# 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

Joseph Monzo