# 23-limit

In 23-limit just intonation, all ratios contain no prime factors higher than 23. The prime 23 is significant as being the start of a record prime gap ending at 29, the previous record prime gap being the one corresponding to the 7-limit. Thus, it is arguably a potential ideal stopping point for prime limits due to it corresponding to the full 27-odd-limit and thus corresponding to mode 14 of the harmonic series, which is to say that all of the first 28 harmonics are in the 23-limit.

## Edo approximations

A list of edos with progressively better tunings for 23-limit intervals: 80, 87, 94, 111, 121i, 130, 140, 152fg, 159, 183, 190g, 193, 217, 243e, 270, 282, 311, 373g, 422, 525, 566g, 581, 718, 742i, 814, 935, 954h, 1106, 1178, 1308, 1323, 1395, 1506hi, 1578, 1889, 2000, 2460 and so on.

Another list of edos which provides relatively good tunings for 23-limit intervals (relative error < 5%): 94, 190g, 193, 217, 243e, 270, 282, 311, 328h, 373g, 388, 422, 436, 460, 525, 540, 566g, 581, 624, 639h, 643i, 653, 692i, 718, 742i, 764(h), 814, 860, 882, 908, 935, 954h, 997, 1012, 1046dgh, 1075, 1106, 1125, 1178, 1205g, 1224, 1236(h), 1258, 1282, 1308, 1323, 1357efhi, 1385, 1395, 1419, 1448(g), 1506hi, 1578, 1600, 1646, 1672h, 1677e, 1696, 1718, 1730(g), 1759, 1768gi, 1817hi, 1821ef, 1889, 1920, 1966, 2000, 2038, 2041, 2072, 2087h, 2103, 2113, 2132eh, 2159, 2217, 2231, 2243e, 2270i, 2311, 2320, 2414, 2460 and so on.

Note: wart notation is used to specify the val chosen for the edo.

94edo is the first edo to be consistent in the 23-odd-limit. The smallest edo where the 23-odd-limit is distinctly consistent, meaning each element of the tonality diamond is distinguished, is 282edo, although 311edo may be preferred for excellent consistency in much larger odd limits, and thus is a good choice if you want the 23-odd-limit to be distinctly consistent and the 27-odd-limit (and higher) to be consistent.

## 23-odd-limit intervals

Ratios of 23 in the 23-odd-limit are:

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

24/23 | 73.681¢ | 23u1 | twethu 1sn | lesser vicesimotertial semitone |

23/22 | 76.956¢ | 23o1u2 | twetholu 2nd | greater vicesimotertial semitone |

23/21 | 157.493¢ | 23or2 | twethoru 2nd | large vicesimotertial neutral second |

26/23 | 212.253¢ | 23u3o2 | twethutho 2nd | vicesimotertial whole tone |

23/20 | 241.961¢ | 23og3 | twethogu 3rd | vicesimotertial inframinor third |

23/19 | 330.761¢ | 23o19u3 | twethonu 3rd | vicesimotertial supraminor third |

28/23 | 340.552¢ | 23uz3 | twethuzo 3rd | vicesimotertial neutral third |

23/18 | 424.364¢ | 23o4 | twetho 4th | vicesimotertial diminished fourth |

30/23 | 459.994¢ | 23uy3 | twethuyo 3rd | vicesimotertial ultramajor third |

23/17 | 523.319¢ | 23o17u4 | twethosu 4th | vicesimotertial acute fourth |

32/23 | 571.726¢ | 23u4 | twethu 4th | vicesimotertial narrow tritone |

23/16 | 628.274¢ | 23o5 | twetho 5th | vicesimotertial wide tritone |

34/23 | 676.681¢ | 23u17o5 | twethuso 5th | vicesimotertial grave fifth |

23/15 | 740.006¢ | 23og6 | twethogu 6th | vicesimotertial ultraminor sixth |

36/23 | 775.636¢ | 23u5 | twethu 5th | vicesimotertial augmented fifth |

23/14 | 859.448¢ | 23or6 | twethoru 6th | vicesimotertial neutral sixth |

38/23 | 869.239¢ | 23u19o6 | twethuno 6th | vicesimotertial submajor sixth |

40/23 | 958.039¢ | 23uy6 | twethuyo 6th | vicesimotertial ultramajor sixth |

23/13 | 987.747¢ | 23o3u7 | twethothu 7th | vicesimotertial minor seventh |

42/23 | 1042.507¢ | 23uz7 | twethuzo 7th | small vicesimotertial neutral seventh |

44/23 | 1123.044¢ | 23u1o7 | twethulo 7th | vicesimotertial major seventh |

23/12 | 1126.391¢ | 23o8 | twetho 8ve | vicesimotertial major seventh |

## Trivia

- Unlike most other prime limits, the smallest superparticular ratio of HC23 is larger than the smallest one of HC19. 23 is the first prime limit to show this phenomenon. The ratio is, in fact, larger than the second smallest one of HC19. See List of superparticular intervals.