11859211/11859210
11859211/11859210, the tredekisma, is the smallest superparticular ratio in the 19-limit and 23-limit.
| Interval information |
19o^43o1u^4zg4,
Quadno-athoquadlu-azogu comma
reduced
It can be seen as the difference between a stack of a 25/16 plus a minisma plus a Saquadnu comma, and a diminished fifth plus two Axirabian limmas:
(25/16)(4096/4095)(131072/130321)÷(1024/729)(128/121)^2
Its S-expression is (S76)(S441)÷(S77)(S324)
Temperaments
Only being 0.00014 cents, it is an insanely, unfathomably narrow comma; hundreds of times narrower than well known atomic commas like the Chalmersia and Kirnberger's atom. As such, tempering out or fudging this alone either way is aurally identical to JI to human ears. It is tempered out in notable edos like 24, 84, 94, 118, 152, 176, 364, 388, 624, 742, 3125, 8269, 8539... among others. It is also tempered out in any union between these edos, such as vishnu and gariwizmic.
Tredekic
Subgroup: 2.3.5.7.11.13.19
| [⟨ | 1 | 0 | 0 | 0 | 0 | 1 | 0 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 3 | -1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | -4 | 1 | ]] |
mapping generators: ~2, ~3, ~5, ~7, ~11, ~133/33
Optimal tunings: TBA
Optimal ET sequence: TBA
Badness (Sintel): TBA
Tredekismic
Subgroup: 2.3.5.7.11.13.17.19
| [⟨ | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 3 | 0 | -1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ]] |
mapping generators: ~2, ~3, ~5, ~7, ~11, ~17, ~133/33
Optimal tunings: TBA
Optimal ET sequence: TBA
Badness (Sintel): TBA
Etymology
The name tredekisma comes from tredek, which is one step of 270edo, which is an excellent 2.3.5.7.11.13.19 subgroup system that tempers it out.