Twelfth root of two
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Its value is 1.05946309435929..., which is slightly more than 18⁄17 ≈ 1.0588. Better approximations are 196⁄185 ≈ 1.059459 or 18904⁄17843 ≈ 1.0594630948.
The equal-tempered chromatic scale
(to six places)
The final A (880 Hz) is twice the frequency of the lower A (440 Hz), that is, one octave higher.
Since the frequency ratio of a semitone is close to 106%, increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down one 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 the same results, ranging from cents up to several half-steps.
The twelfth root of two was calculated accurately by the Chinese court astronomer, historian, physicist and mathematician Zhu Zaiyu, Prince of Zheng of the Ming Dynasty. In 1584, Zhu published a work 律呂精義 A clear explanation of that which concerns the 律 [equal temperament]. Prince Zhu made note of the difference between his ideal mathematically-tuned 呂 (ancient music instrument), which gave the theoretical music instrument lengths for 12-tone equal temperament correct to 25 places, implemented with an 81-column abacus and calculated the cubic root of the square root of the square root of 2, obtaining which coincidentally applied a form of Pythagorean tuning.
- Just Intonation's history of temperaments.
- Music and mathematics
- Piano key frequencies
- Scientific pitch notation
- Well-Tempered Clavier
- Musical tuning
- Nth root
- Barbour, J.M.. A Sixteenth Century Approximation for Pi, The American Mathematical Monthly, Vol. 40, no. 2, 1933. Pp. 69–73.
- Ellis, Alexander and Hermann Helmholtz. On the Sensations of Tone. Dover Publications, 1954. ISBN 0-486-60753-4
- Partch, Harry. Genesis of a Music. Da Capo Press, 1974. ISBN 0-306-80106-X