Hemimean family: Difference between revisions

The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  

The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49).

If we write this equivalently using S-expressions we obtain ([[128/125|S4/S5]])/([[50/49|S5/S7]]) which can be rearranged to [[16/15|S4]]*[[49/48|S7]]/[[25/24|S5]]².

 

Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = [[36/35|S6]]*[[49/48|S7]]*[[64/63|S8]]; substituting this in and simplifying yields:

 

S6*S7*S8*S7/S5² from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])².

* 7- and [[9-odd-limit]]

: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7

As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]] (which combine to make [[5/4]] and whose difference is [[81/80|S9]]), it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)² = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.  

Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two.  

: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7.11/5

: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.11/5