19-limit: Difference between revisions

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In 19-limit [[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: {{EDOs| 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"
Line 15: Line 28:
! Interval Name
! Interval Name
|-
|-
| 20/19
| [[20/19]]
| 88.801
| 88.801
| 19uy1
| 19uy1
Line 21: Line 34:
| small undevicesimal semitone
| small undevicesimal semitone
|-
|-
| 19/18
| [[19/18]]
| 93.603
| 93.603
| 19o2
| 19o2
Line 27: Line 40:
| large undevicesimal semitone
| large undevicesimal semitone
|-
|-
| 21/19
| [[21/19]]
| 173.268
| 173.268
| 19uz2
| 19uz2
Line 33: Line 46:
| small undevicesimal whole tone
| small undevicesimal whole tone
|-
|-
| 19/17
| [[19/17]]
| 192.558
| 192.558
| 19o17u2
| 19o17u2
Line 39: Line 52:
| large undevicesimal whole tone, quasi-meantone
| large undevicesimal whole tone, quasi-meantone
|-
|-
| 22/19
| [[22/19]]
| 253.805
| 253.805
| 19u1o2
| 19u1o2
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| undevicesimal second-third
| undevicesimal second-third
|-
|-
| 19/16
| [[19/16]]
| 297.513
| 297.513
| 19o3
| 19o3
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| undevicesimal minor third
| undevicesimal minor third
|-
|-
| 24/19
| [[24/19]]
| 404.442
| 404.442
| 19u3
| 19u3
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| small undevicesimal major third
| small undevicesimal major third
|-
|-
| 19/15
| [[19/15]]
| 409.244
| 409.244
| 19og4
| 19og4
Line 63: Line 76:
| large undevicesimal major third
| large undevicesimal major third
|-
|-
| 19/14
| [[19/14]]
| 528.687
| 528.687
| 19or4
| 19or4
Line 69: Line 82:
| undevicesimal acute fourth
| undevicesimal acute fourth
|-
|-
| 26/19
| [[26/19]]
| 543.015
| 543.015
| 19u3o4
| 19u3o4
Line 75: Line 88:
| 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
Line 93: Line 106:
| small undevicesimal minor sixth
| small undevicesimal minor sixth
|-
|-
| 19/12
| [[19/12]]
| 795.558
| 795.558
| 19o6
| 19o6
Line 99: Line 112:
| large undevicesimal minor sixth
| large undevicesimal minor sixth
|-
|-
| 32/19
| [[32/19]]
| 902.487
| 902.487
| 19u6
| 19u6
Line 105: Line 118:
| undevicesimal major sixth
| undevicesimal major sixth
|-
|-
| 19/11
| [[19/11]]
| 946.195
| 946.195
| 19o1u7
| 19o1u7
Line 111: Line 124:
| undevicesimal sixth-seventh
| undevicesimal sixth-seventh
|-
|-
| 34/19
| [[34/19]]
| 1007.442
| 1007.442
| 19u17o7
| 19u17o7
Line 117: Line 130:
| small undevicesimal minor seventh
| small undevicesimal minor seventh
|-
|-
| 38/21
| [[38/21]]
| 1026.732
| 1026.732
| 19or7
| 19or7
Line 123: Line 136:
| large undevicesimal minor seventh
| large undevicesimal minor seventh
|-
|-
| 36/19
| [[36/19]]
| 1106.397
| 1106.397
| 19u7
| 19u7
Line 129: Line 142:
| small undevicesimal major seventh
| small undevicesimal major seventh
|-
|-
| 19/10
| [[19/10]]
| 1111.199
| 1111.199
| 19og8
| 19og8
Line 136: Line 149:
|}
|}


== See Also ==
== Music ==
; [[Domin]]
* [https://www.youtube.com/watch?v=WTo5YihoLqs ''Asuttan''] (2024)
* [https://www.youtube.com/watch?v=OPt3Y9VSliU ''Asuttan Bouta''] (2024)


* [[Harmonic limit]]
; [[Joseph Monzo]]
* [[19-odd-limit]]
* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)


[[Category:19-limit| ]] <!-- main article -->
[[Category:19-limit| ]] <!-- main article -->
[[Category:Limit]]
[[Category:Prime limit]]

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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