# Odd prime sum limit

The ** n-odd-prime-sum-limit** (abbreviated

**) is the collection of all just ratios where the no-twos sum of prime factors with repetition of both the numerator and the denominator does not exceed the integer**

*n*-OPSL*n*.

This concept was noted by Tristan Bay as a way to measure how accurately an edo approximates just intonation with lower primes weighted more heavily. Specifically, the idea is to use OPSLs as an alternative metric for consistency limit either instead of or alongside odd limits.

## Comparison with odd limit

The 1- and 2-odd-prime-sum-limit are equivalent to the 1-odd-limit, which only contains a single interval pair {1/1, 2/1}. The 3- and 4-odd-prime-sum-limit are equivalent to the 3-odd-limit, which adds {3/2, 4/3}. All edos are consistent in those limits.

The 5-odd-prime-sum-limit is also equivalent to the 5-odd-limit, adding {5/4, 8/5} and {5/3, 6/5} to the 4-OPSL, and the 6-odd-prime-sum-limit adds {9/8, 16/9} and {9/5, 10/9}. The 7-odd-prime-sum-limit is equivalent to the 9-odd-limit, so it is the first OPSL that differs from the corresponding odd limit. It adds {7/4, 8/7}, {7/6, 12/7}, {7/5, 10/7}, and {9/7, 14/9} to the 6-OPSL. The 8-odd-prime-sum-limit adds {15/8, 16/15} and {15/14, 28/15}, the 9-odd-prime-sum-limit adds {27/16, 32/27}, {27/14, 28/27}, and {27/20, 40/27}, and the 10-odd-prime-sum-limit adds {21/16, 32/21}, {21/20, 40/21}, {25/16, 32/25}, {25/24, 48/25}, {25/14, 28/25}, {25/18, 36/25}, {25/21, 42/25}, and {27/25, 50/27}.

## Minimal OPSL-consistent edos

OPSL | Smallest Consistent Edo* |
---|---|

1 | 1 |

2 | 1 |

3 | 1 |

4 | 1 |

5 | 3 |

6 | 3 |

7 | 5 |

8 | 12 |

9 | 12 |

10 | 12 |

11 | 31 |

12 | 72 |

13 | 72 |

14 | 130 |

15 | 270 |

16 | 270 |

17 | 954 |

18 | 1236 |

19 | 1578 |

20 | 1578 |

21 | 3395 |

22 | 3395 |

23 | 6079 |

24 | 8539 |

25 | 8539 |

26 | 8539 |

27 | 8539 |

28 | 102557 |

29 | 102557 |

30 | 102557 |

31 | 102557 |

32 | 102557 |

33 | 258008 |

34 | 258008 |

35 | 258008 |

36 | 258008 |

*apart from 0edo

## Whole-interval OPSL

The *n*-whole-interval-OPSL, or ** n-WOPSL**, is slightly different from the

*n*-OPSL. This is the collection of all just ratios with a no-twos Wilson height that does not exceed the integer

*n*. When using it to measure consistency in the same way as odd limits, lower primes are favored even more strongly than for OPSLs. It was confused with the original definition for

*n*-OPSL (where the numerator and denominator are compared with

*n*separately) at the time of this Wiki article's creation, but has since been corrected.

### Comparison between odd-limit and WOPSL

Just like with OPSLs, the 1- and 2-WOPSL are equivalent to the 1-odd-limit, which only contains a single interval pair {1/1, 2/1}, and the 3- and 4-WOPSL are equivalent to the 3-odd-limit, which adds {3/2, 4/3}.

The 5-WOPSL adds {5/4, 8/5} without {5/3, 6/5} from the 5-odd-limit, so it is the first WOPSL that differs from the corresponding odd limit. The 6-WOPSL adds {9/8, 16/9}. The 7-WOPSL adds {7/4, 8/7} without {7/6, 12/7}, and the 8-WOPSL adds {5/3, 6/5} as well as {15/8, 16/15}. The 9-WOPSL adds {27/16, 32/27}, and the 10-WOPSL adds {7/6, 12/7}, {21/16, 32/21}, and {25/16, 32/25}. The 11-WOPSL adds {11/8, 16/11}, {9/5, 10/9}, and {45/32, 64/45}. The 12-WOPSL adds {7/5, 10/7}, {35/32, 64/35} and {81/64, 128/81}.