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.
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.
|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|