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]]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; 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 ==
== 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
[[Subgroup]]: 2.3.5.7.11.13
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| ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]]
| ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]]
|}
|}
: mapping generators: ~2, ~3, ~5, ~7, ~351/280
: mapping generators: ~2, ~3, ~5, ~7, ~351/280


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9539, ~5/4 = 386.3145, ~7/4 = 3368.8265, ~351/280 = 391.2462
* [[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 = 1\1, ~3/2 = 701.9536, ~5/4 = 386.3140, ~7/4 = 3368.8259, ~351/280 = 391.2461
* [[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, … }}
{{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 ==
== 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>.  
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.''
<blockquote>The remarkable 123201/123200 might be named the chalmersia, since John Chalmers is presumably the first to see it.</blockquote>
 
—Gene Ward Smith
—Gene Ward Smith


== See also ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


== Notes ==
== References ==


[[Category:Chalmersic]]
[[Category:Chalmersic]]
[[Category:Commas named after music theorists]]
[[Category:Commas named after music theorists]]

Latest revision as of 15:14, 19 February 2026

Interval information
Ratio 123201/123200
Factorization 2-6 × 36 × 5-2 × 7-1 × 11-1 × 132
Monzo [-6 6 -2 -1 -1 2⟩
Size in cents 0.01405217¢
Name chalmersia
Color name Lathotholurugugu comma
FJS name [math]\displaystyle{ \text{d1}^{13,13}_{5,5,7,11} }[/math]
Special properties square superparticular,
reduced
Tenney norm (log2 nd) 33.8213
Weil norm (log2 max(n, d)) 33.8213
Wilson norm (sopfr(nd)) 84
Comma size unnoticeable
S-expressions S351,
S78/S80
Open this interval in xen-calc

The chalmersia is an unnoticeable 13-limit comma with a ratio of 123201/123200 and a size of approximately 0.014 cents. It is the smallest 13-limit superparticular comma.

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:

[⟨ 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 tunings:

  • WE: ~2 = 1200.000379 ¢, ~3/2 = 701.953671 ¢, ~5/4 = 386.313637 ¢, ~7/4 = 968.825646 ¢, ~351/280 = 391.246147 ¢
  • CWE: ~2 = 1200.000000 ¢, ~3/2 = 701.953639 ¢, ~5/4 = 386.313976 ¢, ~7/4 = 968.825869 ¢, ~351/280 = 391.246091 ¢

Optimal ET sequence: 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[1].

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

—Gene Ward Smith

See also

References

  1. ↑ Yahoo! Tuning Group | Nameable 13-limit
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