19-limit: Difference between revisions

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 thus a superset of the [[17-limit]] and a subset of the [[23-limit]].  

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 need.

Intervals of [[harmonic class|HC19]] are relatively clear in terms of their categories. [[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.  

Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals (decreasing [[TE error]]): {{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.  

: '''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.