Chalmersia: Difference between revisions
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changed the phrase "but these are probably mistakes" to "though at least some of these are mistakes" as judging from from some of the archived conversations on this wiki about this comma, it seems that some people actually think that the ending of the name "chalmersia" is a bit weird |
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 123201/123200 | | Ratio = 123201/123200 | ||
| Name = chalmersia | | Name = chalmersia | ||
| Color name = Lathotholurugugu comma | | Color name = Lathotholurugugu comma | ||
| | | 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. | |||
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 == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
== References == | == References == | ||
[[Category: | [[Category:Chalmersic]] | ||
[[Category: | [[Category:Commas named after music theorists]] | ||
Latest revision as of 15:14, 19 February 2026
| Interval information |
reduced
S78/S80
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
| [⟨ | 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
- 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