3136/3125: Difference between revisions

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{{Infobox Interval

| Name = hemimean comma, didacus comma

| Color name = zzg53, Zozoquingu comma

| Color name = zzg53, zozoquingu 3rd, Zozoquingu comma

| Comma = yes

}}

'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[~~small~~ comma]] measuring about 6.1{{cent}}. It is the difference between five classic major thirds ([[5/4]]) and two ~~subminor sevenths (~~[[7/4~~]]); it is also the difference between the septimal semicomma~~ (~~[[126~~/~~125]]~~) ~~and the septimal kleisma ([[225/224~~]]). Perhaps ~~most~~ importantly, it is ([[28/25]])2/([[5/4]]) and ~~(because~~ [[28/25]] = ([[7/5]])/([[5/4]])~~) therefore~~ also ([[28/25]])3/([[7/5]]) which means ~~that~~ its square is equal to the difference between ([[28/25]])5 and [[7/4]]. ~~This~~ has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.

'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1{{cent}}. It is the difference between a stack of five [[5/4|classic major thirds (5/4)]] and a stack of two [[7/4|subminor sevenths (7/4)]]. Perhaps more importantly, it is ([[28/25]])2/([[5/4]]), and in light of the fact that [[28/25]] = ([[7/5]])/([[5/4]]), it is also ([[28/25]])3/([[7/5]]), which means its square is equal to the difference between ([[28/25]])5 and [[7/4]]. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.

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.

== Temperaments ==

=== Didacus (2.5.7) ===

Tempering out this comma in its minimal prime subgroup of 2.5.7 leads to [[~~Hemimean clan #Didacus|~~didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator of [[28/25]].

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]] ~~of temperaments~~, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. (Note that if we temper ~~both of those commas individually~~ we get [[septimal meantone]].)

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.

Note that if we temper 126/125 and/or 225/224 we get [[septimal meantone]].

=== Orion ===

As [[28/25]] is close to [[19/17]] and as the latter is the mediant of [[9/8]] and [[10/9]] (which together make [[5/4]]), it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] ([[5/4]])/([[19/17]])2 = [[1445/1444]], which together imply tempering out 3136/3125 and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].

[[Subgroup]]: 2.5.7.17.19

[[Comma list]]: 476/475, 1445/1444

: sval mapping generators: ~2, ~56/25, ~17

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434

{{Optimal ET sequence|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}

[[Badness]]: 0.0150

==== Hemimean orion ====

As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.

See [[Hemimean family #Hemimean orion]].

=== ~~Orion (2.5.7.17.19)~~ ===

==== Semiorion ====

As [[28/25]] ~~is close to~~ [[19/17]] ~~and as the latter is a precise approximation of half of~~ [[5/4]]~~, it is natural~~ to ~~temper (~~[[28/25]])/([[19/17]]) = [[~~476~~/~~475~~]] ~~and the~~ [[~~square superparticular|semiparticular~~]] ([[5/4]]~~)/(~~[[19/17]]~~)2 =~~ [[~~1445~~/~~1444~~]] ~~which together imply tempering~~ [[~~3136~~/~~3125~~]] ~~and~~ [[~~2128~~/~~2125~~]], ~~resulting in~~ a ~~rank 3 temperament~~.

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.

[[~~Mapping~~]]~~: ~~

See [[Hemimean family #Semiorion]].

~~[{{val| 1 0 -3 0 -1 }} ~~

~~{{val| 0 2 5 0 1 }} ~~

~~{{val| 0 0 0 1 1 }}]~~

[[~~CTE~~]] ~~generators~~: 2/~~1, ~28~~/~~25 = 193~~.~~642~~, ~~~17/16 = 104~~.~~434~~

== Etymology ==

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''.

[[~~Val~~]~~]s: {{Val list| 12, 25, 31, 37, 43, 50, 56, 68~~, ~~93}}~~

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.

== ~~See also~~ ==

== Notes ==

* [[Hemimean family]], the rank-3 family where it is tempered out

* [[Hemimean clan]], the rank-2 clan where it is tempered out

[[Category:Hemimean]]

[[Category:Commas named by combining multiple temperament names]]

[[Category:Commas named after individuals]]

[[Category:Commas named after composers]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Name = hemimean comma, didacus comma
-| Color name = zzg<sup>5</sup>3, Zozoquingu comma
+| Color name = zzg<sup>5</sup>3, zozoquingu 3rd,<br>Zozoquingu comma
 | Comma = yes
 }}
-'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[small comma]] measuring about 6.1{{cent}}. It is the difference between five classic major thirds ([[5/4]]) and two subminor sevenths ([[7/4]]); it is also the difference between the septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224]]). Perhaps most importantly, it is ([[28/25]])<sup>2</sup>/([[5/4]]) and (because [[28/25]] = ([[7/5]])/([[5/4]])) therefore also ([[28/25]])<sup>3</sup>/([[7/5]]) which means that its square is equal to the difference between ([[28/25]])<sup>5</sup> and [[7/4]]. This has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.
+'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1{{cent}}. It is the difference between a stack of five [[5/4|classic major thirds (5/4)]] and a stack of two [[7/4|subminor sevenths (7/4)]]. Perhaps more importantly, it is ([[28/25]])<sup>2</sup>/([[5/4]]), and in light of the fact that [[28/25]] = ([[7/5]])/([[5/4]]), it is also ([[28/25]])<sup>3</sup>/([[7/5]]), which means its square is equal to the difference between ([[28/25]])<sup>5</sup> and [[7/4]]. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.
+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]]<sup>2</sup>. 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*S7<sup>2</sup>*S8/S5<sup>2</sup>, from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])<sup>2</sup>, meaning we split [[49/45]] into two [[25/24]]'s in the resulting temperament.
 == Temperaments ==
 === Didacus (2.5.7) ===
-Tempering out this comma in its minimal prime subgroup of 2.5.7 leads to [[Hemimean clan #Didacus|didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator of [[28/25]].
+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]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. (Note that if we temper both of those commas individually we get [[septimal meantone]].)
+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.
+Note that if we temper 126/125 and/or 225/224 we get [[septimal meantone]].
+=== Orion ===
+As [[28/25]] is close to [[19/17]] and as the latter is the mediant of [[9/8]] and [[10/9]] (which together make [[5/4]]), it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]], which together imply tempering out 3136/3125 and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].
+[[Subgroup]]: 2.5.7.17.19
+[[Comma list]]: 476/475, 1445/1444
+{{Mapping|legend=2| 1 0 -3 0 -1 | 0 2 5 0 1 | 0 0 0 1 1 }}
+: sval mapping generators: ~2, ~56/25, ~17
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
+{{Optimal ET sequence|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}
+[[Badness]]: 0.0150
+==== Hemimean orion ====
+As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.
+See [[Hemimean family #Hemimean orion]].
-=== Orion (2.5.7.17.19) ===
+==== Semiorion ====
-As [[28/25]] is close to [[19/17]] and as the latter is a precise approximation of half of [[5/4]], it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]] and the [[square superparticular|semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]] which together imply tempering [[3136/3125]] and [[2128/2125]], resulting in a rank 3 temperament.
+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.
-[[Mapping]]:<br>
+See [[Hemimean family #Semiorion]].
-[{{val| 1 0 -3 0 -1 }}<br>
-{{val| 0 2 5 0 1 }}<br>
-{{val| 0 0 0 1 1 }}]
-[[CTE]] generators: 2/1, ~28/25 = 193.642, ~17/16 = 104.434
+== Etymology ==
+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''.
-[[Val]]s: {{Val list| 12, 25, 31, 37, 43, 50, 56, 68, 93}}
+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.
-== See also ==
+== Notes ==
-* [[Hemimean family]], the rank-3 family where it is tempered out
-* [[Hemimean clan]], the rank-2 clan where it is tempered out
 [[Category:Hemimean]]
+[[Category:Commas named by combining multiple temperament names]]
+[[Category:Commas named after individuals]]
+[[Category:Commas named after composers]]