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In 19-limit [[Just_intonation|Just Intonation]], all ratios in the system will contain no primes higher than 19.
{{Prime limit navigation|19}}
The '''19-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s 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-prime-limit 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.
The 19-limit is a [[rank and codimension|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.


[[EDO]]s which provides an excellent tuning for 19-limit intervals are: 80, 94, 111, 121, 217, 270, 282, 311, 320, 364, 388, 400, 422, 436, 460, 525, 581, 597, 624, 643, 653, 692, 718, 742, 771, 860, 867, 882, 908, 925, 935, 954, and 997 among others.
These things are contained by the 19-limit, but not the 17-limit:
* The [[19-odd-limit|19-]] and [[21-odd-limit]];
* Mode 10 and 11 of the harmonic or subharmonic series.
 
== Terminology and notation ==
[[Interval_region|Interval categories]] of [[harmonic class|HC19]] are relatively clear. [[19/16]] is most commonly considered a minor third, as 1–19/16–3/2 is an important {{w|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 [[edo]]s with progressively better tunings for 19-limit intervals ([[monotonicity limit]] ≥ 19 and decreasing [[TE error]]): {{EDOs| 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%): {{EDOs| 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 ==
== Intervals ==
Here are all the 21-odd-limit intervals of 19:  
 
Here are all the [[21-odd-limit]] intervals of 19-limit:  
 
{| class="wikitable"
{| class="wikitable"
!Ratio
! Ratio
!Cents Value
! Cents Value
! colspan="2" |Color Name
! colspan="2" | Color Name
!Interval Name
! Interval Name
|-
|-
|20/19
| [[20/19]]
|88.801
| 88.801
|19uy1
| 19uy1
|nuyo 1son
| nuyo 1son
|small undevicesimal semitone
| small undevicesimal semitone
|-
|-
|19/18
| [[19/18]]
|93.603
| 93.603
|19o2
| 19o2
|ino 2nd
| ino 2nd
|large undevicesimal semitone
| large undevicesimal semitone
|-
|-
|21/19
| [[21/19]]
|173.268
| 173.268
|19uz2
| 19uz2
|nuzo 2nd
| nuzo 2nd
|small undevicesimal whole tone
| small undevicesimal whole tone
|-
|-
|19/17
| [[19/17]]
|192.558
| 192.558
|19o17u2
| 19o17u2
|nosu 2nd
| nosu 2nd
|large undevicesimal whole tone, quasi-meantone
| large undevicesimal whole tone, quasi-meantone
|-
|-
|22/19
| [[22/19]]
|253.805
| 253.805
|19u1o2
| 19u1o2
|nulo 2nd
| nulo 2nd
|undevicesimal second-third
| undevicesimal second-third
|-
|-
|19/16
| [[19/16]]
|297.513
| 297.513
|19o3
| 19o3
|ino 3rd
| ino 3rd
|undevicesimal minor third
| undevicesimal minor third
|-
|-
|24/19
| [[24/19]]
|404.442
| 404.442
|19u3
| 19u3
|inu 3rd
| inu 3rd
|small undevicesimal major third
| small undevicesimal major third
|-
|-
|19/15
| [[19/15]]
|409.244
| 409.244
|19og4
| 19og4
|nogu 4th
| nogu 4th
|large undevicesimal major third
| large undevicesimal major third
|-
|-
|19/14
| [[19/14]]
|528.687
| 528.687
|19or4
| 19or4
|noru 4th
| noru 4th
|undevicesimal acute fourth
| undevicesimal acute fourth
|-
|-
|26/19
| [[26/19]]
|543.015
| 543.015
|19u3o4
| 19u3o4
|nutho 4th
| nutho 4th
|undevicesimal super fourth
| undevicesimal super fourth
|-
|-
|19/13
| [[19/13]]
|656.985
| 656.985
|19o3u5
| 19o3u5
|nothu 5th
| nothu 5th
|undevicesimal sub fifth
| undevicesimal subfifth
|-
|-
|28/19
| [[28/19]]
|671.313
| 671.313
|19uz5
| 19uz5
|nuzo 5th
| nuzo 5th
|undevicesimal grave fifth
| undevicesimal gravefifth
|-
|-
|30/19
| [[30/19]]
|790.756
| 790.756
|19uy5
| 19uy5
|nuyo 5th
| nuyo 5th
|small undevicesimal minor sixth
| small undevicesimal minor sixth
|-
|-
|19/12
| [[19/12]]
|795.558
| 795.558
|19o6
| 19o6
|ino 6th
| ino 6th
|large undevicesimal minor sixth
| large undevicesimal minor sixth
|-
|-
|32/19
| [[32/19]]
|902.487
| 902.487
|19u6
| 19u6
|inu 6th
| inu 6th
|undevicesimal major sixth
| undevicesimal major sixth
|-
|-
|19/11
| [[19/11]]
|946.195
| 946.195
|19o1u7
| 19o1u7
|nolu 7th
| nolu 7th
|undevicesimal sixth-seventh
| undevicesimal sixth-seventh
|-
|-
|34/19
| [[34/19]]
|1007.442
| 1007.442
|19u17o7
| 19u17o7
|nuso 7th
| nuso 7th
|small undevicesimal minor seventh
| small undevicesimal minor seventh
|-
|-
|38/21
| [[38/21]]
|1026.732
| 1026.732
|19or7
| 19or7
|noru 7th
| noru 7th
|large undevicesimal minor seventh
| large undevicesimal minor seventh
|-
|-
|36/19
| [[36/19]]
|1106.397
| 1106.397
|19u7
| 19u7
|inu 7th
| inu 7th
|small undevicesimal major seventh
| small undevicesimal major seventh
|-
|-
|19/10
| [[19/10]]
|1111.199
| 1111.199
|19og8
| 19og8
|nogu 8ve
| nogu 8ve
|large undevicesimal major seventh
| large undevicesimal major seventh
|}
|}


== See Also ==
== Music ==
* [[Harmonic_Limit|Harmonic Limit]]
; [[Domin]]
* [[19-odd-limit]]
* [https://www.youtube.com/watch?v=WTo5YihoLqs ''Asuttan''] (2024)
[[Category:19-limit]]
* [https://www.youtube.com/watch?v=OPt3Y9VSliU ''Asuttan Bouta''] (2024)
[[Category:limit]]
 
[[Category:prime_limit]]
; [[Joseph Monzo]]
[[Category:rank_8]]
* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)
 
[[Category:19-limit| ]] <!-- main article -->
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