Twelfth root of two

The twelfth root of two or 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. Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.^[1]

Numerical value

The twelfth root of two to 20 significant figures is 1.0594630943592952646. Fraction approximations in order of accuracy are ¹⁸⁄₁₇, ¹⁹⁶⁄₁₈₅, and ¹⁸⁹⁰⁴⁄₁₇₈₄₃.

As of December 2013, its numerical value has been computed to at least twenty billion decimal digits.^[2]

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)	~ Ratio
A	440.00	2^⁰⁄₁₂	1.000000	1
A♯/B♭	466.16	2^¹⁄₁₂	1.059463
B	493.88	2^²⁄₁₂	1.122462	~ ⁹⁄₈
C	523.25	2^³⁄₁₂	1.189207
C♯/D♭	554.37	2^⁴⁄₁₂	1.259921	~ ⁵⁄₄
D	587.33	2^⁵⁄₁₂	1.334839	~ ⁴⁄₃
D♯/E♭	622.25	2^⁶⁄₁₂	1.414213
E	659.26	2^⁷⁄₁₂	1.498307	~ ³⁄₂
F	698.46	2^⁸⁄₁₂	1.587401
F♯/G♭	739.99	2^⁹⁄₁₂	1.681792	~ ⁵⁄₃
G	783.99	2^{¹⁰⁄₁₂}	1.781797
G♯/A♭	830.61	2^{¹¹⁄₁₂}	1.887748
A	880.00	2^{¹²⁄₁₂}	2.000000	2

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 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). 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.

DJ turntables can have an adjustment up to ±20%, but this is more often used for beat synchronization between songs than for pitch adjustment, which is mostly useful only in transitions between beatless and ambient parts. For beatmatching music of high melodic content the DJ would primarily try to look for songs that sound harmonic together when set to equal tempo.

History

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

References

↑ Christensen, Thomas, The Cambridge history of Western music theory (2002) - page 205
↑ Komsta, Lukasz. "Computations page".

Barbour, J. M. (1933). "A Sixteenth Century Approximation for π". American Mathematical Monthly. 40 (2): 69–73. JSTOR 2300937.
Ellis, Alexander; Helmholtz, Hermann (1954). On the Sensations of Tone. Dover Publications. ISBN 0-486-60753-4.
Partch, Harry (1974). Genesis of a Music. Da Capo Press. ISBN 0-306-80106-X.

Algebraic numbers

Algebraic integer Chebyshev nodes Constructible number Conway's constant ³√2 Eisenstein integer Gaussian integer $φ$ Kummer ring Perron number Pisot–Vijayaraghavan number Quadratic irrational number ℚ Root of unity Salem number $δ$ _$S$ √2 √3 √5 ¹²√2		ℚ

Irrational numbers

Prime ( $ρ$ ) Euler–Mascheroni ( $γ$ ) Logarithm of 2 Twelfth root of two (¹²√2) Apéry's ( $ζ (3)$ ) Sophomore's dream Plastic ( $ρ$ ) Square root of 2 (√2) Erdős–Borwein ( $E$ ) Golden ratio ( $φ$ ) Square root of 3 (√3) Square root of 5 (√5) Silver ratio ( $δ S$ ) Second Feigenbaum ( $α$ ) Euler's ( $e$ ) Pi ( $π$ ) First Feigenbaum ( $δ$ )

Schizophrenic Transcendental Trigonometric

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