3136/3125: Difference between revisions

| Name = hemimean comma, didacus comma

| Color name = zzg⁵3, zozoquingu 3rd,<br>Zozoquingu comma

| Comma = yes

In terms of commas, it is the difference between the [[126/125|septimal semicomma (126/125)]] and the [[225/224|septimal kleisma (225/224)]], or between the [[128/125|augmented comma (128/125)]] and the [[50/49|jubilisma (50/49)]]. Examining the latter expression we can observe that this gives us a relatively simple [[S-expression]] of ([[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]] = ([[6/5]])/([[9/8]]); substituting this in and simplifying yields: S6*S7²*S8/S5², from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])², meaning we split [[49/45]] into two [[25/24]]'s in the resulting temperament.

Tempering out this comma in its minimal prime [[subgroup]] of 2.5.7 leads to [[didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator representing [[28/25]]. See [[hemimean clan]] for extensions of didacus.  

=== Hemimean (2.3.5.7) ===

Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family #Hemimean|hemimean]] temperament, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. See [[hemimean family]] for the family of rank-3 temperaments where it is tempered out.

 

 

=== Orion ===

 

 

 

 

 

 

 

 

 

See [[Hemimean family #Hemimean orion]].

 

As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can extend orion to include prime 3 in its subgroup by tempering both [[289/288|S17]] and [[361/360|S19]]. However, note that (because of tempering [[289/288|S17]]) this splits the period in half, representing a [[17/12]]~[[24/17]] half-octave. This has the consequence that the [[17/16]] generator can be described as a [[3/2]] because [[17/16]] up from [[24/17]] is [[3/2]]. As a result, this equates the generators of hemimean orion and orion up to period-equivalence and is a weak extension of both.

 

See [[Hemimean family #Semiorion]].

 

This comma was first named as ''parahemwuer'' by [[Gene Ward Smith]] in 2005 as a contraction of ''[[parakleismic]]'' and ''[[hemiwürschmidt]]''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12900.html Yahoo! Tuning Group | ''Seven limit comma names from pairs of temperament names'']</ref>. It is not clear how it later became ''hemimean'', but the root of ''hemimean'' is obvious, being a contraction of ''hemiwürschmidt'' and ''meantone''.

 

The name ''didacus'' seems to be first attested in September 2016 ([https://en.xen.wiki/index.php?title=Subgroup_temperaments&diff=next&oldid=26776 here]), and the name was created by Gene Ward Smith. It is unclear what the origin of this name is; [https://en.wikipedia.org/wiki/Didacus_of_Alcalá St. Didacus] was a Spanish missionary after whom the city of San Diego was named, but there seems to be no relation between this individual and musical temperament.