Equal temperament divides the octave into equal logarithmic steps, trading slight detuning for unlimited modulation

Equal temperament divides the octave into a chosen number of logarithmically equal steps. Twelve-tone equal temperament (12-TET) uses 12 equal semitones, each exactly 100 cents (a frequency ratio of 2^(1/12) ≈ 1.0595). The octave’s 2:1 ratio is preserved exactly, but every other interval is a slightly mistuned compromise relative to just intonation — the perfect fifth is about 2 cents flat, the major third about 14 cents sharp. The payoff is that all keys are equally in tune, so unlimited modulation is possible without re-tuning. The choice of 12 is not the only option and is not uniquely ‘natural’: the octave can be divided into any number of equal steps (19-TET, 31-TET, 7-TET, 10-TET), some more consonant for certain timbres or scales — the foundation of microtonal music. 12-TET became dominant in Western music only in the early 20th century; before that, unequal but closed ‘well temperaments’ were used.

Examples

A piano tuned to 12-TET plays in all major and minor keys without re-tuning. 19-TET keeps the same 2:1 octave but every interval inside it is a different size than in 12-TET. An a cappella barbershop quartet drifts toward just intonation, because beatless thirds and fifths give sonic feedback — incompatible with a fixed 12-TET keyboard.

Assessment

Why is 12-TET called ‘equal’ temperament, and what is preserved exactly versus what is mistuned? A 12-TET keyboard and a string quartet play the same piece together — which group tends toward just intervals and why? Name one other equal division of the octave and state the trade-off between 12-TET and just intonation (modulation capability vs. sensory consonance).