Chalmersia: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Ratio = 123201/123200

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

~~| Cents = 0.01405~~

| Name = chalmersia

| 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]]).

== 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~~]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
 | Ratio = 123201/123200
-| Monzo = -6 6 -2 -1 -1 2
-| Cents = 0.01405
 | Name = chalmersia
 | 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]]).
+== 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]]