: [⟨1 0 1 2 4], ⟨0 6 5 3 -2]]

POTE generator: ~6/5 = 317.576

Tuning ranges:

  • valid range: [315.789, 320.000] (5\19 to 4\15)
  • nice range: [308.744, 324.341]
  • strict range: [315.789, 320.000]

Vals: Template:Val list

Badness: 0.027410

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 100/99

Mapping: [⟨1 0 1 2 4 5], ⟨0 6 5 3 -2 -5]]

POTE generator: ~6/5 = 316.611

Tuning ranges:

  • valid range: 315.789 (5\19)
  • nice range: [303.597, 324.341]
  • strict range: 315.789

Vals: Template:Val list

Badness: 0.029749

Kema

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 100/99

Mapping: [⟨1 0 1 2 4 0], ⟨0 6 5 3 -2 14]]

POTE generator: ~6/5 = 317.423

Tuning ranges:

  • valid range: [315.789, 320.000] (5\19 to 4\15)
  • nice range: [308.744, 324.341]
  • strict range: [315.789, 320.000]

Vals: Template:Val list

Badness: 0.022749

Kumbaya

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 56/55, 66/65

Mapping: [⟨1 0 1 2 4 4], ⟨0 6 5 3 -2 -1]]

POTE generator: ~6/5 = 318.595

Vals: Template:Val list

Badness: 0.031633

Qeema

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 126/125

Mapping: [⟨1 0 1 2 -1], ⟨0 6 5 3 17]]

POTE generator: ~6/5 = 314.730

Vals: Template:Val list

Badness: 0.040056

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 126/125

Mapping: [⟨1 0 1 2 -1 0], ⟨0 6 5 3 17 14]]

POTE generator: ~6/5 = 315.044

Vals: Template:Val list

Badness: 0.029419

Darjeeling

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 77/75

Mapping: [⟨1 0 1 2 0], ⟨0 6 5 3 13]]

POTE generator: ~6/5 = 317.656

Vals: Template:Val list

Badness: 0.027648

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 66/65, 77/75

Mapping: [⟨1 0 1 2 0 0], ⟨0 6 5 3 13 14]]

POTE generator: ~6/5 = 317.298

Vals: Template:Val list

Badness: 0.021445

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping: [⟨1 0 1 6], ⟨0 6 5 -12]]

Wedgie: ⟨⟨ 6 5 -12 -6 -36 -42 ]]

POTE generator: ~6/5 = 318.267

Template:Val list

Badness: 0.094872

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1331/1323

Mapping: [⟨1 0 1 6 4], ⟨0 6 5 -12 -2]]

POTE generator: ~6/5 = 318.282

Vals: Template:Val list

Badness: 0.036894

Catakleismic

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping: [⟨1 0 1 -3], ⟨0 6 5 22]]

Wedgie: ⟨⟨ 6 5 22 -6 18 37 ]]

POTE generator: ~6/5 = 316.732

Tuning ranges:

  • valid range: [315.789, 317.647] (5\19 to 9\34)
  • nice range: [315.641, 317.263]
  • strict range: [315.789, 317.263]

Template:Val list

Badness: 0.021501

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4375/4374

Mapping: [⟨1 0 1 -3 9], ⟨0 6 5 22 -21]]

POTE generator: ~6/5 = 316.719

Tuning ranges:

  • valid range: [315.789, 316.981] (5\19 to 14\53)
  • nice range: [315.641, 317.263]
  • strict range: [315.789, 316.981]

Vals: Template:Val list

Badness: 0.021849

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 540/539

Mapping: [⟨1 0 1 -3 9 0], ⟨0 6 5 22 -21 14]]

POTE generator: ~6/5 = 316.738

Tuning ranges:

  • valid range: [315.789, 316.981] (5\19 to 14\53)
  • nice range: [315.641, 318.309]
  • strict range: [315.789, 316.981]

Vals: Template:Val list

Badness: 0.016883

Cataclysmic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2200/2187

Mapping: [⟨1 0 1 -3 -5], ⟨0 6 5 22 32]]

POTE generator: ~6/5 = 317.042

Vals: Template:Val list

Badness: 0.039954

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 275/273

Mapping: [⟨1 0 1 -3 -5 0], ⟨0 6 5 22 32 14]]

POTE generator: ~6/5 = 317.036

Vals: Template:Val list

Badness: 0.022555

Catalytic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4374

Mapping: [⟨1 0 1 -3 -10], ⟨0 6 5 22 51]]

POTE generator: ~6/5 = 316.653

Vals: Template:Val list

Badness: 0.030422

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 1716/1715

Mapping: [⟨1 0 1 -3 -10 0], ⟨0 6 5 22 51 14]]

POTE generator: ~6/5 = 316.639

Vals: Template:Val list

Badness: 0.022337

Cataleptic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 864/847

Mapping: [⟨1 0 1 -3 4], ⟨0 6 5 22 -2]]

POTE generator: ~6/5 = 317.083

Vals: Template:Val list

Badness: 0.044335

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 676/675

Mapping: [⟨1 0 1 -3 4 0], ⟨0 6 5 22 -2 14]]

POTE generator: ~6/5 = 317.118

Vals: Template:Val list

Badness: 0.027343

Bikleismic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4375/4356

Mapping: [⟨2 0 2 -6 -1], ⟨0 6 5 22 15]]

POTE generator: ~6/5 = 316.721

Vals: Template:Val list

Badness: 0.029319

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 325/324

Mapping: [⟨2 0 2 -6 -1 0], ⟨0 6 5 22 15 14]]

POTE generator: ~6/5 = 316.726

Vals: Template:Val list

Badness: 0.021814

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping: [⟨1 0 1 11], ⟨0 6 5 -31]]

Wedgie: ⟨⟨ 6 5 -31 -6 -66 -86 ]]

POTE generator: ~6/5 = 317.121

Template:Val list

Badness: 0.052129

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3388/3375

Mapping: [⟨1 0 1 11 -5], ⟨0 6 5 -31 32]]

POTE generator: ~6/5 = 317.162

Vals: Template:Val list

Badness: 0.039770

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 625/624

Mapping: [⟨1 0 1 11 -5 0], ⟨0 6 5 -31 32 14]]

POTE generator: ~6/5 = 317.162

Vals: Template:Val list

Badness: 0.020156

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping: [⟨1 0 1 -12], ⟨0 6 5 56]]

Wedgie: ⟨⟨ 6 5 56 -6 72 116 ]]

POTE generator: ~6/5 = 317.314

Template:Val list

Badness: 0.163519

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 14700/14641

Mapping: [⟨1 0 1 -12 -5], ⟨0 6 5 56 32]]

POTE generator: ~6/5 = 317.311

Vals: Template:Val list

Badness: 0.048570

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 625/624

Mapping: [⟨1 0 1 -12 -5 0], ⟨0 6 5 56 32 14]]

POTE generator: ~6/5 = 317.311

Vals: Template:Val list

Badness: 0.024371

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping: [⟨1 0 1 4], ⟨0 12 10 -9]]

Wedgie: ⟨⟨ 12 10 -9 -12 -48 -49 ]]

POTE generator: ~35/32 = 158.649

Template:Val list

Badness: 0.052054

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 4000/3969

Mapping: [⟨1 0 1 4 2], ⟨0 12 10 -9 11]]

POTE generator: ~11/10 = 158.677

Vals: Template:Val list

Badness: 0.038023

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 275/273, 325/324

POTE generator: ~11/10 = 158.655

Mapping: [⟨1 0 1 4 2 0], ⟨0 12 10 -9 11 28]]

Vals: Template:Val list

Badness: 0.026005

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping: [⟨1 6 6 12], ⟨0 -12 -10 -25]]

Mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 12 10 25 -12 6 30 ]]

POTE generator: ~9/7 = 441.335

Minimax tuning:

[[1 0 0 0⟩, [6/25 0 0 12/25⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Eigenmonzos: 2, 7

Algebraic generator: real root of 5x3 - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

Template:Val list

Badness: 0.047261

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3136/3125

Mapping: [⟨1 6 6 12 -5], ⟨0 -12 -10 -25 23]]

POTE generator: ~9/7 = 441.355

Vals: Template:Val list

Badness: 0.047417

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 625/624

POTE generator: ~9/7 = 441.363

Mapping: [⟨1 6 6 12 -5 14], ⟨0 -12 -10 -25 23 -28]]

Template:Val list

Badness: 0.026842

Tritikleismic

See also: Gamelismic clan § Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

Mapping generators: ~63/50, ~6/5

Wedgie: ⟨⟨ 18 15 -6 -18 -60 -56 ]]

POTE generator: ~6/5 = 316.872

Minimax tuning:

[[1 0 0 0⟩, [2 0 6/7 -6/7⟩, [8/3 0 5/7 -5/7⟩, [8/3 0 -2/7 2/7⟩]
Eigenmonzos: 2, 7/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [46/21 5/7 0 -5/14⟩, [20/7 -2/7 0 1/7⟩]
Eigenmonzos: 2, 9/7

Template:Val list

Badness: 0.056337

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Mapping generators: ~44/35, ~6/5

POTE generator: ~6/5 = 316.881

Minimax tuning:

  • 11-odd-limit
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [46/21 5/7 0 -5/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [71/21 3/7 0 -3/14 0⟩]
Eigenmonzos: 2, 9/7

Vals: Template:Val list

Badness: 0.019333

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

Mapping: [⟨3 0 3 10 8 0], ⟨0 6 5 -2 3 14]]

Mapping generators: ~44/35, ~6/5

POTE generator: ~6/5 = 316.9585

Vals: Template:Val list

Badness: 0.015652

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [⟨3 0 3 10 8 0 -2], ⟨0 6 5 -2 3 14 18]]

Mapping generators: ~34/27, ~6/5

POTE generator: ~6/5 = 316.9082

Vals: Template:Val list

Badness: 0.013551

Quadritikleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping: [⟨4 0 4 7], ⟨0 6 5 4]]

Wedgie: ⟨⟨ 24 20 16 -24 -42 -19 ]]

POTE generator: ~6/5 = 316.9999

Template:Val list

Badness: 0.039231

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 6250/6237

Mapping: [⟨4 0 4 7 17], ⟨0 6 5 4 -3]]

POTE generator: ~6/5 = 316.925

Vals: Template:Val list

Badness: 0.023406

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 1573/1568

Mapping: [⟨4 0 4 7 17 0], ⟨0 6 5 4 -3 14]]

POTE generator: ~6/5 = 316.989

Vals: Template:Val list

Badness: 0.018731

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping: [⟨1 6 6 6], ⟨0 -18 -15 -13]]

Wedgie: ⟨⟨ 18 15 13 -18 -30 -12 ]]

POTE generator: ~25/21 = 294.303

Template:Val list

Badness: 0.076460

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 3125/3087

Mapping: [⟨1 6 6 6 14], ⟨0 -18 -15 -13 -43]]

POTE generator: ~25/21 = 294.181

Vals: Template:Val list

Badness: 0.052805

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 275/273, 325/324, 540/539

Mapping: [⟨1 6 6 6 14 14], ⟨0 -18 -15 -13 -43 -42]]

POTE generator: ~13/11 = 294.187

Vals: Template:Val list

Badness: 0.031074

Marfifths

The marfifths temperament (19&140, named by Xenllium) tempers out the hemimage comma, 10976/10935. It splits the interval of major tenth (~10/3) into three marvelous fifth (112/75) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 15625/15552

Mapping: [⟨1 -6 -4 -17], ⟨0 18 15 47]]

Wedgie: ⟨⟨ 18 15 47 -18 24 67 ]]

POTE generator: ~75/56 = 505.705

Template:Val list

Badness: 0.063448

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 10976/10935

Mapping: [⟨1 -6 -4 -17 22], ⟨0 18 15 47 -44]]

POTE generator: ~75/56 = 505.684

Vals: Template:Val list

Badness: 0.058902

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 10976/10935

Mapping: [⟨1 -6 -4 -17 22 -14], ⟨0 18 15 47 -44 42]]

POTE generator: ~75/56 = 505.686

Vals: Template:Val list

Badness: 0.030082

Diatessic

The marfifths temperament (121&140, named by Xenllium) is closely related to the diatess tuning (generator: 505.727281 cents).

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2200/2187, 5632/5625

Mapping: [⟨1 -6 -4 -17 -37], ⟨0 18 15 47 96]]

POTE generator: ~75/56 = 505.740

Vals: Template:Val list

Badness: 0.061172

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 625/624, 1375/1372

Mapping: [⟨1 -6 -4 -17 -37 -14], ⟨0 18 15 47 96 42]]

POTE generator: ~75/56 = 505.740

Vals: Template:Val list

Badness: 0.028671

Marf

The marf temperament (19&121, named by Xenllium) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 15625/15552

Mapping: [⟨1 -6 -4 -17 14], ⟨0 18 15 47 -25]]

POTE generator: ~75/56 = 505.769

Vals: Template:Val list

Badness: 0.075112

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539, 625/624, 896/891

Mapping: [⟨1 -6 -4 -17 14 -14], ⟨0 18 15 47 -25 42]]

POTE generator: ~75/56 = 505.771

Vals: Template:Val list

Badness: 0.038317

Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping: [⟨9 0 9 11], ⟨0 6 5 6]]

Wedgie: ⟨⟨ 54 45 54 -54 -66 -1 ]]

POTE generator: ~6/5 = 317.005

Template:Val list

Badness: 0.1934

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4000/3993, 15625/15552

Mapping: [⟨9 0 9 11 24], ⟨0 6 5 6 3]]

POTE generator: ~6/5 = 317.010

Template:Val list

Badness: 0.05172

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 4000/3993

Mapping: [⟨9 0 9 11 24 0], ⟨0 6 5 6 3 14]]

POTE generator: ~6/5 = 317.086

Template:Val list

Badness: 0.0391

Sqrtphi

The just value of sqrt (φ) is 416.545 cents.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 16875/16807

Mapping: [⟨1 12 11 16], ⟨0 -30 -25 -38]]

POTE generator: ~125/98 = 416.603

Template:Val list

Badness: 0.0704

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4375/4356

Mapping: [⟨1 12 11 16 17], ⟨0 -30 -25 -38 -39]]

POTE generator: ~14/11 = 416.604

Template:Val list

Badness: 0.0255

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624, 1375/1372

Mapping: [⟨1 12 11 16 17 28], ⟨0 -30 -25 -38 -39 -70]]

POTE generator: ~14/11 = 416.585

Template:Val list

Badness: 0.0200

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

Mapping: [⟨1 12 11 16 17 28 27], ⟨0 -30 -25 -38 -39 -70 -66]]

POTE generator: ~14/11 = 416.585

Template:Val list

Badness: 0.0130

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

Mapping: [⟨1 12 11 16 17 28 27 -2], ⟨0 -30 -25 -38 -39 -70 -66 18]]

POTE generator: ~14/11 = 416.580

Template:Val list

Badness: 0.0147

Scales

Music

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