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 (decreasing TE error): 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.

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, "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