cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-14 of 14 results.

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

Views

Author

Jianing Song, Nov 08 2019

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    First[RealDigits[2^(5/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)
  • PARI
    default(realprecision, 100); 2^(5/12)

Formula

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

Views

Author

Jianing Song, Nov 08 2019

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    First[RealDigits[2^(11/12), 10, 100]] (* Paolo Xausa, Apr 28 2024 *)
  • PARI
    default(realprecision, 100); 2^(11/12)

Formula

Equals 2/A010774.
Equals Product_{k>=0} (1 + (-1)^k/(12*k + 1)). - Amiram Eldar, Jul 29 2020

A341113 Numerators of continued fraction convergents to 2^(1/12).

Original entry on oeis.org

1, 1, 17, 18, 89, 196, 1461, 1657, 3118, 7893, 18904, 140221, 579788, 720009, 2019806, 2739815, 166408706, 169148521, 673854269, 843002790, 2359859849, 19721881582, 100969267759, 120691149341, 342351566441, 463042715782, 805394282223, 1268436998005, 7147579272248
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2021

Keywords

Examples

			Convergents to [1; 16, 1, 4, 2, 7, 1, 1, 2, 2, ...]: 1, 17/16, 18/17, 89/84, 196/185, 1461/1379, 1657/1564, 3118/2943, 7893/7450, 18904/17843, ...

Links

Crossrefs

For denominators see A341114.
Cf. A001333, A010774, A103922.

Programs

  • Mathematica
    Join[{1}, Numerator[Convergents[2^(1/12), 28]]] (* Amiram Eldar, Apr 28 2021 *)

Formula

a(0) = 1, a(1) = 1, a(n) = A103922(n-1) * a(n-1) + a(n-2) for n >1.

A341114 Denominators of continued fraction convergents to 2^(1/12).

Original entry on oeis.org

0, 1, 16, 17, 84, 185, 1379, 1564, 2943, 7450, 17843, 132351, 547247, 679598, 1906443, 2586041, 157068903, 159654944, 636033735, 795688679, 2227411093, 18614977423, 95302298208, 113917275631, 323136849470, 437054125101, 760190974571, 1197245099672
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2021

Keywords

Links

Crossrefs

For numerators see A341113.
Cf. A002351, A010774, A103922.

Programs

  • Mathematica
    Join[{0}, Denominator[Convergents[2^(1/12), 27]]] (* Amiram Eldar, Feb 05 2021 *)

Formula

a(0) = 0, a(1) = 1, a(n) = A103922(n-1) * a(n-1) + a(n-2) for n > 1.
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.