3136/3125: Difference between revisions
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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]] | 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 [[ | [[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 [[ | 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 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~28/25 = 193.642{{c}}, ~17/16 = 104.434{{c}} | |||
[[Optimal tuning]] ([[CTE]]): ~2 = | |||
{{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 [[ | 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]] | 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]]. | ||
== 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 | 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. | ||
== | == References == | ||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||
Latest revision as of 10:10, 23 February 2026
| Interval information |
didacus comma
Zozoquingu comma
3136/3125, the hemimean comma or didacus comma, is a small 7-limit comma measuring about 6.1 ¢. 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.
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).
Examining the latter expression we can observe that this gives us a relatively simple S-expression of (S4/S5)/(S5/S7) which can be rearranged to S4⋅S7/S52. Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = S6⋅S7⋅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 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 temperament, which splits the 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 out 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 The Milky Way realm.
Subgroup: 2.5.7.17.19
Comma list: 476/475, 1445/1444
Subgroup-val mapping: [⟨1 0 -3 0 -1], ⟨0 2 5 0 1], ⟨0 0 0 1 1]]
- mapping generators: ~2, ~56/25, ~17
Optimal tuning (CTE): ~2 = 1200.000 ¢, ~28/25 = 193.642 ¢, ~17/16 = 104.434 ¢
Optimal ET sequence: 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch
Badness (Smith): 0.0150
Hemimean orion
As tempering out either 1216/1215 (S16/S18) or 1701/1700 (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.
Semiorion
As 1445/1444 (S17/S19) we can extend orion to include prime 3 in its subgroup by tempering out both 289/288 (S17) and 361/360 (S19). 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.
Etymology
This comma was first named as parahemwuer by Gene Ward Smith in 2005 as a contraction of parakleismic and hemiwürschmidt[1]. 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, and the name was created by Gene Ward Smith[2]. It is unclear what the origin of this name is; 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.