Kleismic family: Difference between revisions

The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots [[9/8]]. As a result, all kleismic temperaments trisect 9/8, for which 1 third tone is 25/24 and two are 27/25.  

The kleisma has the [[S-expression]] S25²*S26 which explains how the tone is trisected, thus tempering out its constituyents S25 = [[625/624]] and S26 = [[676/675]] is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] = S25*S26 it is also tempered out. Cata finds the third-tritave [[³√3]] to be [[13/9]] as its comma [[2197/2187]] also has the expression S25*S26², and the half-tritave [[√3]] to be [[26/15]], whose difference is the basis S26.  

The third tone is now 25/24 ~ 26/25 ~ 27/26, which means 27/25 ~ [[13/12]] are made equal.  

For a version of cata with a 1\5 period, see [[Thunderclysmic]].  

* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}[[error map]]: {{val|+0.121 +0.690 -0.655 -1.022}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}[[error map]]: {{val|+0.000 +0.597 -0.854 -1.240}}

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. In here, the JI intervals 25/24 - 27/25 - 9/8 - 7/6 are made equidistant.