Chalmersia

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.

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

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.

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

—Gene Ward Smith

See also

• Unnoticeable comma
• List of superparticular intervals