equaltempered

Equal temperament, or equal-tempered tuning, is a musical tuning system in which the octave is divided into equal steps. In the standard Western form, the octave is split into 12 equal semitones (12-TET). The frequency ratio between adjacent notes is the twelfth root of 2, so after 12 steps you reach a frequency double the starting one. Each semitone is defined as 100 cents, making the octave 1200 cents in total.

In equal temperament, all semitones have the same size, which allows easy modulation to any key. However, most intervals are only approximations of their pure counterparts in just intonation. For example, a major third in 12-TET is 400 cents, rather than the just 386.31 cents, and a perfect fifth is 700 cents rather than about 701.96. The trade-off is a consistent tuning system that enables flexible harmony across keys, at the expense of perfectly pure intervals.

History and usage: The idea of dividing the octave into equal parts appeared in various tuning experiments, with 12-TET gaining prominence in the 18th and 19th centuries. It became the standard tuning system for most Western keyboard instruments and, by extension, much of Western classical, jazz, and popular music. Pitch reference standards, such as A4 = 440 Hz, are separate from the tuning system itself and vary by context and history.

Variants: While 12-TET is the most common form, equal temperaments with other numbers of divisions exist (for example, 19-TET, 24-TET, 31-TET, 53-TET). These distribute the octave into equal steps as well, offering different degrees of pitch resolution while preserving equal spacing.