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 1-5 of 5 results.

A221363 Decimal expansion of the Pythagorean comma.

Original entry on oeis.org

1, 0, 1, 3, 6, 4, 3, 2, 6, 4, 7, 7, 0, 5, 0, 7, 8, 1, 2, 5
Offset: 1

Views

Author

Jonathan Sondow, Jan 19 2013

Keywords

Comments

In musical tuning, the Pythagorean comma is 12 fifths / 7 octaves = (3/2)^12 / 2^7.

Examples

			3^12 / 2^19 = 531441/524288 = 1.0136432647705078125

References

  • Larry Baggett, In the Dark on the Sunny Side: A Memoir of an Out-of-Sight Mathematician, Mathematical Association of America, 2012, p. 78.
  • Dave Benson, Music: A Mathematical Offering. Cambridge: Cambridge University Press (2006): 164.
  • J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 257.

Links

Crossrefs

Cf. A005663, A005664, A010774, A046102.

Programs

  • Mathematica
    RealDigits[N[31441/524288, 50]][[1]]

Formula

A229948/A229943 - Omar E. Pol, Oct 25 2013

A229948 Decimal expansion of 2187/2048, the Pythagorean apotome.

Original entry on oeis.org

1, 0, 6, 7, 8, 7, 1, 0, 9, 3, 7, 5
Offset: 1

Views

Author

Omar E. Pol, Oct 25 2013

Keywords

Comments

The Pythagorean apotome 2187/2048 (also called the apotome Pythagorica) is one of the musical intervals.

Examples

			1.06787109375

Links

Crossrefs

Cf. A010774, A221363, A229943, A230437.

Formula

A221363 * A229943 = (3^12/2^19)*(2^8/3^5) = 3^7/2^11 = 2187/2048.

A210621 Decimal expansion of 256/81.

Original entry on oeis.org

3, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2
Offset: 1

Views

Author

N. J. A. Sloane, Mar 24 2012

Keywords

Comments

According to Maor (1994), the Rhind Papyrus asserts that a circle has the same area as a square with a side that is 8/9 the diameter of the circle. From this we can determine that 256/81 is one of the ancient Egyptian approximations of Pi. - Alonso del Arte, Jun 12 2012

Examples

			3.1604938271604938271604938271604938271604938271604938271604...

References

  • Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 12.
  • Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 88.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 237.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, ยง3.6 The Quest for Pi, p. 89.
  • Carl Theodore Heisel, Behold! The grand problem no longer unsolved: The circle squared beyond refutation, c. 1935. (proposes Pi = 3 + 13/81)
  • Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994): 41, 47 note 1.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 48.

Links

Crossrefs

Cf. A068028, A229943, A255910, A268315.

Programs

Formula

256/81 = (4/3)^4.
Equals 3*A229943 = A255910^2 = A268315/3. - Hugo Pfoertner, Jun 26 2024

Extensions

Offset corrected by Rick L. Shepherd, Jan 06 2014

A230437 Decimal expansion of (2/(3 - 2^(1/2)))^(1/4).

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 25 2013

Keywords

Comments

Approximation to the frequency ratio of a semitone in equal temperament, suggested by Marin Mersenne.
This constant is close to A010774, the frequency ratio of a semitone in equal temperament. The difference is 0.0002695778428445...

Examples

			1.059732672202139808746969376992583251211677...

Links

Crossrefs

Cf. A010774, A221363, A229943, A229948.

Programs

  • Mathematica
    RealDigits[Surd[2/(3 - 2^(1/2)), 4], 10, 120][[1]] (* Amiram Eldar, May 16 2023 *)

A358029 Decimal expansion of the ratio between step sizes of the diatonic and chromatic semitones produced by a circle of 12 perfect fifths in Pythagorean tuning.

Original entry on oeis.org

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

Views

Author

Eliora Ben-Gurion, Oct 25 2022

Keywords

Comments

Pythagorean tuning is a form of tuning produced by repeated stacking of the perfect fifth, which has the frequency ratio of 3:2. A circle of 12 perfect fifths is approximately equal to the tuning system predominantly in use in the world today. If the perfect fifth is stacked 12 times and the resulting sequence is octave-reduced, then this divides the octave into 5 chromatic semitones which are equal to 2187/2048 (A229948), and 7 diatonic semitones which are equal to 256/243 (A229943). Diatonic semitones are those which are derived from a circle of 7 perfect fifths, the diatonic scale, and 5 chromatic semitones are a byproduct of an addition of 5 more perfect fifths, that is, another rotation, to the scale.

Examples

			1.2600167526708245359312761226039242337181155792327678334106520161...

Links

Crossrefs

Cf. A131071, A221363, A258054, A257811.
Cf. A229943, A229948.

Programs

  • Mathematica
    RealDigits[(7*Log[3] - 11*Log[2])/(8*Log[2] - 5*Log[3]), 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

Formula

Equals log(2187/2048) / log(256/243).
Equals log(A229948) / log(A229943).
Equals (7*log(3) - 11*log(2))/(8*log(2) - 5*log(3)).
Showing 1-5 of 5 results.

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