Kleismic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,171 edits
correct per Milne's comments
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m fix heading structure
Line 7: Line 7:
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 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)<sup>5</sup> = 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]].
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields {{multival| 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)<sup>5</sup> = 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 =
== Hanson ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 32: Line 32:
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>


== Seven limit extensions ==
=== Seven limit extensions ===
The second comma of the [[Normal lists|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.
The second comma of the [[Normal lists|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 =
== Keemun ==
{{main|Keemun}}
{{main|Keemun}}


Line 57: Line 57:
[[Badness]]: 0.027408
[[Badness]]: 0.027408


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


Line 75: Line 75:
Badness: 0.027410
Badness: 0.027410


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


Line 93: Line 93:
Badness: 0.029749
Badness: 0.029749


=== Kema ===
==== Kema ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 111: Line 111:
Badness: 0.022749
Badness: 0.022749


=== Kumbaya ===
==== Kumbaya ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 124: Line 124:
Badness: 0.031633
Badness: 0.031633


== Qeema ==
=== Qeema ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 137: Line 137:
Badness: 0.040056
Badness: 0.040056


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


Line 150: Line 150:
Badness: 0.029419
Badness: 0.029419


== Darjeeling ==
=== Darjeeling ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 163: Line 163:
Badness: 0.027648
Badness: 0.027648


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


Line 176: Line 176:
Badness: 0.021445
Badness: 0.021445


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


Line 191: Line 191:
[[Badness]]: 0.094872
[[Badness]]: 0.094872


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


Line 204: Line 204:
Badness: 0.036894
Badness: 0.036894


= Catakleismic =
== Catakleismic ==
{{main | Catakleismic }}
{{main | Catakleismic }}


Line 226: Line 226:
[[Badness]]: 0.021501
[[Badness]]: 0.021501


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


Line 244: Line 244:
Badness: 0.021849
Badness: 0.021849


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


Line 262: Line 262:
Badness: 0.016883
Badness: 0.016883


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


Line 275: Line 275:
Badness: 0.039954
Badness: 0.039954


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


Line 288: Line 288:
Badness: 0.022555
Badness: 0.022555


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


Line 301: Line 301:
Badness: 0.030422
Badness: 0.030422


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


Line 314: Line 314:
Badness: 0.022337
Badness: 0.022337


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


Line 327: Line 327:
Badness: 0.044335
Badness: 0.044335


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


Line 340: Line 340:
Badness: 0.027343
Badness: 0.027343


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


Line 353: Line 353:
Badness: 0.029319
Badness: 0.029319


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


Line 366: Line 366:
Badness: 0.021814
Badness: 0.021814


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


Line 381: Line 381:
[[Badness]]: 0.052129
[[Badness]]: 0.052129


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


Line 394: Line 394:
Badness: 0.039770
Badness: 0.039770


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


Line 407: Line 407:
Badness: 0.020156
Badness: 0.020156


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


Line 422: Line 422:
[[Badness]]: 0.163519
[[Badness]]: 0.163519


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


Line 435: Line 435:
Badness: 0.048570
Badness: 0.048570


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


Line 448: Line 448:
Badness: 0.024371
Badness: 0.024371


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


Line 463: Line 463:
[[Badness]]: 0.052054
[[Badness]]: 0.052054


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


Line 476: Line 476:
Badness: 0.038023
Badness: 0.038023


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


Line 489: Line 489:
Badness: 0.026005
Badness: 0.026005


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


Line 513: Line 513:
[[Badness]]: 0.047261
[[Badness]]: 0.047261


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


Line 526: Line 526:
Badness: 0.047417
Badness: 0.047417


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


Line 539: Line 539:
Badness: 0.026842
Badness: 0.026842


= Tritikleismic =
== Tritikleismic ==
{{see also| Gamelismic clan #Tritikleismic }}
{{see also| Gamelismic clan #Tritikleismic }}


Line 566: Line 566:
[[Badness]]: 0.056337
[[Badness]]: 0.056337


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


Line 586: Line 586:
Badness: 0.019333
Badness: 0.019333


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


Line 601: Line 601:
Badness: 0.015652
Badness: 0.015652


== 17-limit ==
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 616: Line 616:
Badness: 0.013551
Badness: 0.013551


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


Line 631: Line 631:
[[Badness]]: 0.039231
[[Badness]]: 0.039231


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


Line 644: Line 644:
Badness: 0.023406
Badness: 0.023406


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


Line 657: Line 657:
Badness: 0.018731
Badness: 0.018731


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


Line 672: Line 672:
[[Badness]]: 0.076460
[[Badness]]: 0.076460


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


Line 685: Line 685:
Badness: 0.052805
Badness: 0.052805


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


Line 698: Line 698:
Badness: 0.031074
Badness: 0.031074


= Marfifths =
== Marfifths ==
The ''marfifths'' temperament (19&amp;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.
The ''marfifths'' temperament (19&amp;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.


Line 715: Line 715:
[[Badness]]: 0.063448
[[Badness]]: 0.063448


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


Line 728: Line 728:
Badness: 0.058902
Badness: 0.058902


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


Line 741: Line 741:
Badness: 0.030082
Badness: 0.030082


== Diatessic ==
=== Diatessic ===
The ''diatessic'' temperament (121&amp;140, named by [[User:Xenllium|Xenllium]]) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
The ''diatessic'' temperament (121&amp;140, named by [[User:Xenllium|Xenllium]]) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).


Line 756: Line 756:
Badness: 0.061172
Badness: 0.061172


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


Line 769: Line 769:
Badness: 0.028671
Badness: 0.028671


== Marf ==
=== Marf ===
The ''marf'' temperament (19&amp;121, named by [[User:Xenllium|Xenllium]]) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
The ''marf'' temperament (19&amp;121, named by [[User:Xenllium|Xenllium]]) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.


Line 784: Line 784:
Badness: 0.075112
Badness: 0.075112


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


Line 797: Line 797:
Badness: 0.038317
Badness: 0.038317


= Marthirds =
== Marthirds ==
The ''marthirds'' temperament (19&amp;193, named by [[User:Xenllium|Xenllium]]) tempers out the sesquiquartisma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.
The ''marthirds'' temperament (19&amp;193, named by [[User:Xenllium|Xenllium]]) tempers out the sesquiquartisma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.


Line 814: Line 814:
[[Badness]]: 0.104253
[[Badness]]: 0.104253


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


Line 827: Line 827:
Badness: 0.075624
Badness: 0.075624


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


Line 840: Line 840:
Badness: 0.043728
Badness: 0.043728


= Novemkleismic =
== Novemkleismic ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 855: Line 855:
[[Badness]]: 0.193429
[[Badness]]: 0.193429


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


Line 868: Line 868:
Badness: 0.051730
Badness: 0.051730


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


Line 881: Line 881:
Badness: 0.039072
Badness: 0.039072


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


Line 896: Line 896:
[[Badness]]: 0.070378
[[Badness]]: 0.070378


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


Line 909: Line 909:
Badness: 0.025515
Badness: 0.025515


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


Line 922: Line 922:
Badness: 0.020040
Badness: 0.020040


== 17-limit ==
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 935: Line 935:
Badness: 0.013028
Badness: 0.013028


== 19-limit ==
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 948: Line 948:
Badness: 0.014748
Badness: 0.014748


== Scales ==
=== Scales   ===
* [[Sqrtphi17]]
* [[Sqrtphi17]]
* [[Sqrtphi23]]
* [[Sqrtphi23]]
* [[Sqrtphi49]]
* [[Sqrtphi49]]


== Music ==
=== Music   ===
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 Prelude for Piano in Square root of Phi Tuning] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 Prelude for Piano in Square root of Phi Tuning] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 A Fight for Phi] by [[Vito Sicurella]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 A Fight for Phi] by [[Vito Sicurella]]

Revision as of 15:28, 31 May 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:

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:

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:

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:

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:

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

Main article: 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:

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:

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:

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 diatessic 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

Marthirds

The marthirds temperament (19&193, named by Xenllium) tempers out the sesquiquartisma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third (56/45) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 2460375/2458624

Mapping: [⟨1 -6 -4 -19], ⟨0 24 20 69]]

Wedgie: ⟨⟨ 24 20 69 -24 42 104 ]]

POTE generator: ~56/45 = 379.252

Template:Val list

Badness: 0.104253

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 15625/15552, 19712/19683

Mapping: [⟨1 -6 -4 -19 -43], ⟨0 24 20 69 147]]

POTE generator: ~56/45 = 379.257

Vals: Template:Val list

Badness: 0.075624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 19712/19683

Mapping: [⟨1 -6 -4 -19 -43 -14], ⟨0 24 20 69 147 56]]

POTE generator: ~56/45 = 379.256

Vals: Template:Val list

Badness: 0.043728

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: 0.193429

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

Vals: Template:Val list

Badness: 0.051730

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

Vals: Template:Val list

Badness: 0.039072

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: 0.070378

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

Vals: Template:Val list

Badness: 0.025515

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

Vals: Template:Val list

Badness: 0.020040

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

Vals: Template:Val list

Badness: 0.013028

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

Vals: Template:Val list

Badness: 0.014748

Scales

Music

Retrieved from "https://en.xen.wiki/index.php?title=Kleismic_family&oldid=71223"