19-limit
The 19-limit consists of just intonation intervals whose ratios contain no prime factors higher than 19. It is the 8th prime limit and is thus a superset of the 17-limit and a subset of the 23-limit.
The 19-limit is a rank-8 system, and can be modeled in a 7-dimensional lattice, with the primes 3, 5, 7, 11, 13, 17, and 19 represented by each dimension. The prime 2 does not appear in the typical 19-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, an eighth dimension is need.
Edo approximations
A list of edos with progressively better tunings for 19-limit intervals: 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h and so on.
Another list of edos which provides relatively good tunings for 19-limit intervals (relative error < 5%): 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and so on.
- Note: wart notation is used to specify the val chosen for the edo. In the above list, "152fg" means taking the second closest approximation of harmonics 13 and 17.
Intervals
Here are all the 21-odd-limit intervals of 19-limit:
| Ratio | Cents Value | Color Name | Interval Name | |
|---|---|---|---|---|
| 20/19 | 88.801 | 19uy1 | nuyo 1son | small undevicesimal semitone |
| 19/18 | 93.603 | 19o2 | ino 2nd | large undevicesimal semitone |
| 21/19 | 173.268 | 19uz2 | nuzo 2nd | small undevicesimal whole tone |
| 19/17 | 192.558 | 19o17u2 | nosu 2nd | large undevicesimal whole tone, quasi-meantone |
| 22/19 | 253.805 | 19u1o2 | nulo 2nd | undevicesimal second-third |
| 19/16 | 297.513 | 19o3 | ino 3rd | undevicesimal minor third |
| 24/19 | 404.442 | 19u3 | inu 3rd | small undevicesimal major third |
| 19/15 | 409.244 | 19og4 | nogu 4th | large undevicesimal major third |
| 19/14 | 528.687 | 19or4 | noru 4th | undevicesimal acute fourth |
| 26/19 | 543.015 | 19u3o4 | nutho 4th | undevicesimal super fourth |
| 19/13 | 656.985 | 19o3u5 | nothu 5th | undevicesimal subfifth |
| 28/19 | 671.313 | 19uz5 | nuzo 5th | undevicesimal gravefifth |
| 30/19 | 790.756 | 19uy5 | nuyo 5th | small undevicesimal minor sixth |
| 19/12 | 795.558 | 19o6 | ino 6th | large undevicesimal minor sixth |
| 32/19 | 902.487 | 19u6 | inu 6th | undevicesimal major sixth |
| 19/11 | 946.195 | 19o1u7 | nolu 7th | undevicesimal sixth-seventh |
| 34/19 | 1007.442 | 19u17o7 | nuso 7th | small undevicesimal minor seventh |
| 38/21 | 1026.732 | 19or7 | noru 7th | large undevicesimal minor seventh |
| 36/19 | 1106.397 | 19u7 | inu 7th | small undevicesimal major seventh |
| 19/10 | 1111.199 | 19og8 | nogu 8ve | large undevicesimal major seventh |