6656/6561: Difference between revisions
m →Tetris |
|||
| Line 23: | Line 23: | ||
=== No-5's Leapday === | === No-5's Leapday === | ||
In regular [[13-limit]] [[leapday]], the mapping for prime 5 is very complex at -25 gens, which is also in the opposite direction of how all other primes in the [[13-limit]] are reached. Furthermore, adding prime 5 to rank 3 [[parapythic]] is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable [[patent val]] tuning not appearing in the [[optimal ET sequence]] is [[80edo]], which is approximately the just-13's tuning (as [[10edo]] is used as a [[consistent circle]] of [[~]][[16/13]]'s therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup. In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which this case is also a sign of leapday being very efficient. | In regular [[13-limit]] [[leapday]], the mapping for prime 5 is very complex at -25 gens, which is also in the opposite direction of how all other primes in the [[13-limit]] are reached. Furthermore, adding prime 5 to rank 3 [[parapythic]] is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable [[patent val]] tuning not appearing in the [[optimal ET sequence]] is [[80edo]], which is approximately the just-13's tuning (as [[10edo]] is used as a [[consistent circle]] of [[~]][[16/13]]'s therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which this case is also a sign of leapday being very efficient. | ||
[[Subgroup]]: 2.3.7.11.13 | [[Subgroup]]: 2.3.7.11.13 | ||
Revision as of 00:49, 22 October 2024
| Interval information |
The tetris comma is the amount by which 13/8 exceeds (9/8)4, that is, the tetratone.
Temperaments
When tempered, it implies a sharp fifth or a doubly as sharp tone, and it features as the important comma that reduces rank 3 parapythic to no-5's leapday, which is notable as having much lower badness, as discussed there on this page.
Tetris
If tempered only on the 2.3.13 subgroup, you get tetris. 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 17 + 29 = 46edo, 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 no-5's 13-limit parapythic temperament documented below as no-5's leapday; 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
No-5's Leapday
In regular 13-limit leapday, the mapping for prime 5 is very complex at -25 gens, which is also in the opposite direction of how all other primes in the 13-limit are reached. Furthermore, adding prime 5 to rank 3 parapythic is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable patent val tuning not appearing in the optimal ET sequence is 80edo, which is approximately the just-13's tuning (as 10edo is used as a consistent circle of ~16/13's therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which this case is also a sign of leapday being very efficient.
Subgroup: 2.3.7.11.13
Comma list: 896/891, 352/351, 6656/6561
Mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]
- mapping generators: ~2, ~3, ~7, ~11, ~13
Optimal tuning (CTE): 2 = 1\1, ~3/2 = 704.633
Optimal ET sequence: 12de, 17, 46, 63
Badness (Dirichlet): 0.436