Twelfth root of two

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 equal temperament.

Contents

Numerical value

Its value is 1.05946309435929..., which is slightly more than 18⁄17 ≈ 1.0588. Better approximations are 196⁄185 ≈ 1.059459 or 18904⁄17843 ≈ 1.0594630948.

The equal-tempered chromatic scale

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

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 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. Modern recording studios utilize digital pitch shifting to achieve the same results, ranging from cents up to several half-steps.

History

The twelfth root of two was calculated accurately by the Chinese court astronomer, historian, physicist and mathematician Zhu Zaiyu, Prince of Zheng of the Ming Dynasty. In 1584, Zhu published a work 律呂精義 A clear explanation of that which concerns the 律 [equal temperament]. Prince Zhu made note of the difference between his ideal mathematically-tuned 呂 (ancient music instrument), which gave the theoretical music instrument lengths for 12-tone equal temperament correct to 25 places, implemented with an 81-column abacus and calculated the cubic root of the square root of the square root of 2, obtaining \sqrt[3]{\sqrt{\sqrt{2}}} = \sqrt [12] {2}=1.059463094359295264561825, which coincidentally applied a form of Pythagorean tuning.

Calculated again 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

References