19-limit: Difference between revisions

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The '''19-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 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 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 ~~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 approximations ==~~

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

~~A list of~~ [[~~edo~~]]~~s with progressively better tunings for 19~~-limit ~~intervals: {{EDOs| 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~~.

* The [[19-odd-limit|19-]] and [[21-odd-limit]];

* Mode 10 and 11 of the harmonic or subharmonic series.

~~Another list~~ of ~~edos which provides~~ relatively ~~good tunings for 19-limit intervals (~~[[~~TE relative error|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~~.

== 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.

: '''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.

== 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 ==

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|}

== ~~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| ]]

@@ Line 1: / Line 1: @@
 {{Prime limit navigation|19}}
-The '''19-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 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 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 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 approximations ==
+These things are contained by the 19-limit, but not the 17-limit:
-A list of [[edo]]s with progressively better tunings for 19-limit intervals: {{EDOs| 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.
+* The [[19-odd-limit|19-]] and [[21-odd-limit]];
+* Mode 10 and 11 of the harmonic or subharmonic series.
-Another list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error|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.
+== 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.
-: '''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.
+== 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 ==
@@ Line 142: / 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 -->