Equal Temperament

Also known as: 12-TET; ET

framework · music theory · formal-scientific

Tuning framework dividing the octave into twelve mathematically equal semitones, optimizing key flexibility at the cost of pure intervals.

Equal Temperament is the tuning framework dividing the octave into twelve mathematically equal semitones (each frequency ratio of 2^(1/12) ≈ 1.0595), optimizing key flexibility and modulation freedom at the cost of pure intervals — only the octave is acoustically pure, and all other intervals are slightly mistuned relative to their just-intonation counterparts. The system was theoretically described as early as Vincenzo Galilei (1581) and Zhu Zaiyu's Lülü jingyi (1584), with Werckmeister III (1691) providing a close approximation, and full equal temperament became standard through gradual adoption from the 18th through mid-19th centuries — substantially later than the popular myth of J.S. Bach's Wohltemperierte Klavier as the founding equal-tempered work suggests. The 12-tone equal-tempered system (12-TET) is now near-universal in Western fixed-pitch instruments (piano, guitar, fretted strings) and underlies digital audio synthesis, MIDI, and most of the world's amplified popular music.

Originators

Vincenzo Galilei (1581 early proposal); Zhu Zaiyu (1584 independent mathematical description in Ming-dynasty China); Andreas Werckmeister (Werckmeister III 1691, close-approximation well-tempered system); subsequent gradual adoption through 18th-19th centuries; precise mathematical articulation by Sauveur and others high

Year / Decade

1581 (Galilei early proposal); 1584 (Zhu Zaiyu); 1691 (Werckmeister III); mid-19th century (full Western adoption) high

Primary sources

Galilei, V. (1581). Dialogo della musica antica et della moderna, Werckmeister, A. (1691). Musicalische Temperatur, Jorgensen, O. (1991). Tuning: Containing the Perfection of Eighteenth-Century Temperament, the Lost Art of Nineteenth-Century Temperament, and the Science of Equal Temperament, Duffin, R.W. (2007). How Equal Temperament Ruined Harmony high

Core components

12 equal semitones per octave (frequency ratio 2^(1/12) ≈ 1.05946)
Closed cycle of fifths with no wolf intervals
Enharmonic equivalence (G♯ = A♭, etc.)
All intervals except octave slightly mistuned relative to just intonation: perfect fifth ~2 cents flat, major third ~14 cents sharp, minor third ~16 cents flat
Cents notation (1200 cents per octave, 100 cents per semitone)
Microtonal extensions: 19-TET, 24-TET (quarter-tone), 31-TET, 53-TET
Mathematical equivalence to dividing the logarithm of 2 by 12 along the frequency continuum

Primary use case

Standard tuning for Western fixed-pitch instruments (piano, guitar, fretted strings, organ); foundation for MIDI, digital audio synthesis, and electronic music production; tuning premise for nearly all 20th- and 21st-century Western popular music; pedagogical framework for music-theory teaching of Western tonal and post-tonal music; framework underlying the closed-loop Circle of Fifths visualization and neo-Riemannian Tonnetz; engineering reference for instrument design and tuning.

Common criticisms

Equal temperament has been substantially critiqued aesthetically — Owen Jorgensen's Tuning (1991) documents the gradual displacement of well-temperaments and unequal temperaments through the 19th century, and Ross Duffin's How Equal Temperament Ruined Harmony (2007) provides a polemical critique arguing that 12-TET impoverishes harmonic experience compared to historical alternatives
the historical claim that equal temperament was in widespread use during Bach's lifetime is empirically false — Bach's Wohltemperierte Klavier was almost certainly intended for one of several well-temperaments (with key-color variation) rather than 12-TET, an issue that pedagogy and popular music history have long obscured
the major third in equal temperament is approximately 14 cents sharp of just intonation, an audible mistuning particularly in sustained chords
performers on flexible-pitch instruments (strings, voice, brass) routinely adjust intonation contextually away from 12-TET, suggesting equal temperament is a keyboard-design compromise rather than an aesthetic ideal
xenharmonic and microtonal communities have argued 12-TET imposes a Procrustean bed on musical possibility, with alternative equal divisions (19-TET, 31-TET, 53-TET) producing useful harmonic resources that 12-TET excludes
cross-cultural application is limited because most non-Western traditions have substantially different tuning frameworks (Indian shruti, Indonesian gamelan, Arabic maqam) that 12-TET fits poorly.

Lineage

Siblings: Just Intonation, Circle of Fifths