1029/1024: Difference between revisions

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{{Interwiki

| en = 1029/1024

| de = 1029/1024

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{{Infobox Interval

| Name = slendric comma, gamelisma, gamelan residue

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| Comma = yes

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~~{{interwiki~~

'''1029/1024''', the '''slendric comma''' or '''gamelisma''', is a [[small comma|small]] [[7-limit]] (also [[2.3.7 subgroup|2.3.7-subgroup]]) [[comma]] measuring about 8.4 [[cent]]s. It is the amount by which a stack of three [[8/7]]'s falls short of [[3/2]], and the ratio between [[49/48]] ({{S|7}}) and [[64/63]] ({{S|8}}), which gives it the [[S-expression]] of S7/S8, making it an ultraparticular comma.

~~| de = 1029/1024~~

~~| en = 1029/1024~~

}}

'''1029/1024''', the '''slendric comma''' or '''gamelisma''', is a [[small comma|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 8.4 [[cent]]s. It is the amount by which a stack of three [[8/7]]'s falls short of [[3/2]], and the ratio between ~~S7 =~~ [[49/48]] and ~~S8 =~~ [[64/63]], which gives it the [[S-expression]] of S7/S8, making it an ultraparticular comma~~. It is also the amount by which a stack of three [[garischisma]]s falls short of a [[countercomp comma]]~~.

== Commatic relations ==

This comma ~~factorizes into~~ [[~~superparticular~~]]s ~~as:~~

This comma is the difference between a [[Pythagorean limma]] and a stack of three septimal commas, as well as the difference between a [[Pythagorean countercomma]] and a stack of three [[septimal schisma]]s.

* [[~~217/216~~]] × [[~~3969~~/~~3968~~]] (~~subgroup:~~ [[~~31-limit|2.3.7.31~~]])

* [[~~225~~/~~224~~]] × [[~~2401~~/~~2400~~]] (subgroup: ~~[[7-limit|~~2.3.5.7]])

In the full 7-limit it factorizes into [[superparticular]]s as ([[225/224]])⋅([[2401/2400]]). It also factorizes into the following constituent superparticulars in the higher limits:

* [[~~273~~/~~272~~]] × [[~~833~~/~~832~~]] (subgroup: ~~[[17-limit|~~2.3.7.~~13.17]]~~)

* [[385/384]] and [[441/440]] (subgroup: 2.3.5.7.11)

* [[~~343~~/~~342~~]] × [[~~513~~/~~512~~]] (subgroup: ~~[[19-limit|~~2.3.7.~~19]]~~)

* [[343/342]] and [[513/512]] (subgroup: 2.3.7.19)

* [[~~385~~/~~384~~]] × [[~~441~~/~~440~~]] (subgroup: ~~[[11-limit|~~2.3.5.7.~~11]]~~).

* [[273/272]] and [[833/832]] (subgroup: 2.3.7.13.17)

* [[217/216]] and [[3969/3968]] (subgroup: 2.3.7.31)

Tempering out these constituent commas adds new intervals (outside of the 2.3.7 subgroup) to the chain of 8/7s while doing minimal additional damage to 2.3.7 itself.

Tempering out these constituent commas adds new intervals (outside of the 2.3.7 subgroup) to the chain of 8/7's while doing minimal additional damage to 2.3.7 itself.

== Temperaments ==

Tempering out this comma alone in the [[2.3.7 subgroup]] leads to the rank-2 [[slendric]] temperament, or in the full 7-limit, the rank-3 [[gamelismic]] temperament. In either case, it enables the [[slendric pentad]], and the perfect fifth is split into three equal parts, one for 8/7 and two for [[21/16]]. In addition, the ~~[[256/243|~~Pythagorean limma ~~(256/243)]]~~ is also split into three, one for 64/63[[~]]49/48 and two for [[28/27]]. It therefore provides the little interval known as a [[quark]].

[[Tempering out]] this comma alone in the 2.3.7 subgroup leads to the rank-2 [[slendric]] temperament, or in the full 7-limit, the rank-3 [[gamelismic]] temperament. In either case, it enables the [[slendric pentad]], and the perfect fifth is split into three equal parts, one for 8/7 and two for [[21/16]]. In addition, the Pythagorean limma is also split into three, one for 64/63[[~]]49/48 and two for [[28/27]]. It therefore provides the little interval known as a [[quark]].

See [[Gamelismic family]] for the rank-3 family where it is tempered out. See [[Gamelismic clan]] for the rank-2 clan where it is tempered out.

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+{{Interwiki
+| en = 1029/1024
+| de = 1029/1024
+}}
 {{Infobox Interval
 | Name = slendric comma, gamelisma, gamelan residue
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 | Comma = yes
 }}
-{{interwiki
+'''1029/1024''', the '''slendric comma''' or '''gamelisma''', is a [[small comma|small]] [[7-limit]] (also [[2.3.7 subgroup|2.3.7-subgroup]]) [[comma]] measuring about 8.4 [[cent]]s. It is the amount by which a stack of three [[8/7]]'s falls short of [[3/2]], and the ratio between [[49/48]] ({{S|7}}) and [[64/63]] ({{S|8}}), which gives it the [[S-expression]] of S7/S8, making it an ultraparticular comma.
-| de = 1029/1024
-| en = 1029/1024
-}}
-'''1029/1024''', the '''slendric comma''' or '''gamelisma''', is a [[small comma|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 8.4 [[cent]]s. It is the amount by which a stack of three [[8/7]]'s falls short of [[3/2]], and the ratio between S7 = [[49/48]] and S8 = [[64/63]], which gives it the [[S-expression]] of S7/S8, making it an ultraparticular comma. It is also the amount by which a stack of three [[garischisma]]s falls short of a [[countercomp comma]].
 == Commatic relations ==
-This comma factorizes into [[superparticular]]s as:
+This comma is the difference between a [[Pythagorean limma]] and a stack of three septimal commas, as well as the difference between a [[Pythagorean countercomma]] and a stack of three [[septimal schisma]]s.
-* [[217/216]] × [[3969/3968]] (subgroup: [[31-limit|2.3.7.31]])
-* [[225/224]] × [[2401/2400]] (subgroup: [[7-limit|2.3.5.7]])
+In the full 7-limit it factorizes into [[superparticular]]s as ([[225/224]])⋅([[2401/2400]]). It also factorizes into the following constituent superparticulars in the higher limits:
-* [[273/272]] × [[833/832]] (subgroup: [[17-limit|2.3.7.13.17]])
+* [[385/384]] and [[441/440]] (subgroup: 2.3.5.7.11)
-* [[343/342]] × [[513/512]] (subgroup: [[19-limit|2.3.7.19]])
+* [[343/342]] and [[513/512]] (subgroup: 2.3.7.19)
-* [[385/384]] × [[441/440]] (subgroup: [[11-limit|2.3.5.7.11]]).
+* [[273/272]] and [[833/832]] (subgroup: 2.3.7.13.17)
+* [[217/216]] and [[3969/3968]] (subgroup: 2.3.7.31)
-Tempering out these constituent commas adds new intervals (outside of the 2.3.7 subgroup) to the chain of 8/7s while doing minimal additional damage to 2.3.7 itself.
+Tempering out these constituent commas adds new intervals (outside of the 2.3.7 subgroup) to the chain of 8/7's while doing minimal additional damage to 2.3.7 itself.
 == Temperaments ==
-Tempering out this comma alone in the [[2.3.7 subgroup]] leads to the rank-2 [[slendric]] temperament, or in the full 7-limit, the rank-3 [[gamelismic]] temperament. In either case, it enables the [[slendric pentad]], and the perfect fifth is split into three equal parts, one for 8/7 and two for [[21/16]]. In addition, the [[256/243|Pythagorean limma (256/243)]] is also split into three, one for 64/63[[~]]49/48 and two for [[28/27]]. It therefore provides the little interval known as a [[quark]].
+[[Tempering out]] this comma alone in the 2.3.7 subgroup leads to the rank-2 [[slendric]] temperament, or in the full 7-limit, the rank-3 [[gamelismic]] temperament. In either case, it enables the [[slendric pentad]], and the perfect fifth is split into three equal parts, one for 8/7 and two for [[21/16]]. In addition, the Pythagorean limma is also split into three, one for 64/63[[~]]49/48 and two for [[28/27]]. It therefore provides the little interval known as a [[quark]].
 See [[Gamelismic family]] for the rank-3 family where it is tempered out. See [[Gamelismic clan]] for the rank-2 clan where it is tempered out.