3136/3125 - Revision history

FloraC: Wording, paragraphing, and style

2026-02-23T10:10:54Z

Wording, paragraphing, and style

← Older revision		Revision as of 10:10, 23 February 2026
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	'''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.		'''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: ~~S6S7~~<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.		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 ==		== Temperaments ==
	=== Didacus (2.5.7) ===		=== Didacus (2.5.7) ===
	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.		[[Tempering out]] this comma in its minimal prime [[subgroup]] of [[2.5.7 subgroup\|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) ===		=== 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]] 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]].		Note that if we temper out 126/125 and/or 225/224 we get [[septimal meantone]].

	=== Orion ===		=== Orion ===
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	{{Mapping\|legend=2\| 1 0 -3 0 -1 \| 0 2 5 0 1 \| 0 0 0 1 1 }}		{{Mapping\|legend=2\| 1 0 -3 0 -1 \| 0 2 5 0 1 \| 0 0 0 1 1 }}
			: mapping generators: ~2, ~56/25, ~17

	~~: sval mapping generators: ~2, ~56/25, ~17~~		[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~28/25 = 193.642{{c}}, ~17/16 = 104.434{{c}}

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

	[[Badness]]: 0.0150		[[Badness]] (Smith): 0.0150

	==== Hemimean orion ====		==== 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.		As tempering out either [[1216/1215]] ([[S-expression\|S16/S18]]) or [[1701/1700]] ([[S-expression\|S18/S20]]) 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]].		See [[Hemimean family #Hemimean orion]].

	==== Semiorion ====		==== Semiorion ====
	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.		As [[1445/1444]] ([[S-expression\|S17/S19]]) we can extend orion to include prime 3 in its subgroup by tempering out both [[289/288]] ({{S\|17}}) and [[361/360]] ({{S\|19}}). However, note that from the vanish of 289/288 this splits the period in half, representing a [[17/12]]~[[24/17]] half-octave. This has the implication 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]].		See [[Hemimean family #Semiorion]].

FloraC: Formatting

2026-02-23T10:00:21Z

Formatting

← Older revision		Revision as of 10:00, 23 February 2026
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	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'' (or ''hemithirds'') and ''meantone''.		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'' (or ''hemithirds'') 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.		The name ''didacus'' seems to be first attested in September 2016, and the name was created by Gene Ward Smith<ref>[https://en.xen.wiki/index.php?title=Subgroup_temperaments&diff=next&oldid=26776 Xenharmonic Wiki \| ''Subgroup temperaments (Revision as of 16:44, 23 September 2016 by Wikispaces>genewardsmith)'']</ref>. It is unclear what the origin of this name is; {{w\|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~~ ==		== References ==

	[[Category:Hemimean]]		[[Category:Hemimean]]

Lériendil: /* Etymology */

2026-02-22T06:42:33Z

Etymology

← Older revision		Revision as of 06:42, 22 February 2026
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	== Etymology ==		== 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''.		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'' (or ''hemithirds'') 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.		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.

Lériendil at 09:52, 17 April 2025

2025-04-17T09:52:10Z

← Older revision		Revision as of 09:52, 17 April 2025
Line 47:		Line 47:
	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''.		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; [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.		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 ==		== Notes ==

Lériendil at 09:51, 17 April 2025

2025-04-17T09:51:59Z

← Older revision		Revision as of 09:51, 17 April 2025
Line 47:		Line 47:
	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''.		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.		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; [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 ==		== Notes ==

Lériendil: trying to find etymology for "didacus"

2025-04-17T02:49:57Z

trying to find etymology for "didacus"

← Older revision		Revision as of 02:49, 17 April 2025
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	== Etymology ==		== 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''.		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.

	== Notes ==		== Notes ==

BudjarnLambeth at 02:06, 4 November 2024

2024-11-04T02:06:55Z

← Older revision		Revision as of 02:06, 4 November 2024
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	[[Category:Commas named by combining multiple temperament names]]		[[Category:Commas named by combining multiple temperament names]]
	[[Category:Commas named after individuals]]		[[Category:Commas named after individuals]]
			[[Category:Commas named after composers]]

BudjarnLambeth at 12:57, 3 November 2024

2024-11-03T12:57:14Z

← Older revision		Revision as of 12:57, 3 November 2024
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	[[Category:Hemimean]]		[[Category:Hemimean]]
	[[Category:Commas named ~~for their regular~~ temperament ~~properties~~]]		[[Category:Commas named by combining multiple temperament names]]
	[[Category:Commas named after individuals]]		[[Category:Commas named after individuals]]

BudjarnLambeth at 12:51, 3 November 2024

2024-11-03T12:51:23Z

← Older revision		Revision as of 12:51, 3 November 2024
Line 50:		Line 50:

	[[Category:Hemimean]]		[[Category:Hemimean]]
			[[Category:Commas named for their regular temperament properties]]
			[[Category:Commas named after individuals]]

FloraC: Improve readability and cleanup

2024-02-07T08:03:42Z

Improve readability and cleanup

← Older revision		Revision as of 08:03, 7 February 2024
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	\| Comma = yes		\| 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: S6S7<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.

	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:

	S6S7<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 ==		== Temperaments ==
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	=== Hemimean (2.3.5.7) ===		=== 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 ===		=== Orion ===
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	[[Comma list]]: 476/475, 1445/1444		[[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		[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
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	== Etymology ==		== 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]]''.		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''.

	== Notes ==		== Notes ==

	[[Category:Hemimean]]		[[Category:Hemimean]]