Twelfth root of two

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The twelfth root of two or \sqrt[12]{2} is an algebraic irrational number. It is most important in music theory, where it represents the frequency ratio of a semitone in Twelve-tone equal temperament.

Numerical value[edit]

Its value is approximately 1.0594630943592952645, which is slightly more than 1817 ≈ 1.0588. As of December 2013, its numerical value in decimal has been computed to at least twenty billion digits.[1]

The equal-tempered chromatic scale[edit]

Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts.

Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches:

Note
 
Frequency
Hz
Multiplier
 
Coefficient
(to six places)
A 440.00 20/12 1.000000
A/B 466.16 21/12 1.059463
B 493.88 22/12 1.122462
C 523.25 23/12 1.189207
C/D 554.37 24/12 1.259921
D 587.33 25/12 1.334839
D/E 622.25 26/12 1.414213
E 659.26 27/12 1.498307
F 698.46 28/12 1.587401
F/G 739.99 29/12 1.681792
G 783.99 210/12 1.781797
G/A 830.61 211/12 1.887748
A 880.00 212/12 2.000000

The final A (880 Hz) is twice the frequency of the lower A (440 Hz), that is, one octave higher.

Pitch adjustment[edit]

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 by about 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). This is the technique used to create "Alvin and the Chipmunks" (the recording tempo was slowed accordingly, so that when the speed was increased on playback [nearly doubled, raising pitch about an octave - while simultaneously halving the duration], the now-familiar sound of the chipmunks was created). Modern recording studios utilize digital pitch shifting to achieve similar results, (although the "beat" is not affected—the moment-to-moment frequency is sampled, then altered; this allows the tempo to remain unchanged) ranging from cents up to several half-steps.

History[edit]

Calculated in 1636 by the French mathematician Marin Mersenne, and as the techniques for calculating logarithms develop, the original approach for calculation would eventually become trivial.

See also[edit]

Further reading[edit]

  • 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

References[edit]

  1. ^ Lukasz Komsta: Computations page