**Audio: equal temperament (0:07)**

INTERVAL | SYMBOL | SEMITONES | INTERVAL | SYMBOL | SEMITONES |
---|---|---|---|---|---|

Perfect unison | P1 | 0 | Perfect octave | P8 | 12 |

Minor second | m2 | 1 | Major seventh | M7 | 11 |

Major second | M2 | 2 | Minor seventh | m7 | 10 |

Minor third | m3 | 3 | Major sixth | M6 | 9 |

Major third | M3 | 4 | Minor sixth | m6 | 8 |

Perfect fourth | P4 | 5 | Perfect fifth | P5 | 7 |

Augmented fourth | aug4 | 6 | Diminished fifth | dim5 | 6 |

**equal temperament** plays a melody on a piano in Twelve Tone Equal Temperament tuning (12TET). The *equal temperament* figure shows that each bar in the score contains an interval and its inversion. The *equal temperament* table lists the name of each interval, its symbol and the number of semitones it contains. An interval and its inversion occupy the same row.

Equal Temperament (ET) is a tuning system in which the frequency ratio between two adjacent notes is fixed.

Twelve Tone Equal Temperament, abbreviated to 12TET, is a tuning system in which the frequency ratio between two adjacent notes is a semitone.

A semitone is the twelfth root of 2. The frequency ratio of a semitone cannot be expressed as a simple fraction. It is 1.059463, correct to six decimal places. The frequency ratio of the semitone is permanently fixed and does not change. The frequency ratio between any two consecutive notes in 12TET is always a semitone and there are twelve semitones in an octave in 12TET.

12TET is similar to Pythagorean tuning and just intonation in that all three systems produce twelve notes, the names of the notes and the names of the intervals are the same, the octave has a frequency ratio of 2:1 and unison is a frequency ratio of 1:1. The difference is that equal temperament does not use whole numbers such as 2, 3 and 5 to derive frequency ratios. Instead, it starts at the other end of the scale, as it were, and asks: how many notes do you want? If the answer is 12, then the octave is divided into 12 to produce 12TET. The division is not an arithmetic twelfth but a geometric twelfth root of two.

You need a computer or a calculator to calculate the frequency ratio of an interval in 12TET. First, count the number of semitones in the interval, then calculate the twelfth root of 2 to the power of this number. For example, the interval of a fifth contains seven semitones. Its frequency ratio is the twelfth root of 2 to the power of 7. This produces the value, 1.4983, which is pretty close to the value of 1.5 (3:2) obtained from Pythagorean tuning and just intonation.

Equal temperament feels as though it has been around for ages but it really came into effect around the beginning of the eighteenth century. Nowadays it has largely transplanted Pythagorean tuning and just intonation as the tuning system of choice. It is the most widely used tuning system in music and many musical instruments are manufactured to conform to 12TET. Nearly all the melody and harmony in the guide is written in 12TET.