Chalmersia: Difference between revisions

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After doing a little digging, it appears that all of the names for this comma are derived from the temperament name, and so far, it seems that temperament was named after John H. Chalmers- perhaps this is the key to resolving the argument over this comma's names
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{{Infobox Interval
{{Infobox Interval
| Icon =
| Ratio = 123201/123200
| 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
| 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 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.  
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]].


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 are thus etymologically derived from the temperament name.
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]]<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


== See also ==
== See also ==
Line 18: Line 46:
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:13-limit]]
== Notes ==
[[Category:Unnoticeable comma]]
 
[[Category:Ratio]]
[[Category:Chalmersic]]
[[Category:Superparticular]]
[[Category:Commas named after music theorists]]
[[Category:Chalmers]]

Latest revision as of 17:34, 25 December 2024

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 height (log2 nd) 33.8213
Weil height (log2 max(n, d)) 33.8213
Wilson height (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 ¢. 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 40/39s; 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:

[⟨ 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:

  • 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: 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[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

Notes

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