Kleismic family: Difference between revisions
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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&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&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&140, named by [[User:Xenllium|Xenllium]]) is closely related to the '''diatess tuning''' (generator: 505.727281 cents). | The ''diatessic'' temperament (121&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&121, named by [[User:Xenllium|Xenllium]]) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75. | The ''marf'' temperament (19&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&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&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]] | ||