**Audio: pythagorean tuning (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 |

Major sixth | 27:16 | Minor third | 32:27 |

Major third | 81:64 | Minor sixth | 128:81 |

Major seventh | 243:128 | Minor second | 256:243 |

Augmented fourth | 729:512 | Diminished fifth | 1024:729 |

**pythagorean tuning** plays a melody in Pythagorean tuning. It consists of twelve different notes in succession followed by a note an octave above the first note. The frequency ratio between each pair of notes is either 3:2 or 2:3. The *pythagorean tuning* figure shows the score and the table lists the names of the intervals and their frequency ratios in Pythagorean tuning.

Pythagorean tuning is a system of tuning which constructs notes using a frequency ratio of 3:2.

Around 540BC in Greece, Pythagoras started to systematically construct the tuning system which now bears his name. The frequency ratio of the octave, 2:1, was already known, so the starting point for Pythagorean tuning was to divide the octave in half. Half an octave is the frequency ratio 3:2. Using this simple ratio results in the set of intervals shown in the *Pythagorean tuning* table.