| Twelfth Root Of Two |
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The smallest interval in the equal tempered chromatic scale is a semitone, which is numerically equal to a multiplication with: or approximately 1.05946309436. 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 Prince Chu Tsai-Yu of the Ming Dynasty . In 1596, he published a work, ''Lu Lu Ching I'' ("A clear explanation of that which concerns the ''lu'' ( Musical Pipes )"), which gave pipe lengths for 12-tone equal temperament "with absolute correctness to nine places". The calculation must have involved starting with the number 2 followed by at least 108 zeros, taking the square root twice, and then the cube root. Prince Chu noticed the difference between his ideal mathematically-tuned ''lu'' and traditional pipes, which used a form of Pythagorean Tuning . SEE ALSO
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:"Mathematically, ... each successive pitch is related to the previous pitch by a factor of the twelfth root of 2. That is, the ratio between the frequencies of any two successive pitches in either standard is 1.05946309436."
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