19-limit

Prime 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 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 needed.

These things are contained by the 19-limit, but not the 17-limit:

The 19- and 21-odd-limit;
Mode 10 and 11 of the harmonic or subharmonic series.

Terminology and notation

Interval categories of HC19 are relatively clear. 19/16 is most commonly considered a minor third, as 1–19/16–3/2 is an important tertian chord (the Functional Just System and Helmholtz–Ellis notation agree). However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if 17/16 is considered an augmented unison and/or if 23/16 is considered an augmented fourth. Perhaps most interestingly, Sagittal notation provides an accidental to enharmonically spell intervals of HC19 this way.

Edo approximation

Here is a list of edos with progressively better tunings for 19-limit intervals (monotonicity limit ≥ 19 and decreasing TE error): 34dh, 38df, 41, 50, 53, 58h, 68, 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. For a more comprehensive list, see Sequence of equal temperaments by error.

Here is a list of edos which provides relatively good tunings for 19-limit intervals (TE 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, "34dh" means taking the second closest approximations of harmonics 7 and 19.

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