847/845

Interval information

Ratio	847/845
Factorization	5-1 × 7 × 112 × 13-2
Monzo	[0 0 -1 1 2 -2⟩
Size in cents	4.092754¢
Name	cuthbert comma
Color name	3uu1oozg1, thuthulolozogu unison
FJS name	[math]\displaystyle{ \text{P1}^{7,11,11}_{5,13,13} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	19.449
Weil norm (log2 max(n, d))	19.4524
Wilson norm (sopfr(nd))	60
Comma size	small
S-expression	S11/S13
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847/845, the cuthbert comma, is a small 13-limit (also 5.7.11.13-subgroup) comma measuring about 4.09 ¢. It is the difference between 7/5 and a stack of two 13/11's. It is also the difference between 125/121 and 175/169.

In terms of full 13-limit commas, it is the difference between the following superparticular pairs:

• 66/65 and 78/77
• 121/120 and 169/168
• 352/351 and 2080/2079
• 385/384 and 4225/4224

Meanwhile, it can be factorized as (1001/1000)⋅(2200/2197) or (441/440)⋅(10648/10647).

In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, 637/625 is about as simple but much larger, and 2941225/2924207 is significantly more complex yet still twice as large.

Temperaments

Tempering out this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the cuthbert chords.

Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament, and tempering it out in the 7/5.11/5.13/5 subgroup leads to a rank-2 temperament containing the 5:7:11:13 chord. Both have a natural weak extension to prime 3 by tempering out 1575/1573, enabling 5:7:11:13:15.