Twelfth root of two
The twelfth root of two or 12√2 is an algebraic irrational number. It is most important in music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament. Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.
The twelfth root of two to 20 significant figures is 1.0594630943592952646. Fraction approximations in order of accuracy are 18⁄17, 196⁄185, and 18904⁄17843.
As of December 2013[update], its numerical value has been computed to at least twenty billion decimal digits.
Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts.
Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches:
The final A (880 Hz) is exactly twice the frequency of the lower A (440 Hz), that is, one octave higher.
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