Chalmersia: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Ratio = 123201/123200

~~| Monzo = -6 6 -2 -1 -1 2~~

| Name = chalmersia

~~| Cents = 0.01405~~

| Name = chalmersia~~, chalmersma, chalmersima, chalmerisma, chalmersisma~~

| Color name = Lathotholurugugu comma

| ~~FJS name =~~

| Comma = yes

~~| Sound~~ =

}}

The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[comma]] with a [[ratio]] of '''123201/123200''' and a size of approximately 0.014 [[cent]]s. It is the smallest 13-limit [[superparticular]] comma.

~~The '''chalmersia''' is an [~~[~~unnoticeable comma|unnoticeable]]~~ [~~[13~~-limit]] ~~comma with a ratio of '''~~123201/123200''' and a value of approximately 0.014 [[cent]]s. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two – these are features highly characteristic of '''chalmers temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] ~~augmented fifth plus a~~ [[77/64]] ~~minor third~~.

It factors into the two smallest [[17-limit]] superparticular ratios: 123201/123200 = ([[194481/194480]])⋅([[336141/336140]]).

~~Although "chalmersia" is~~ the ~~name that is primarily used for this article as~~ it ~~is the name for this comma on~~ the [~~http~~://~~www~~.~~huygens-fokker~~.~~org~~/~~docs~~/~~intervals~~.~~html Huygens-Fokker Foundation's interval list~~], ~~there are other similar names that this comma sometimes goes by~~, ~~including '''chalmersma'''~~, ~~'''chalmersima'''~~, ~~'''chalmerisma''' and '''chalmersisma'''~~. ~~Regardless of the exact name used for this comma~~, ~~it seems that the temperament itself~~ was named in ~~honor of~~ [[John H. Chalmers]], ~~and that all~~ the ~~various names for this comma are thus etymologically derived from the temperament name~~.

== Temperaments ==

[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''. It equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]'s; it splits [[11/7]] into two [[351/280]]'s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].

[[Subgroup]]: 2.3.5.7.11.13

[[Mapping]]:

{| class="right-all"

| [⟨ || 1 || 1 || 2 || 2 || 2 || 4 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || -3 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]]

|}

: mapping generators: ~2, ~3, ~5, ~7, ~351/280

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.000379{{c}}, ~3/2 = 701.953671{{c}}, ~5/4 = 386.313637{{c}}, ~7/4 = 968.825646{{c}}, ~351/280 = 391.246147{{c}}

* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.953639{{c}}, ~5/4 = 386.313976{{c}}, ~7/4 = 968.825869{{c}}, ~351/280 = 391.246091{{c}}

{{Optimal ET sequence|legend=1| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}

[[Badness]] (Sintel): 0.0267

== Etymology ==

The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html Yahoo! Tuning Group | ''Nameable 13-limit'']</ref>.

<blockquote>The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.</blockquote>

—Gene Ward Smith

== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

~~[[Category:13-limit]]~~

== References ==

~~[[Category:Unnoticeable comma]]~~

~~[[Category:Ratio]]~~

[[Category:Chalmersic]]

[[Category:~~Superparticular~~]]

[[Category:Commas named after music theorists]]

[[Category:~~Chalmers~~]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
 | Ratio = 123201/123200
-| Monzo = -6 6 -2 -1 -1 2
+| Name = chalmersia
-| Cents = 0.01405
-| Name = chalmersia, <br> chalmersma, <br> chalmersima, <br> chalmerisma, <br> chalmersisma
 | Color name = Lathotholurugugu comma
-| FJS name =
+| Comma = yes
-| Sound =
 }}
+The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[comma]] with a [[ratio]] of '''123201/123200''' and a size of approximately 0.014 [[cent]]s. It is the smallest 13-limit [[superparticular]] comma.
-The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a value of approximately 0.014 [[cent]]s. It is the smallest 13-limit [[superparticular]] comma. Tempering it out equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two – these are features highly characteristic of '''chalmers temperaments'''. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third.
+It factors into the two smallest [[17-limit]] superparticular ratios: 123201/123200 = ([[194481/194480]])⋅([[336141/336140]]).
-Although "chalmersia" is the name that is primarily used for this article as it is the name for this comma on the [http://www.huygens-fokker.org/docs/intervals.html Huygens-Fokker Foundation's interval list], there are other similar names that this comma sometimes goes by, including '''chalmersma''', '''chalmersima''', '''chalmerisma''' and '''chalmersisma'''.  Regardless of the exact name used for this comma, it seems that the temperament itself was named in honor of [[John H. Chalmers]], and that all the various names for this comma are thus etymologically derived from the temperament name.
+== Temperaments ==
+[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''. It equates [[351/350]] and [[352/351]], thus splitting [[176/175]] into two, and equates 385/351 and 351/320, thus splitting [[77/64]] into two. In addition, it equates a stack consisting of a [[729/512]] tritone plus a [[169/128]] grave fourth with a stack consisting of a [[25/16]] augmented fifth plus a [[77/64]] minor third; it splits [[81/77]] into two [[40/39]]'s; it splits [[11/7]] into two [[351/280]]'s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].
+[[Subgroup]]: 2.3.5.7.11.13
+[[Mapping]]: <br>
+{| class="right-all"
+| [⟨ || 1 || 1 || 2 || 2 || 2 || 4 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || -3 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~351/280
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.000379{{c}}, ~3/2 = 701.953671{{c}}, ~5/4 = 386.313637{{c}}, ~7/4 = 968.825646{{c}}, ~351/280 = 391.246147{{c}}
+* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.953639{{c}}, ~5/4 = 386.313976{{c}}, ~7/4 = 968.825869{{c}}, ~351/280 = 391.246091{{c}}
+{{Optimal ET sequence|legend=1| 12f, 19e, 22, 27e, 31, 46, 53, 58, 80, 104c, 111, 159, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1236, 1448, 1506, 2190, 2684, 3395, 4079, 4349, 4843, 5585, 6079, 8269, 8539, … }}
+[[Badness]] (Sintel): 0.0267
+== Etymology ==
+The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html Yahoo! Tuning Group | ''Nameable 13-limit'']</ref>.
+<blockquote>The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.</blockquote>
+—Gene Ward Smith
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-[[Category:13-limit]]
+== References ==
-[[Category:Unnoticeable comma]]
-[[Category:Ratio]]
+[[Category:Chalmersic]]
-[[Category:Superparticular]]
+[[Category:Commas named after music theorists]]
-[[Category:Chalmers]]