1716/1715

The lummic comma, 1716/1715, is an unnoticeable 13-limit superparticular comma measuring about 1.0 ¢. It often arises as the difference between simpler commas: (176/175)/(196/195), (351/350)/(441/440), (1573/1568)/(385/384), (540/539)/(1575/1573), and (1001/1000)/(2401/2400). It factors into (2080/2079)⋅(9801/9800).

Interval information

Ratio	1716/1715
Factorization	22 × 3 × 5-1 × 7-3 × 11 × 13
Monzo	[2 1 -1 -3 1 1⟩
Size in cents	1.009172¢
Name	lummic comma
Color name	3o1or3g-2, tholotriru-agu negative 2nd
FJS name	[math]\displaystyle{ \text{M}{-2}^{11,13}_{5,7,7,7} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	21.4888
Weil norm (log2 max(n, d))	21.4897
Wilson norm (sopfr(nd))	57
Comma size	unnoticeable
Open this interval in xen-calc

Temperaments

Tempering out this comma leads to the rank-5 lummic temperament. It is tempered out by several notable edos including 72edo, 270edo and 400edo among others.

Subgroup: 2.3.5.7.11.13

Mapping:

[⟨	1	0	0	0	0	-2	],
⟨	0	1	0	0	0	-1	],
⟨	0	0	1	0	0	1	],
⟨	0	0	0	1	0	3	],
⟨	0	0	0	0	1	-1	]]

mapping generators: ~2, ~3, ~5, ~7, ~11

Optimal tunings:

WE: ~2 = 1199.9814 ¢, ~3/2 = 701.9502 ¢, ~5/4 = 386.4011 ¢, ~7/4 = 969.0830 ¢, ~11/8 = 551.2624 ¢

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9462 ¢, ~5/4 = 386.3868 ¢, ~7/4 = 969.0766 ¢, ~11/8 = 551.2373 ¢

Optimal ET sequence: 22f, 26, 27e, 31, 45ef, 46, 58, 72, 103, 121, 130, 193, 198, 224, 270, 494, 764, 1137, 1258, 1289, 1419, 1559, 1689, 2183d

Badness (Sintel): 0.318

Etymology

This comma was named after Carl Lumma. Perhaps interestingly, a lummisma was first proposed by Gene Ward Smith in 2011 as a name for 1001/1000, but Carl Lumma pointed out that 1716/1715 was "better", which might ultimately led to this comma being named so.