Twelfth root of two
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The twelfth root of two is a quantity representing the frequency ratio between any two consecutive notes of a modern chromatic scale in equal temperament. Systems of unequal temperament lacked the ability for free tonal modulation, as intervals in some keys were intolerably bad (which were referred to as "wolf fifths", as an allusion to howling). Equal temperament solves this problem by dividing up the wolf-ness equally across the scale, thus making each interval only slightly "tempered" away from a pure harmonic relationship.
The smallest interval in the equal tempered chromatic scale is a semitone, which is numerically equal to a multiplication with ![\sqrt[12]{2}](../../../math/7/0/b/70b8b8fc763c20423a65bd934e378085.png)
, or approximately 1.0594630943593.
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 x12 = 2, or the twelfth root of two.
[edit] History
The twelfth root of two was first calculated accurately by the Chinese mathematician 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 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 of Pythagorean 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.
[edit] See also
- Equal temperament
- Just Intonation's history of temperaments.
- Piano key frequencies
- Well-Tempered Clavier
- Musical tuning
[edit] 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
