Glossary, Twelve-tone Equal Temperament
   
Glossary
Acoustic
Acoustics
Ancohemitonic

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Atonal Theory

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Atritonic

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Avoid Note
Bebop
Blues
Cardinality

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Cardinality Equivalence

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Cent
Chord
Chord Formula
Chord Type
Chromatic Cluster

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Chromatic Scale
Clock Diagram

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Cluster-free

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Cohemitonic

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Common Practice
Compatibility
Complement

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Consonance
Diatonic
Double Augmented Hexatonic
Double Diminished (Octatonic)
Eleventh
Enharmonic Equivalent
Evenness

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Fifth
Forte Number

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Fourth
Guitar
Harmonic Major
Harmonic Minor
Harmony
Interval
Interval Class

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Interval Content

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Inversion
Involution

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Jazz
Jazz Theory
Key
Keyboard
Lewin-Quinn FC-components

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Limited Transposition

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M-Relation

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Major
Melody
Minor
Mode
Ninth
Note
OC-Equivalence

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OPC-Equivalence

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OPTC-Equivalence

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OPTIC-Equivalence

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OPTIC/K-Equivalence

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OTC-Equivalence

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Octatonic
Octave
Octave-Equivalence

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Other Scales
Parallel Key
Pentatonic
Permutation Equivalence

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Piano
Pitch
Pitch Class

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Playing Outside
Prime Form

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Quartal

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Reharmonization
Relative Key
Rhythm
Roman Numeral Function
Root
Scale
Second
Semitone
Set Class

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Seventh
Sixth
Slash Chords
Symmetry

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Tenth
Tertiary
Third
Thirteenth
Tonality
Tonic
Transposition
Triad
Tritone
Tritonic

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Tuning Systems
Twelfth
Twelve-tone Equal Temperament
Unison
Voice Leading
Whole Tone
Whole-Tone Scale
Z-Relation

Set Theory



Twelve-tone equal temperament or 12-TET is the ubiquitous modern tuning system in Western music that makes certain tradeoffs compared to its tuning-system predecessors, eliminating one major restriction (limited keys per piece) by redistributing impurity more evenly to all intervals and all keys, making major and minor third intervals more impure and making the equal-tempered semitone and tritone very impure (see table).

In the Western musical world, theoretical research into equal temperament (kicked off by the work of Chinese polymath Zhu Zaiyu) dates back to the sixteenth or seventeenth centuries, but the tuning system did not become prevalent until at least the eighteenth century (after Johann Sebastian Bach) when composers began relying on it in their compositions with modulations between keys in a single piece.

Music for other equal-temperament tuning systems exists, for example Indonesian gamelans tuned to 5-TET, a Thai xylophone tuned to 7-TET, and the quarter-tone (24-TET) Arab tone system known as gadwal.

Table of 12-TET intervals and errors compared to various just intonation intervals

Just Intonation v. 12-TET. Comparison of each possible ratio of small whole numbers (grid points equal just intonation) to its nearest ET interval (colored lines with irrational slopes), where larger circles represent lower error and smaller circles are worse matches.

Lowest ratios are shown in the table below, with the difference in cents. See also Interval Classes.

In­ter­val 12-TET Ra­tio Cents Just In­ton­a­tion Cents Dif­fer­ence (Cents)
Semi­tone 21/12 = 1.059463 100 16/15 = 1.0666… 111.73 11.73
Whole-tone 22/12) = 1.122462 200 9/8 = 1.125 203.91 3.91
Minor third 23/12 = 1.189207 300 6/5 = 1.2 315.64 15.64
Major third 24/12 = 1.259921 400 5/4 = 1.25 386.31 13.69
Perfect fourth 25/12 = 1.334840 500 4/3 = 1.333333… 498.04 1.96
Tritone 26/12 = 1.414214 600 7/5 = 1.4 582.51 17.49

For more about equal temperament, see List of pitch intervals and Comparison to just intonation.