Kleismic family: Difference between revisions
Sorting |
→Catakleismic: move music to the dedicated article Tag: Manual revert |
||
| (6 intermediate revisions by 3 users not shown) | |||
| Line 35: | Line 35: | ||
=== Overview to extensions === | === Overview to extensions === | ||
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[875/864]], the keemic comma, gives keemun. [[ | The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan. | ||
[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. 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. | [[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. 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. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator. | ||
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts. | |||
The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]]. | |||
== Catakleismic == | == Catakleismic == | ||
{{Main| Catakleismic }} | {{Main| Catakleismic }} | ||
Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. | |||
Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]], [[225/224|S15 = S25⋅S26⋅S27]], [[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}. | |||
=== 7-limit === | === 7-limit === | ||
| Line 64: | Line 72: | ||
==== 2.3.5.7.13 subgroup ==== | ==== 2.3.5.7.13 subgroup ==== | ||
Subgroup: 2.3.5.7.13 | Subgroup: 2.3.5.7.13 | ||
| Line 717: | Line 723: | ||
[[Badness]] (Sintel): 1.43 | [[Badness]] (Sintel): 1.43 | ||
Music | ; Music | ||
* | * [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]] | ||
=== 11-limit === | === 11-limit === | ||
| Line 771: | Line 777: | ||
== Marfifths == | == Marfifths == | ||
Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 821: | Line 827: | ||
=== Diatessic === | === Diatessic === | ||
Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents). | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 853: | Line 859: | ||
=== Marf === | === Marf === | ||
Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 996: | Line 1,002: | ||
== Marthirds == | == Marthirds == | ||
Named by [[Xenllium]] in 2021, marthirds tempers out 2460375/2458624, the [[breeze comma]], and may be described as the {{nowrap| 19 & 193 }} temperament. It is generated by a marvel-comma-flat classical major third, [[56/45]], four of which minus an octave make the hanson generator of [[6/5]]. Its [[ploidacot]] is zeta-24-cot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,048: | Line 1,054: | ||
{{Main| Sqrtphi }} | {{Main| Sqrtphi }} | ||
The just value of sqrt (φ) is 416.545 cents. | Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it. | ||
Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,128: | Line 1,136: | ||
== Quartkeenlig == | == Quartkeenlig == | ||
Named by [[Eliora]] in 2022, quartkeenlig uses a generator that is a quartertone of [[33/32]][[~]][[36/35]] tempered together in the [[11-limit]], and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with [[7/6]], keenanisma, [[385/384]], tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). As six quartertones make the hanson generator of ~[[6/5]], its [[ploidacot]] is alpha-36-cot. It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,227: | Line 1,235: | ||
== Subgroup extensions == | == Subgroup extensions == | ||
=== Kleismic (2.3.5.13) a.k.a. cata === | === Kleismic (2.3.5.13) a.k.a. cata === | ||
Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = | Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]]. | ||
Subgroup: 2.3.5.13 | Subgroup: 2.3.5.13 | ||
| Line 1,237: | Line 1,245: | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}} | * WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}} | ||
* CWE: ~2 = 1200. | * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }} | {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }} | ||
| Line 1,254: | Line 1,262: | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}} | * WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}} | ||
* CWE: ~2 = 1200. | * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }} | {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }} | ||