A214832 -id:A214832

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A010774

Decimal expansion of 12th root of 2.

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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

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

FORMULA

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A103922, A010768.

Cf. A214832, A254531.

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved

A131071

12-note scale in Hertz (rounded to integers).

+10
4

261, 275, 293, 309, 330, 348, 366, 391, 412, 440, 464, 495, 521

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..13.

Wikipedia, Pythagorean tuning.

Index entries for sequences related to music.

FORMULA

The scale involves 9/8 and 256/243 as fractions and the start is A = 440 Hz.

The initial term (rounded frequency of the C) is calculated as 16/27 * 440 Hz = 260.74 Hz, cf. the Wikipedia page on Pythagorean tuning for the ratios of the frequencies. - M. F. Hasler, Oct 07 2011

CROSSREFS

Cf. A131062 for the corresponding C major scale. [M. F. Hasler, Oct 07 2011]

Cf. A214832.

KEYWORD

nonn

AUTHOR

Hans Isdahl, Sep 24 2007

STATUS

approved

A254531

a(n) is the position of the piano key whose frequency is closest to n Hz, start with A0 = the 1st key.

+10
4

1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22

(list; graph; refs; listen; history; text; internal format)

OFFSET

27,3

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 27..4308

Wikipedia, Piano Key Frequencies

Wikipedia, Twelfth root of two

Index entries for sequences based on music

FORMULA

a(n) = round(12*log_2(n/440)) + 49, 27 <= n <= 4308.

a(A214832(k)) = k for k = 1..88.

EXAMPLE

. | Frequency [Hz] | Piano key | Pitch

. i | f = A079731(i) | a(f) |

. ---+----------------+-----------+------

. 0 | 28 | 1 | A0

. 1 | 55 | 13 | A1

. 2 | 110 | 25 | A2

. 3 | 220 | 37 | A3

. 4 | 440 | 49 | A4 A440

. 5 | 880 | 61 | A5

. 6 | 1760 | 73 | A6

. 7 | 3520 | 85 | A7 .

PROG

(Haskell)

a254531 = (+ 49) . round . (* 12) . logBase 2 . (/ 440) . fromIntegral

(PARI) a(n) = round(12*log(n/440)/log(2))+49 \\ Jianing Song, Oct 14 2019

CROSSREFS

Cf. A214832, A079731, A010774.

KEYWORD

nonn,fini,full

AUTHOR

Reinhard Zumkeller, Feb 01 2015

EXTENSIONS

Corrected by Jianing Song, Oct 14 2019

STATUS

approved

A079731

Fundamental piano frequencies in Hertz rounded to nearest integer.

+10
2

28, 55, 110, 220, 440, 880, 1760, 3520

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..7.

J. Campbell, The Equal Tempered Scale and Peculiarities of Piano Tuning.

Index entries for sequences based on music

FORMULA

a(n)=round(27.5*2^n), n=0...7.

CROSSREFS

Cf. A214832, A254531.

KEYWORD

easy,fini,full,nonn

AUTHOR

Ralf Stephan, Feb 18 2003

STATUS

approved

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