Twelfth root of two
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The twelfth root of two or 12√ is an algebraic irrational number. It is most important in music theory, where it represents the frequency ratio of a semitone ( Play (help·info)) 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. It allows measurement and discussion of equal intervals (frequency ratios) as equally spaced. A semitone is 100 cents (1 cent = 1200√).
The twelfth root of two to 20 significant figures is 4630943592952646. Fraction approximations in order of accuracy are 1.05918/, 196/, and 18904/.
The equal-tempered chromatic scale
|Note||Standard interval name(s)
relating to A 440
(to six places)
|A♯/B♭||Minor second/Half step/Semitone||466.16||21⁄12||463 1.059||≈ 16⁄15|
|B||Major second/Full step/Whole tone||493.88||22⁄12||462 1.122||≈ 9⁄8|
|C||Minor third||523.25||23⁄12||207 1.189||≈ 6⁄5|
|C♯/D♭||Major third||554.37||24⁄12||921 1.259||≈ 5⁄4|
|D||Perfect fourth||587.33||25⁄12||839 1.334||≈ 4⁄3|
|D♯/E♭||Augmented fourth/Diminished fifth/Tritone||622.25||26⁄12||213 1.414||≈ 7⁄5|
|E||Perfect fifth||659.26||27⁄12||307 1.498||≈ 3⁄2|
|F||Minor sixth||698.46||28⁄12||401 1.587||≈ 8⁄5|
|F♯/G♭||Major sixth||739.99||29⁄12||792 1.681||≈ 5⁄3|
|G||Minor seventh||783.99||210⁄12||797 1.781||≈ 9⁄5|
|G♯/A♭||Major seventh||830.61||211⁄12||748 1.887||≈ 15⁄8|
The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.
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, ranging from cents up to several half-steps (note that reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not).
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.
- Just intonation § Practical difficulties
- Music and mathematics
- Piano key frequencies
- Scientific pitch notation
- The Well-Tempered Clavier
- Musical tuning
- nth root
- Barbour, J. M. (1933). "A Sixteenth Century Approximation for π". American Mathematical Monthly. 40 (2): 69–73. doi:10.2307/2300937. 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.