6656/6655
| Interval information |
reduced
6656/6655, the jacobin comma, is an unnoticeable 13-limit (also 2.5.11.13 subgroup) superparticular comma of about 0.26 ¢. It is the difference between a stack of three 11/8 superfourths and one 13/10 naiadic plus an octave (13/5).
In terms of commas, it is the difference between 364/363 and 385/384, between 2080/2079 and 3025/3024 as well as between 4096/4095 and 10648/10647. In the 17-limit, it factors neatly into (12376/12375)⋅(14400/14399).
Temperaments
By tempering it out, the jacobin temperament is defined. Interestingly, 1789edo is an edo that supports the jacobin temperament - and is consistent to distance 2 on the subgroup 2.5.11.13, the subgroup of the comma. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament. Miscellaneous temperaments tempering out this comma are collected in The Jacobins.
The 17-limit factorization shows us a natural path of extension by adding 12376/12375. Equally, it can be conceptualized as a rank-2 temperament in 2.13/10.11 subgroup, in which case it is called jacobin-naiadic, after 13/10 being called a naiadic by interval class.
See The Jacobins#Jacobin for technical data.
Etymology
This comma was apparently named by Gene Ward Smith in 2014 for reasons unknown[1]. An hypothesis is that "jacobin" refers to the Jacobins, a political group founded in 1789 during the French Revolution, since this comma is tempered out by 1789edo, a tuning which, furthermore, is consistent to distance 2 in the 2.5.11.13-subgroup-limited 13-odd-limit, reinforcing the importance of this comma in this tuning.