# Tuning Systems

##### Glossary

Historically, other tuning systems besides 12-tone equal temperament have been popular. A few important tuning systems are discussed briefly, with pointers outside of *Harmonious* where you can learn more.

Every tuning system mentioned below employs perfect octaves with a ratio of 2:1, matching acoustic perception of the human ear.

### Just Intonation

In **just intonation**, intervals are tuned to whole number ratios to reflect the underlying acoustic consonances, such as 3:2 for fifths (= 4:3 for fourths), 5:4 for major thirds, 6:5 for minor thirds, etc. However, stacking two or more of any perfect interval, say, perfect fifths, and subtracting out the octaves, will build up small tuning errors, which become audible even before returning to the starting pitch. (See Comma.)

It is impossible to have pure (just) octaves (2:1) and build a tuning system from repeated just intervals (just fifths, just thirds, just anything) without building up errors (without some sort of adaptive tuning).

### Pythagorean Tuning

**Pythagorean tuning** uses just intervals where the whole numbers in the ratios must be of the form 2^{x} × 3^{y}. This tuning system dates back several millennia to ancient Greece and was independently codified in ancient China around the same time. Since it is a form of just intonation, it shares the same tradeoffs.

### Meantone Temperament

**Meantone temperament** compromises the tuning of the fifths to improve the tuning of the thirds. Quarter-comma meantone was popular from the early 1500’s to the late 1800’s.

### Well Temperament

**Well temperament** allows a single instrument to play a piece in all 24 major and minor keys without changing tuning between movements, an effect employed by Johann Sebastian Bach’s important work *The Well-Tempered Clavier*, first published in 1722. Well temperament or **good temperament** refers to the use of irregular intervals where the fifths of different keys are different sizes, but no key has particularly impure intervals.

### Equal Temperament

**Equal temperament** abandons whole-number-ratio intervals as the basis of the chromatic scale and instead divides the octave into a fixed number of equal semitone intervals, resulting in irrational interval sizes. (See 12-TET, Semitone and Cent.) By making the fifth slightly impure, stacked equal-tempered fifths no longer accrue error but meet back up at the octave. However all other intervals are more impure than the fourth/fifth.

### Other Equal-Tempered Tuning Systems

*Harmonious* provides an exhaustive theoretical breakdown of 12-TET, the twelve-tone chromatic scale, and it is possible to generalize this musical set theory analysis to any equal temperament system: 5-TET, 7-TET, 10-TET, 11-TET, 13-TET, up to 24-TET, an exercise left for the reader. (To get you started, here are some charts and tables:)

Number of OPTIC/K-equivalent objects.Shows n-TET on the x-axis, and the number of musical objects (a count of all possible chords and scales) for each tuning system on the y-axis. The blue line uses the log scale on the left, while the green bars use the scale on the right. The same data are shown in the table below.

n-TET | OPTIC/K eqv. objects |
---|---|

5 | 3 |

6 | 7 |

7 | 8 |

8 | 17 |

9 | 22 |

10 | 43 |

11 | 62 |

12 | 122 |

13 | 189 |

14 | 368 |

15 | 611 |

16 | 1,193 |

17 | 2,055 |

18 | 4,040 |

19 | 7,154 |

20 | 14,114 |

21 | 25,481 |

22 | 50,399 |

23 | 92,204 |

24 | 182,737 |

### Microtonal Music

The internet allows communities of enthusiasts to find one another and bond over obscure bits of knowledge. The studious proponents of **microtonal music** are no exception. If you are interested in going down the **musical tuning** systems rabbit hole, see List of pitch intervals or Isomorphic keyboard, visit the Microtonal community wiki on Miraheze, or search the web for **microtonal**.