- Twelfth root of two
The twelfth root of two or sqrt [12] {2} is an algebraic
irrational number , representing thefrequency ratio between any two consecutivenote s of a modernchromatic scale inequal temperament ; that is, the interval of asemitone .Its
decimal value is approximately 1.05946309435929... . This value can be computed using thecontinued fraction OEIS|id=A103922::1 + frac{1}{16 + frac{1}{1 + frac{1}{4 + frac{1}{2 + frac{1}{7 +frac{1}{1 + frac{1}{1+ frac{1}{2 + frac{1}{ddots}
Since a musical interval is a ratio of frequencies, and the equal tempered chromatic scale is a way of dividing the
octave (which has a ratio of 2:1) into twelve equal parts, the semitone must be that ratio which when multiplied by itself twelve times will be equal to two. Therefore it is the positive real solution for x in the equation x^{12} = 2, or the twelfth root of two.History
The twelfth root of two was first calculated accurately by the Chinese mathematician Prince
Chu Tsai-Yu of theMing Dynasty . In 1596, he published a work, "Lu lu ching i" ("A clear explanation of that which concerns the "lu" (musical pipes)"), which gave theoretical pipe lengths for 12-tone equal temperament correct to nine places. Prince Chu made note of the difference between his ideal mathematically-tuned "lu" and traditional pipes, which used a form ofPythagorean tuning .This would be calculated again later in 1636 by the French mathematician
Marin Mersenne , and as the techniques for calculating logarithms became widely known, this calculation would eventually become trivial.ee also
* Just Intonation's history of temperaments.
*Piano key frequencies
*Well-Tempered Clavier
*Musical tuning
*Nth root References
* 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
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