# 13-limit

The **13-limit** or 13-prime-limit consists of just intonation intervals such that the highest prime factor in all ratios is 13. Thus, 40/39 would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but 34/33 would not, since 34 is 2 × 17, and 17 is a prime number higher than 13. The 13-limit is the 6th prime limit and is thus a superset of the 11-limit and a subset of the 17-limit.

The 13-limit is a rank-6 system, and can be modeled in a 5-dimensional lattice, with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a sixth dimension is needed.

## Edo approximation

Here is a list of edos which represent 13-limit intervals better (decreasing TE error): 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 and so on.

Here is a list of edos which tunes the 13-limit well relative to their size (TE relative error < 5.5%): 31, 41, 46, 53, 58, 72, 87, 94, 103, 111, 121, 130, 140, 152f, 159, 183, 190, 198, 212, 217, 224, 270, 282, 296, 301, 311, 320, 328, 342f, 354, 364, 369f, 373, 383, 400, 414, 422, 431, 441, 460, 472, 494 and so on.

**Note**: wart notation is used to specify the val chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

## Intervals

Here are all the 15-odd-limit intervals of 13:

Ratio | Cents Value | Color Name | Name | |
---|---|---|---|---|

14/13 | 128.298 | 3uz2 | thuzo 2nd | tridecimal supraminor second |

13/12 | 138.573 | 3o2 | tho 2nd | tridecimal subneutral second |

15/13 | 247.741 | 3uy2 | thuyo 2nd | tridecimal semifourth |

13/11 | 289.210 | 3o1u3 | tholu 3rd | tridecimal minor third |

16/13 | 359.472 | 3u3 | thu 3rd | tridecimal supraneutral third |

13/10 | 454.214 | 3og4 | thogu 4th | tridecimal naiadic |

18/13 | 563.382 | 3u4 | thu 4th | tridecimal sub-tritone |

13/9 | 636.618 | 3o5 | tho 5th | tridecimal super-tritone |

20/13 | 745.786 | 3uy5 | thuyo 5th | tridecimal cocytic |

13/8 | 840.528 | 3o6 | tho 6th | tridecimal subneutral sixth |

22/13 | 910.790 | 3u1o6 | thulo 6th | tridecimal major sixth |

26/15 | 952.259 | 3og7 | thogu 7th | tridecimal semitwelfth |

24/13 | 1061.427 | 3u7 | thu 7th | tridecimal supraneutral seventh |

13/7 | 1071.702 | 3or7 | thoru 7th | tridecimal submajor seventh |

## Music

*Justification*(2022)

*Bicycle Wheels*(2023)

*Venusian Cataclysms*^{[dead link]}play^{[dead link]}*Chord Progression on the Harmonic Overtone Series*^{[dead link]}play^{[dead link]}

*String Quartet No. 5*(1979) – Bandcamp | YouTube – performed by Kepler Quartet*String Quartet No. 7*(1984)

- performed by Kepler Quartet

*Unlicensed Copy*(2017) – mostly 7-limit with some erstwhile 13-based chromaticisms

*Canon on a ground*– in 2.11.13 subgroup

*Dying Visions of a Lonesome Machine*(2021)*Avenoir*(2022)*I Was a Teenage Boltzmann Brain*(2022)*Atlas Apassionata*(2022)