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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]].  


==19-odd limit Intervals==
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.
 
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 ==
 
Here are all the [[21-odd-limit]] intervals of 19-limit:


{| class="wikitable"
{| class="wikitable"
! Ratio
! Cents Value
! colspan="2" | Color Name
! Interval Name
|-
|-
! | Ratio
| [[20/19]]
! | Cents Value
| 88.801
! | Name
| 19uy1
| nuyo 1son
| small undevicesimal semitone
|-
|-
| | [[20/19|20/19]]
| [[19/18]]
| | 88.801
| 93.603
| | lesser undevicesimal semitone
| 19o2
| ino 2nd
| large undevicesimal semitone
|-
|-
| | [[19/18|19/18]]
| [[21/19]]
| | 93.603
| 173.268
| | greater undevicesimal semitone
| 19uz2
| nuzo 2nd
| small undevicesimal whole tone
|-
|-
| | [[19/17|19/17]]
| [[19/17]]
| | 192.558
| 192.558
| | undevicesimal whole tone ("meantone")
| 19o17u2
| nosu 2nd
| large undevicesimal whole tone, quasi-meantone
|-
|-
| | [[22/19|22/19]]
| [[22/19]]
| | 253.805
| 253.805
| | enneadecimal second–third
| 19u1o2
| nulo 2nd
| undevicesimal second-third
|-
|-
| | [[19/16|19/16]]
| [[19/16]]
| | 297.513
| 297.513
| | undevicesimal minor third
| 19o3
| ino 3rd
| undevicesimal minor third
|-
|-
| | [[24/19|24/19]]
| [[24/19]]
| | 404.442
| 404.442
| | lesser undevicesimal major third
| 19u3
| inu 3rd
| small undevicesimal major third
|-
|-
| | [[19/15|19/15]]
| [[19/15]]
| | 409.244
| 409.244
| | greater undevicesimal major third
| 19og4
| nogu 4th
| large undevicesimal major third
|-
|-
| | [[19/14|19/14]]
| [[19/14]]
| | 528.687
| 528.687
| | undevicesimal acute fourth
| 19or4
| noru 4th
| undevicesimal acute fourth
|-
|-
| | [[26/19|26/19]]
| [[26/19]]
| | 543.015
| 543.015
| | undevicesimal superfourth
| 19u3o4
| nutho 4th
| undevicesimal super fourth
|-
|-
| | [[19/13|19/13]]
| [[19/13]]
| | 656.985
| 656.985
| | undevicesimal subfifth
| 19o3u5
| nothu 5th
| undevicesimal subfifth
|-
|-
| | [[28/19|28/19]]
| [[28/19]]
| | 671.313
| 671.313
| | undevicesimal grave fifth
| 19uz5
| nuzo 5th
| undevicesimal gravefifth
|-
|-
| | [[30/19|30/19]]
| [[30/19]]
| | 790.756
| 790.756
| | lesser undevicesimal minor sixth
| 19uy5
| nuyo 5th
| small undevicesimal minor sixth
|-
|-
| | [[19/12|19/12]]
| [[19/12]]
| | 795.558
| 795.558
| | lesser undevicesimal minor sixth
| 19o6
| ino 6th
| large undevicesimal minor sixth
|-
|-
| | [[32/19|32/19]]
| [[32/19]]
| | 902.487
| 902.487
| | undevicesimal major sixth
| 19u6
| inu 6th
| undevicesimal major sixth
|-
|-
| | [[19/11|19/11]]
| [[19/11]]
| | 946.195
| 946.195
| | enneadecimal sixth–seventh
| 19o1u7
| nolu 7th
| undevicesimal sixth-seventh
|-
|-
| | [[34/19|34/19]]
| [[34/19]]
| | 1007.442
| 1007.442
| | undevicesimal minor seventh
| 19u17o7
| nuso 7th
| small undevicesimal minor seventh
|-
|-
| | [[36/19|36/19]]
| [[38/21]]
| | 1106.397
| 1026.732
| | lesser undevicesimal major seventh
| 19or7
| noru 7th
| large undevicesimal minor seventh
|-
|-
| | [[19/10|19/10]]
| [[36/19]]
| | 1111.199
| 1106.397
| | greater undevicesimal major seventh
| 19u7
| inu 7th
| small undevicesimal major seventh
|-
| [[19/10]]
| 1111.199
| 19og8
| nogu 8ve
| large undevicesimal major seventh
|}
|}


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

Latest revision as of 16:23, 20 August 2025

← 17-limit  19-limit  23-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

Music

Domin
Joseph Monzo
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