2197/2187

The threedie, 2197/2187, is a 3.13-subgroup analog of the 2.5-subgroup diesis. Measuring about 7.9 ¢, it is a small comma.

Interval information

Ratio	2197/2187
Factorization	3-7 × 133
Monzo	[0 -7 0 0 0 3⟩
Size in cents	7.897979¢
Name	threedie
Color name	s3o32, satritho 2nd, Satritho comma
FJS name	[math]\displaystyle{ \text{dd2}^{13,13,13} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	22.1961
Weil norm (log2 max(n, d))	22.2026
Wilson norm (sopfr(nd))	60
Comma size	small
S-expression	S25⋅S262
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Temperaments

Analogous to tempering out the diesis, which divides the octave into three sharp 5/4's, tempering out the threedie divides the tritave into three slightly flat 13/9's, since (13/9)³/3 = 2197/2187. Furthermore, tempering out the threedie divides 9/8 into three instances of 27/26. 596edo provides the optimal patent val for the corresponding rank-5 temperament.

It factors into simpler commas in the 2.3.5.13 subgroup as (325/324)(676/675), and tempering out both of them leads to the 2.3.5.13 kleismic temperament (also known as cata). It can also be expressed in the 2.3.7.13 subgroup as (169/168)/(729/728), i.e. S13/S27.