# 19-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 thus 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 need.

## Terminology and notation

Intervals of 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 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 (decreasing TE error): 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.

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, "152fg" means taking the second closest approximation of harmonics 13 and 17.

## 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

*Asuttan*(2024)

*Theme from Invisible Haircut*(1990)