Octaves (12 semitones) increase exponentially when measured on a linear frequency scale (Hz).

Octaves are equally spaced when measured on a logarithmic scale (cents).

Thetwelfth root of two or${\displaystyle \sqrt[12]{2}}$ (orequivalently${\displaystyle 2^{1/12}}$) is analgebraicirrational number, approximately equal to 1.0594631. It is important in Westernmusic theory, where it represents thefrequencyratio (musical interval) of asemitone (Play^ⓘ) intwelve-tone equal temperament. This number was proposed for the first time in relationship tomusical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones).^[a] A semitone itself is divided into 100cents (1 cent =${\displaystyle \sqrt[1200]{2}=2^{1/1200}}$).

Thetwelfth root oftwo to 20 significant figures is1.0594630943592952646.^[2] Fraction approximations in increasing order of accuracy include⁠18/17⁠,⁠89/84⁠,⁠196/185⁠,⁠1657/1564⁠, and⁠18904/17843⁠.

Amusical interval is a ratio of frequencies and theequal-tempered chromatic scale divides theoctave (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 2^1⁄12 times that of the one below it.^[3]

Applying this value successively to the tones of a chromatic scale, starting fromA abovemiddleC (known asA₄) with a frequency of 440 Hz, produces the following sequence ofpitches:

The finalA (A₅: 880 Hz) is exactly twice the frequency of the lowerA (A₄: 440 Hz), that is, one octave higher.

Other tuning scales use slightly different interval ratios:

• Thejust orPythagorean perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is agrad, the twelfth root of thePythagorean comma ($\sqrt[12]{531441/524288}$).
• The equal temperedBohlen–Pierce scale uses the interval of the thirteenth root of three ($\sqrt[13]{3}$).
• Stockhausen'sStudie II (1954) makes use of the twenty-fifth root of five ($\sqrt[25]{5}$), a compound major third divided into 5×5 parts.
• Thedelta scale is based on ≈$\sqrt[50]{3/2}$.
• Thegamma scale is based on ≈$\sqrt[20]{3/2}$.
• Thebeta scale is based on ≈$\sqrt[11]{3/2}$.
• Thealpha scale is based on ≈$\sqrt[9]{3/2}$.

Since the frequency ratio of a semitone is close to 106% ($100\sqrt[12]{2}\approx 105.946$), 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". Upscalereel-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 digitalpitch shifting to achieve similar results, ranging fromcents up to several half-steps. Reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not.

Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) bySimon Stevin.^[4] In 1581 Italian musicianVincenzo Galilei may be the first European to suggest twelve-tone equal temperament.^[1] The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musicianZhu Zaiyu using an abacus to reach twenty four decimal places accurately,^[1] calculated circa 1605 by Flemish mathematicianSimon Stevin,^[1] in 1636 by the French mathematicianMarin Mersenne and in 1691 by German musicianAndreas Werckmeister.^[5]

• Fret
• Just intonation § Practical difficulties
• Music and mathematics
• Piano key frequencies
• Scientific pitch notation
• Twelve-tone technique
• The Well-Tempered Clavier

• Barbour, J. M. (1933). "A Sixteenth Century Chinese Approximation forπ".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.{{cite book}}:ISBN / Date incompatibility (help)
• Partch, Harry (1974).Genesis of a Music. Da Capo Press.ISBN 0-306-80106-X.

• Mathematical constants
• Algebraic numbers
• Irrational numbers
• Musical tuning

• Pages using the Phonos extension
• Articles with short description
• Short description matches Wikidata
• CS1 errors: ISBN date