This appendix provides visualizations of 4 well known tunings as Lissajous figures. These figures are produced from two perpendicular harmonic motions. By representing a tuning ratio such as 3 / 2, a perfect fifth, with a figure having the relative frequencies 3 and 2 a figure depicting the simplicity or complexity of the ratio is obtained. Purer intervals, those with lower numbers in their ratios, result in more simple curves.

In the case of the tempered tunings the figures move, illustrating the instability of these non ratiometric tunings. For the purposes of illustration and comparison one of the relative frequencies is taken as the denominator of one of the ratiometric tunings and the other relative frequency is the tempered tuning multiplied by that factor.

As with other pages in this essay you can listen to the intervals. Left click to hear the intervals as a sequence of notes and right click for a chord (you need decent speakers or headphones). All the tunings are relative to A = 440 Hz (unless you changed it as described in Appendix B) and A is the major 6th degree of these tunings. Consequently other notes, including the key note, may differ slightly amongst the tunings.

Pythagorean Chromatic (F# x B)

Just Chromatic

Quarter Comma Mean Tone Temperament

Equal Temperament

Pythagorean

Just

Pythagorean (F# x B) |

1 / 1Unison |

256 / 243Minor second |

9 / 8Major second |

32 / 27Minor third |

81 / 64Major third |

4 / 3Perfect fourth |

1024 / 729Tritone |

3 / 2Perfect fifth |

128 / 81Minor sixth |

27 / 16Major sixth |

16 / 9Minor seventh |

243 / 128Major seventh |

2 / 1Octave |

Just |

1 / 1Unison |

16 / 15Minor second |

9 / 8Major second |

6 / 5Minor third |

5 / 4Major third |

4 / 3Perfect fourth |

45 / 32Tritone |

3 / 2Perfect fifth |

8 / 5Minor sixth |

5 / 3Major sixth |

9 / 5Minor seventh |

15 / 8Major seventh |

2 / 1Octave |

Quarter Comma Mean Tone |

1Unison |

8 / 5^{5/4}Minor second |

5^{2/4 }/ 2Major second |

4 / 5^{3/4}Minor third |

5^{4/4 }/ 4Major third |

2 / 5^{1/4}Perfect fourth |

5^{6/4 }/ 8Tritone |

5^{1/4}Perfect fifth |

5^{8/4 }/ 16Minor sixth |

5^{3/4 }/ 2Major sixth |

5^{10/4 }/ 32Minor seventh |

5^{5/4 }/ 4Major seventh |

2Octave |

Equal Temperament |

0 centsUnison |

100 centsMinor second |

200 centsMajor second |

300 centsMinor third |

400 centsMajor third |

500 centsPerfect fourth |

600 centsTritone |

700 centsPerfect fifth |

800 centsMinor sixth |

900 centsMajor sixth |

1000 centsMinor seventh |

1100 centsMajor seventh |

1200 centsOctave |

In the ratiometric tunings (Pythagorean and Just) the interval of a tone (major second) is the same in both tunings, a relatively pure interval. The Pythagorean thirds, of less importance in ancient Greek music, are less satisfactory. The relative purity of the Just major third is striking.

In the tempered tunings (Mean tone and Equal tempered) the stability of the Mean tone major third is striking. The slower moving figures for the fourths and fifths of these tunings illustrates that they are quite close to the ratiometric intervals, Equal Temperament more so than the Quarter Comma Mean Tone.