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.

Showing 1-4 of 4 results.

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

Views

Author

Jianing Song, Nov 08 2019

Keywords

Comments

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.

Examples

			1.78179743...

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... (this sequence)
Major seventh: 2^(11/12) = 1.8877486253... (A329220)
Perfect octave: 2^(12/12) = 2.0000000000

Programs

  • Mathematica
    First[RealDigits[2^(5/6), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)
  • PARI
    default(realprecision, 100); 2^(10/12)

Formula

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

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

A328228 Simple continued fraction expansion of 2^(7/12).

Original entry on oeis.org

1, 2, 147, 5, 1, 3, 5, 4, 4, 1, 1, 159, 6, 1, 1, 1, 4, 1, 2, 1, 2, 3, 1, 8, 15, 47, 1, 103, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 10, 3, 1, 2, 1, 2, 4, 1, 1, 1, 9, 28, 2, 4, 2, 2, 5, 1, 3, 1, 1, 2, 1, 1, 1, 52, 6, 2, 6, 1, 5, 94, 3, 6, 26, 1, 6, 5, 1, 3, 109
Offset: 0

Views

Author

Daniel Hoyt, Oct 08 2019

Keywords

Comments

2^(7/12) is the multiplier with respect to a base frequency to produce a perfect fifth interval in an equal tempered chromatic scale.
This constant is of interest because it is close to the just intonation perfect fifth coefficient of 1.5 (continued fraction [1, 2]). It is the closest to just intonation of the chromatic scale divisions other than the octaves (2*frequency), and unison (1*frequency). The perfect fifth is the most consonant division of the chromatic scale.

Links

Crossrefs

Cf. A005664, A103922, A328229.

Programs

Showing 1-4 of 4 results.

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