Chalmersia: Difference between revisions

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The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. ~~Named in honor of [[John H. Chalmers]], 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.

The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. 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 properties characteristic of '''chalmersic 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 splits [[81/77]] into two [[40/39]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].

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

== Temperaments ==

Tempering out the comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''.

[[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:

* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9539, ~5/4 = 386.3145, ~7/4 = 3368.8265, ~351/280 = 391.2462

* [[CWE]]: ~2 = 1\1, ~3/2 = 701.9536, ~5/4 = 386.3140, ~7/4 = 3368.8259, ~351/280 = 391.2461

{{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, … }}

== Etymology ==

The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]].

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>.

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

~~— Gene~~ Ward Smith~~, [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html ''Nameable 13-limit''].~~

—Gene Ward Smith

== See also ==

Line 18:

Line 46:

* [[List of superparticular intervals]]

[[Category:~~Chalmers~~]]

== Notes ==

[[Category:Chalmersic]]

[[Category:Commas named after music theorists]]

@@ Line 6: / Line 6: @@
 }}
-The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. Named in honor of [[John H. Chalmers]], 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.
+The '''chalmersia''' is an [[unnoticeable comma|unnoticeable]] [[13-limit]] comma with a ratio of '''123201/123200''' and a size of approximately 0.014{{cent}}. 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 properties characteristic of '''chalmersic 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 splits [[81/77]] into two [[40/39]]s; and it splits the pythagorean limma [[256/243]] into [[26/25]] and [[78/77]].
+It factors into the two smallest 17-limit superparticular ratios: 123201/123200 = (194481/194480)(336141/336140).
+== Temperaments ==
+Tempering out the comma in the full 13-limit gives the rank-5 '''chalmersic temperament'''.
+[[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:
+* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9539, ~5/4 = 386.3145, ~7/4 = 3368.8265, ~351/280 = 391.2462
+* [[CWE]]: ~2 = 1\1, ~3/2 = 701.9536, ~5/4 = 386.3140, ~7/4 = 3368.8259, ~351/280 = 391.2461
+{{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, … }}
 == Etymology ==
-The chalmersia was named by [[Gene Ward Smith]] in 2003 after [[John Chalmers]].
+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>.
 :''The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.''
-— Gene Ward Smith, [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7316.html ''Nameable 13-limit''].
+—Gene Ward Smith
 == See also ==
@@ Line 18: / Line 46: @@
 * [[List of superparticular intervals]]
-[[Category:Chalmers]]
+== Notes ==
+[[Category:Chalmersic]]
+[[Category:Commas named after music theorists]]