# 13-limit

The 13-prime-limit refers to a constraint on selecting just intonation intervals such that the highest prime number in all ratios is 13. Thus, 40/39 would be allowable, since 40 is 2*2*2*5 and 39 is 3*13, but 34/33 would not be allowable, since 34 is 2*17, and 17 is a prime number higher than 13. An interval doesn't need to contain a 13 to be considered within the 13-limit. For instance, 3/2 is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a 13 in it is not necessarily within the 13-limit. 23/13 is not within the 13-limit, since 23 is a prime number higher than 13).

The 13-prime-limit 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 need.

Edos good for 13-limit are 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 183, 207, 217, 224, 270, 494, 851, 1075, 1282, 1578, 2159, 2190, 2684, 3265, 3535, 4573, 5004, 5585, 6079, 8269, 8539, 13854, 14124, 16808, 20203, 22887, 28742, 32007, 37011, 50434, 50928, 51629, 54624, 56202, 59467, 64471, 65052, ... .

## Intervals

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

Ratio | Cents Value | Interval name | |
---|---|---|---|

14/13 | 128 | 3uz2 | thuzo 2nd |

13/12 | 139 | 3o2 | tho 2nd |

15/13 | 248 | 3uy2 | thuyo 2nd |

13/11 | 289 | 3o1u3 | tholu 3rd |

16/13 | 359 | 3u3 | thu 3rd |

13/10 | 454 | 3og4 | thogu 4th |

18/13 | 563 | 3u4 | thu 4th |

13/9 | 637 | 3o5 | tho 5th |

20/13 | 746 | 3uy5 | thuyo 5th |

13/8 | 841 | 3o6 | tho 6th |

22/13 | 911 | 3u1o6 | thulo 6th |

26/15 | 952 | 3og7 | thogu 7th |

24/13 | 1061 | 3u7 | thu 7th |

13/7 | 1072 | 3or7 | thoru 7th |

See: Gallery of Just Intervals

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# Music

Venusian Cataclysms play by Dave Hill

Chord Progression on the Harmonic Overtone Series play by Dave Hill