A010774 Decimal expansion of 12th root of 2.

1, 0, 5, 9, 4, 6, 3, 0, 9, 4, 3, 5, 9, 2, 9, 5, 2, 6, 4, 5, 6, 1, 8, 2, 5, 2, 9, 4, 9, 4, 6, 3, 4, 1, 7, 0, 0, 7, 7, 9, 2, 0, 4, 3, 1, 7, 4, 9, 4, 1, 8, 5, 6, 2, 8, 5, 5, 9, 2, 0, 8, 4, 3, 1, 4, 5, 8, 7, 6, 1, 6, 4, 6, 0, 6, 3, 2, 5, 5, 7, 2, 2, 3, 8, 3, 7, 6, 8, 3, 7, 6, 8, 6, 3, 9, 4, 5, 5, 6

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

This number figures in our standard 12-tone tuning of music today.

It represents the frequency ratio of a semitone in equal temperament. The equal-tempered chromatic scale divides the octave, which has a ratio of 2:1, into twelve parts of equal ratio: [2^(n/12), 2^((n+1)/12)), 0 <= n <= 11. - Daniel Forgues, Feb 28 2013

Examples

			2^(1/12) = 1.059463094359295264561825294946341700779204317494...
		

References

• D. Coulter, Digital Audio Processing. Berkeley, California: Focal Press (2000) p. 30
• Ian Stewart, Professor Stewart's Incredible Numbers, London, Profile Books, 2015, pp. 217-228.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Étienne Ghys, Musique..., Images des Mathématiques, CNRS, Mar 15 2020.
• Wikipedia, Twelfth root of two
• Index entries for sequences based on music
• Index entries for algebraic numbers, degree 12

Crossrefs

Cf. A103922, A010768.

Cf. A214832, A254531.

Programs

Mathematica

RealDigits[N[2^(1/12), 100]][[1]] (* Alonso del Arte, Jan 06 2011 *)

PARI

sqrtn(2,12) \\ Charles R Greathouse IV, Apr 14 2014

Formula

Equals Product_{k>=0} (1 + (-1)^k/(12*k + 11)). - Amiram Eldar, Jul 29 2020

Equals sqrt(A010768). - Hugo Pfoertner, May 31 2024