A010768 -id:A010768 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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.

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

Cf. A103922, A010768.

Cf. A214832, A254531.

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

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

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

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

A203145 Decimal expansion of Gamma(5/6).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			1.1287870299081259612609010902588420133267874416647554517520...

Cf. A002161, A019692, A073005, A073006, A175379.

RealDigits[Gamma[5/6], 10, 100][[1]] (* Bruno Berselli, Dec 18 2012 *)

gamma(5/6) \\ Charles R Greathouse IV, Nov 26 2024

A073005 * this * A231863 * A010768 = A073006. - R. J. Mathar, Jan 15 2021
Equals 2*Pi/Gamma(1/6) = A019692 / A175379. - Amiram Eldar, Jul 04 2023
Equals 2^(4/3) * Pi^(3/2) / (sqrt(3) * Gamma(1/3)^2). - Vaclav Kotesovec, Jul 04 2023

A329219 Decimal expansion of 2^(10/12) = 2^(5/6).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(10/12) is the ratio of the frequencies of the pitches in a minor seventh (e.g., D4-C5) in 12-tone equal temperament.

			1.78179743...

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (this sequence)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/6), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PARI
```
default(realprecision, 100); 2^(10/12)
```

Equals 2/A010768.

Equals Product_{k>=0} (1 + (-1)^k/(6*k + 1)). - Amiram Eldar, Jul 25 2020

A329216 Decimal expansion of 2^(5/12).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(5/12) is the ratio of the frequencies of the pitches in a perfect fourth (e.g., D4-G4) in 12-tone equal temperament.

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (this sequence)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (A329219)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)

PARI
```
default(realprecision, 100); 2^(5/12)
```

Equals 2/A328229.

A329220 Decimal expansion of 2^(11/12).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(11/12) is the ratio of the frequencies of the pitches in a major seventh (e.g., D4-C#5) in 12-tone equal temperament.

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (A329219)
Major seventh:   2^(11/12) = 1.8877486253...  (this sequence)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(11/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)

PARI
```
default(realprecision, 100); 2^(11/12)
```

Equals 2/A010774.

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

A271836 Decimal expansion of 3^(1/3) / 2^(1/6).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Apr 15 2016

Used in a formula for a regular octahedron, a = 3^(1/3)/2^(1/6) * V^(1/3), where a is the edge length and V^(1/3) is the cube root of the volume.

An algebraic number of degree 6 and denominator 2; minimal polynomial is 2x^6 - 9. - Charles R Greathouse IV, Apr 18 2016

			1.2848982934253252956716...

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Wikipedia, Octahedron
Index entries for algebraic numbers, degree 6

Cf. A002581, A010768.

Maple
```
Digits:=100: evalf(3^(1/3)/2^(1/6));
```
Mathematica
```
RealDigits[N[3^(1/3)/2^(1/6), 100]]
```

3^(1/3) / 2^(1/6) \\ Altug Alkan, Apr 15 2016

sqrtn(9/2,6) \\ Charles R Greathouse IV, Apr 18 2016

Equals A002581 / A010768.

cp's OEIS Frontend

A010774 Decimal expansion of 12th root of 2.

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A203145 Decimal expansion of Gamma(5/6).

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A329219 Decimal expansion of 2^(10/12) = 2^(5/6).

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A329216 Decimal expansion of 2^(5/12).

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A329220 Decimal expansion of 2^(11/12).

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A271836 Decimal expansion of 3^(1/3) / 2^(1/6).

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