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.

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

A229943 Decimal expansion of 256/243, the Pythagorean semitone.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 25 2013

Keywords

Comments

The Pythagorean diatonic semitone is one of the musical intervals. Has a ratio of 256/243, and is often called the Pythagorean limma. It is also sometimes called the Pythagorean minor semitone.
After the initial term the sequence has period 27, repeat: 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9.

Examples

			1.053497942386831275720164609...

References

  • J. M. Merino de la Fuente, Las vibraciones de la música, Editorial Club Universitario (2006), 133.
  • Alberto Rojo, La física en la vida cotidiana, Siglo Veintiuno Editores (2011), 137.

Links

Crossrefs

Cf. A010774, A221363, A229948, A230437.

Programs

  • Mathematica
    RealDigits[256/243,10,120][[1]] (* Harvey P. Dale, Jul 17 2019 *)

Formula

A229948/A221363 = (3^7/2^11)/(3^12/2^19) = 2^8/3^5 = 256/243.

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 1-4 of 4 results.

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