**Audio: just intonation (0:04)**

INTERVAL | RATIO | INVERSION | RATIO |
---|---|---|---|

Perfect unison | 1:1 | Perfect octave | 2:1 |

Perfect fifth | 3:2 | Perfect fourth | 4:3 |

Major second | 9:8 | Minor seventh | 16:9 |

10:9 | 9:5 | ||

Major third | 5:4 | Minor sixth | 8:5 |

Major sixth | 5:3 | Minor third | 6:5 |

Major seventh | 15:8 | Minor second | 16:15 |

48:25 | 25:24 | ||

Augmented fourth | 45:32 | Diminished fifth | 45:32 |

25:18 | 25:18 |

**just intonation** plays the same melody as in *Pythagorean tuning* but uses just intonation tuning. The *just intonation* figure repeats the *Pythagorean tuning* score and the table lists the names of the intervals and their frequency ratios in just intonation.

Just intonation tuning uses small numbers to construct frequency ratios.

Just intonation is a tuning system that started to take off in the Renaissance period around the turn of the fifteenth century. It attempts to solve the problems of the big numbers that arise in Pythagorean tuning by using small numbers instead. The major consequence of just intonation is that the interval of a third and its inversion, the sixth, became as important in music, if not more important, than the intervals of fifth and fourth.

Just intonation uses the harmonic series to construct a tuning system. The relative frequencies of the first six partials in the harmonic series are 1, 2, 3, 4, 5 and 6. These can easily be combined to form eight intervals: unison (1:1), octave (2:1), fifth (3:2), fourth (4:3), major third (5:4), minor sixth (8:5), minor third (6:5) and major sixth (5:3). The remaining four intervals, the major and minor seconds and sevenths, are all formed from permutations of the numbers 2, 3 and 5. The numbers 2, 3 and 5 are all prime numbers and, for this reason, just intonation is also known as 5 limit tuning.

Just intonation is one of a number of just tuning systems. Meantone temperament and well temperament are two other types of just tuning. There is also a 7-limit just tuning system that uses the four prime numbers 2, 3, 5 and 7. In principle, any number of just tuning systems can be created using any set of prime numbers.

There are two downsides to just tuning. First, there are so many ways to combine prime numbers together that similar intervals may have alternative frequency ratios, as shown by the seconds and sevenths in the *just intonation* table. Second, the pesky Pythagorean comma still continues to make an appearance and there is always a residual gap of some sort that cannot be closed.

Just tuning systems are still in use today and there is also plenty of evidence to show that solo singers and choirs use just tuning to sing. Just tuning continues to have plenty to offer in terms of writing melody.