3136/3125: Difference between revisions

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'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[7-limit]] [[~~small~~ comma]] measuring about 6.1{{cent}}. It is the difference between a stack of five classic major thirds ([[5/4]]) and a stack of two 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.

'''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 septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224]]), or between the augmented comma ([[128/125]]) ~~and the jubilisma~~ ([[50/49]]).

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 ==

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=== 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.

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 ~~both~~ 126/125 and 225/224 we get [[septimal meantone]].

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 [[~~square superparticular|~~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]].

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

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

~~[[Sval]] [[mapping]]: [~~{{~~val~~| 1 0 -3 0 -1 ~~}}, {{val~~| 0 2 5 0 1 ~~}}, {{val~~| 0 0 0 1 1 }}]

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

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

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

{{~~Val list~~|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}

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

[[Badness]]: 0.0150

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See [[Hemimean family #Semiorion]].

== 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''.

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.

== Notes ==

[[Category:Hemimean]]

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

[[Category:Commas named after individuals]]

[[Category:Commas named after composers]]

@@ Line 4: / Line 4: @@
 | 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 a stack of five classic major thirds ([[5/4]]) and a stack of two 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.
+'''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 septimal semicomma ([[126/125]]) and the septimal kleisma ([[225/224]]), or between the augmented comma ([[128/125]]) and the jubilisma ([[50/49]]).
+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 ==
@@ Line 13: / Line 13: @@
 === 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.
+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 both 126/125 and 225/224 we get [[septimal meantone]].
+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 [[square superparticular|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]].
+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
@@ Line 24: / Line 24: @@
 [[Comma list]]: 476/475, 1445/1444
-[[Sval]] [[mapping]]: [{{val| 1 0 -3 0 -1 }}, {{val| 0 2 5 0 1 }}, {{val| 0 0 0 1 1 }}]
+{{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
+: sval mapping generators: ~2, ~56/25, ~17
 [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
-{{Val list|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}
+{{Optimal ET sequence|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}
 [[Badness]]: 0.0150
@@ Line 43: / Line 43: @@
 See [[Hemimean family #Semiorion]].
+== 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''.
+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.
+== Notes ==
 [[Category:Hemimean]]
+[[Category:Commas named by combining multiple temperament names]]
+[[Category:Commas named after individuals]]
+[[Category:Commas named after composers]]