Chalmersia
Ratio | 123201/123200 |
Factorization | 2^{-6} × 3^{6} × 5^{-2} × 7^{-1} × 11^{-1} × 13^{2} |
Monzo | [-6 6 -2 -1 -1 2⟩ |
Size in cents | 0.014052167¢ |
Name | chalmersia |
Color name | Lathotholurugugu comma |
FJS name | [math]\text{d1}^{13,13}_{5,5,7,11}[/math] |
Special properties | square superparticular, reduced |
Tenney height (log_{2} nd) | 33.8213 |
Weil height (log_{2} max(n, d)) | 33.8213 |
Wilson height (sopfr (nd)) | 84 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.44129 bits |
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 features highly 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 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