Twelfth root of two

The twelfth root of two to 20 significant figures is . The continued fraction begins [1: 16, 1, 4, 2, 7, 1, 1, 2, 2, 7, 4, 1, 2, 1, ...], so a simple rational approximation is . ==The equal-tempered chromatic scale==

A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into 12 equal parts. Each note has a frequency that is 2 times that of the one below it. Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A4) with a frequency of 440 Hz, produces the following sequence of pitches: The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher. Other tuning scales Other tuning scales use slightly different interval ratios: • The just or Pythagorean perfect fifth is 3/2, and the difference between the equal-tempered perfect fifth and the just is a grad, the 12th root of the Pythagorean comma (3 / 2^{19/12}). • The equal-tempered Bohlen–Pierce scale uses the interval of the 13th root of three (\sqrt[13]{3}). • Stockhausen's Studie II (1954) makes use of the 25th root of five (\sqrt[25]{5}), a compound major third divided into 5×5 parts. • The delta scale is based on ≈\sqrt[50]{3/2}. • The gamma scale is based on ≈\sqrt[20]{3/2}. • The beta scale is based on ≈\sqrt[11]{3/2}. • The alpha scale is based on ≈\sqrt[9]{3/2}. ==Pitch adjustment==

depicts equal distances between notes (logarithmic) Since the frequency ratio of a semitone is close to 106% (100\sqrt[12]{2} \approx 105.946), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about a semitone, or "half-step". Upscale reel-to-reel magnetic tape recorders typically have pitch adjustments of up to ±6%, generally used to match the playback or recording pitch to other music sources having slightly different tunings (or possibly recorded on equipment that was not running at quite the right speed). 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. ==History==

Historically this number was proposed for the first time in relation to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin. Vincenzo Galilei may have been the first European to suggest 12-tone equal temperament, in 1581. ==See also==