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}}. 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 | 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). | ||
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[[Category:Chalmersic]] | [[Category:Chalmersic]] | ||
[[Category:Commas named after | [[Category:Commas named after music theorists]] | ||
Latest revision as of 17:34, 25 December 2024
| 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 ¢. 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
| [⟨ | 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
- 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