Twelfth root of two
The twelfth root of two or (or equivalently ) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone () in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones).[a] A semitone itself is divided into 100 cents (1 cent = ).
![{\displaystyle {\sqrt[{12}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc835f27425fb3140e1f75a5faa35b1e8b9efc35)
![{\displaystyle {\sqrt[{1200}]{2}}=2^{1/1200}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/221eb1075aacd8a300f1fd9576d825dbcc49b96e)
Numerical value[edit]
The twelfth root of two to 20 significant figures is 1.0594630943592952646.[2] Fraction approximations in increasing order of accuracy include 18/17, 89/84, 196/185, 1657/1564, and 18904/17843.
History[edit]
Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.[4] In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament.[1] The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician Zhu Zaiyu using an abacus to reach twenty four decimal places accurately,[1] calculated circa 1605 by Flemish mathematician Simon Stevin,[1] in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister.[5]