equaltemperaments

Equal temperament is a tuning system in which the octave is divided into equal steps, allowing a musician to play in any key with the same intervallic relationship. The most widely used variant is the 12-tone equal temperament (12-TET), in which the octave is divided into 12 equal semitones. Each semitone multiplies the frequency by the twelfth root of two (2^(1/12)), so the interval between any two adjacent notes is constant. This produces a uniform keyboard or string-fretted instrument layout, where transposition by any number of semitones preserves the same pitch relations.

Historically, temperaments were developed to balance consonance and key flexibility. In Western music, 12-TET became standard between the 18th and 19th centuries due to the practical advantages for modulation and keyboard instruments. Under 12-TET, some intervals diverge slightly from their simple, just-intonation ratios (for example, the perfect fifth is 700 cents in 12-TET, versus about 702 cents in just intonation), trading pure tuning for uniformity across keys.

Beyond 12-TET, many other equal temperaments have been proposed, including 24-TET, 19-TET, and a broad family of equal divisions of the octave (EDOs). In addition, microtonal tunings use even finer divisions. Equal temperament is not exclusive to Western music and has influenced tuning practices worldwide, as well as electronic music, while alternative tuning systems remain central to historical performance and some contemporary compositions.