Twelfth root of two

This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries.

[2] Fraction approximations in increasing order of accuracy include ⁠18/17⁠, ⁠89/84⁠, ⁠196/185⁠, ⁠1657/1564⁠, and ⁠18904/17843⁠.

), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step".

Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps.

Reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not.

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.