6656/6561
| Interval information |
(Shannon, [math]\sqrt{nd}[/math])
The 6656/6561, the tetris comma, is a small 13-limit comma. It is the amount by which 13/8 exceeds (9/8)4, that is, the tetratone.
It is a schisma flat of 65/64, and a Pythagorean comma flat of 1053/1024.
Temperaments
When tempered out, it implies a sharp fifth or a doubly as sharp tone, and it features as the important comma that reduces rank-3 parapythic to leapfrog (aka no-fives leapday), which is notable as having much lower badness.
Tetris
If tempered out only on the 2.3.13 subgroup, you get tetris (while tetric would fit the -ic convention better, it is already used for a MOS pattern). 17edo is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want 13/8 to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, 34edo may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate ~13/9 and still good ~13/12. By contrast, 29edo is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13 ¢ flat. Therefore through the addition of vals we can deduce that the smallest reasonably optimized tuning is 46edo since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related leapfrog temperament; this corresponds to having larger edos in the optimal ET sequence. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.
Subgroup: 2.3.13
Mapping: [⟨1 0 -9], ⟨0 1 8]]
- mapping generators: ~2, ~3, ~13
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 704.822
Optimal ET sequence: 5, 12, 17, 63, 80, 97, 114, 131, 245b
Badness (Dirichlet): 0.522
Leapfrog
As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding ~14/11 as the major third (tempering out 896/891) and ~13/11 as the minor third (tempering out 352/351) as well as tempering out the tetris comma, 6656/6561. We may also find ~23/16 as the tritone, tempering out 736/729. This results in a temperament called leapfrog, which is the no-5's version of leapday.
Etymology
This comma was named by Jerdle and Godtone in 2024 as a contraction of "tetratone" and "tridecimal".