# 3-limit

A **3-limit** interval is either an integer whose only prime factors are 2 and 3, the reciprocal of such an integer, the ratio of a power of 2 to a power of 3, or the ratio of a power of 3 to a power of 2. All 3-limit intervals can be written as 2^a 3^b, where a and b can be any (positive, negative or zero) integer. Some examples of 3-limit intervals are 3/2, 4/3, 9/8. Confining intervals to the 3-limit is known as Pythagorean tuning, and the Pythagorean tuning used in Europe during the Middle Ages is seed out of which grew the common-practice tradition of Western music.

EDOs which do relatively well at approximating 3-limit intervals can be found as the denominators of the convergents and semiconvergents of the continued fraction for the logarithm of 3 base 2. These are 1, 2, 3, 5, 7, 12, 17, 29, 41, 53, 94, 147, 200, 253, 306, ...

Another approach is to find EDOs which have more accurate 3 than all smaller EDOs. This results in 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, ...

3-limit intervals up to odd-limit 19683:

Ratio | cents | Color name | Interval category | ||
---|---|---|---|---|---|

1/1 | 0.000 | w1 | wa unison | unison | C |

2187/2048 | 113.685 | Lw1 | large wa 1sn | aug. unison | C# |

256/243 | 90.225 | sw2 | small wa 2nd | minor 2nd | Db |

9/8 | 203.910 | w2 | wa 2nd | major 2nd | D |

19683/16384 | 317.595 | Lw2 | large wa 2nd | aug. 2nd | D# |

32/27 | 294.135 | w3 | wa 3rd | minor 3rd | Eb |

81/64 | 407.820 | Lw3 | large wa 3rd | major 3rd | E |

8192/6561 | 384.360 | sw4 | small wa 4th | dim. fourth | Fb |

4/3 | 498.045 | w4 | wa 4th | fourth | F |

729/512 | 611.730 | Lw4 | large wa 4th | aug. fourth | F# |

1024/729 | 588.270 | sw5 | small wa 5th | dim. fifth | Gb |

3/2 | 701.955 | w5 | wa 5th | fifth | G |

6561/4096 | 815.640 | Lw5 | large wa 5th | aug. fifth | G# |

128/81 | 792.180 | sw6 | small wa 6th | minor 6th | Ab |

27/16 | 905.865 | w6 | wa 6th | major 6th | A |

32768/19683 | 882.405 | sw7 | small wa 7th | dim. 7th | Bbb |

16/9 | 996.090 | w7 | wa 7th | minor 7th | Bb |

243/128 | 1109.775 | Lw7 | large wa 7th | major 7th | B |

4096/2187 | 1086.315 | sw8 | small wa 8ve | dim. octave | Cb |

2/1 | 1200.000 | w8 | wa 8ve | octave | C |