- Twelfth root of two
The twelfth root of two or 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::
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 in the equation , 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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