Chalmersia: Difference between revisions

Line 5:

| Comma = yes

}}

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 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; it splits [[11/7]] into two [[351/280]]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]]).

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'''.

[[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

Line 31:

Line 30:

[[Optimal tuning]]s:

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

* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9539{{c}}, ~5/4 = 386.3145{{c}}, ~7/4 = 3368.8265{{c}}, ~351/280 = 391.2462{{c}}

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

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9536{{c}}, ~5/4 = 386.3140{{c}}, ~7/4 = 3368.8259{{c}}, ~351/280 = 391.2461{{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, … }}

Line 43:

Line 42:

== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

== ~~Notes~~ ==

== References ==

[[Category:Chalmersic]]

[[Category:Commas named after music theorists]]

@@ Line 5: / Line 5: @@
 | Comma = yes
 }}
+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 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; it splits [[11/7]] into two [[351/280]]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]]).
-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'''.
+[[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
@@ Line 31: / Line 30: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9539, ~5/4 = 386.3145, ~7/4 = 3368.8265, ~351/280 = 391.2462
+* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9539{{c}}, ~5/4 = 386.3145{{c}}, ~7/4 = 3368.8265{{c}}, ~351/280 = 391.2462{{c}}
-* [[CWE]]: ~2 = 1\1, ~3/2 = 701.9536, ~5/4 = 386.3140, ~7/4 = 3368.8259, ~351/280 = 391.2461
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9536{{c}}, ~5/4 = 386.3140{{c}}, ~7/4 = 3368.8259{{c}}, ~351/280 = 391.2461{{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, … }}
@@ Line 43: / Line 42: @@
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-== Notes ==
+== References ==
 [[Category:Chalmersic]]
 [[Category:Commas named after music theorists]]