Kleismic family: Difference between revisions
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=== 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. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]] 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. | 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|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. | [[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. | ||
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=== Diatessic === | === Diatessic === | ||
Diatessic may be described as {{nowrap| 121 & 140 | 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 1,245: | 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 1,254: | ||
Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy. | Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy. | ||
Subgroup: 2.3.5.13.37 | Subgroup: 2.3.5.13.37 | ||
Comma list: 325/324, 481/480, 625/624 | Comma list: 325/324, 481/480, 625/624 | ||
| Line 1,262: | 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 }} | ||