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 1-10 of 14 results. Next

A060528 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, 79335, 111202, 190537, 5446238, 5636775, 5827312, 6017849, 6208386, 6398923, 6589460, 6779997, 6970534, 7161071
Offset: 1

Views

Author

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 12 2001

Keywords

Comments

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 6589460. This is not a perfect recurrent sequence because its self-accumulating nature fails between the 9th and 10th terms, between the 14th and 15th terms, and between the 30th and 31st terms. The examples of recurrence which are present in this sequence are of the same type that is seen in sequences A054540, A060526 and A060527. The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts. - corrected by K. G. Stier, Jan 29 2015
Also the denominators of increasingly better rational approximations to log(3)/log(2) = 1.5849625... (see A020857). The respective numerators are A254351. The reason why the sequence's "self-accumulating nature fails between the 9th and 10th terms, the 14th and 15th terms and the 30th and 31st terms" (see original comment) is simply that 84/53, 1054/665 and 301994/190537 are very good approximations, thus followed by a jump. (E.g., this phenomenon can also be seen in the numerators and denominators of rational approximations to Pi.). - K. G. Stier, Jan 29 2015

Crossrefs

Cf. A054540, A060525, A060526, A060527, A254351.
A005664 is a subsequence, A206788 is a supersequence.

Programs

  • Maxima
    x:bfloat(log(3)/log(2)),fpprec:100, errold:2,for denominator:1 thru 10000 do (numerator:round(x*denominator), errnew:abs(x-numerator/denominator), if errnew < errold then (errold:errnew, print(denominator))); /* K. G. Stier, Jan 29 2015 */
  • PARI
    lista(nn) = {d = 2; v = log(3)/log(2); for (den=1, nn, num = round(v*den); newd = abs(v-num/den); if (newd < d, print1(den, ", "); d = newd;););} \\ after Maxima, Michel Marcus, Feb 28 2015

Extensions

Incorrect term 571611 removed by K. G. Stier, Jan 29 2015
More terms from Jon E. Schoenfield, Feb 06 2015

A005663 Numerators of convergents to log_2(3) = log(3)/log(2).

Original entry on oeis.org

1, 2, 3, 8, 19, 65, 84, 485, 1054, 24727, 50508, 125743, 176251, 301994, 16785921, 17087915, 85137581, 272500658, 357638239, 630138897, 9809721694, 10439860591, 103768467013, 217976794617, 1193652440098, 8573543875303
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			log_2(3) = 1.5849625007211561814537389439...

References

  • R. K. Guy, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005664, A028507, A020857.

Programs

  • Mathematica
    Numerator[Convergents[Log[2,3],30]] (* Harvey P. Dale, Sep 10 2015 *)
  • PARI
    a(n) = component(component(contfracpnqn(contfrac(log(3)/log(2), n)), 1), 1) \\ Michel Marcus, May 20 2013

Extensions

More terms from James Sellers, Sep 16 2000

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

A117559 Equal divisions of the octave of decreasing fifteen-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 7, 8, 24, 58, 111, 130, 224, 270, 494, 2190, 2684, 5585, 6079, 14618, 20203, 81860, 96478
Offset: 0

Views

Author

Gene Ward Smith, Mar 28 2006

Keywords

Comments

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest.

Links

Crossrefs

Cf. A117554, A117555, A117556, A117557, A117559, A005664, A117536, A117538.

A028507 Continued fraction expansion for log_2(3).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 1, 5, 2, 23, 2, 2, 1, 1, 55, 1, 4, 3, 1, 1, 15, 1, 9, 2, 5, 7, 1, 1, 4, 8, 1, 11, 1, 20, 2, 1, 10, 1, 4, 1, 1, 1, 1, 1, 37, 4, 55, 1, 1, 49, 1, 1, 1, 4, 1, 3, 2, 3, 3, 1, 5, 16, 2, 3, 1, 1, 1, 1, 1, 5, 2, 1, 2, 8, 7, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 2, 2, 2, 16, 8, 10, 1, 25, 2, 1
Offset: 0

Views

Author

Tony Smith (tsmith(AT)innerx.net)

Keywords

Examples

			log_2(3) = 1.5849625007211561814537389439...

Links

Crossrefs

Cf. A005663, A005664, A020857 (decimal expansion).

Programs

  • Maple
    Digits := 200: convert(evalf( log(3)/log(2) ),confrac);
  • Mathematica
    ContinuedFraction[Log[2,3],120] (* Harvey P. Dale, Oct 24 2011 *)

Extensions

More terms from James Sellers, Sep 16 2000
Offset changed by Andrew Howroyd, Aug 07 2024

A117554 Equal divisions of the octave of decreasing 5-limit Pepper ambiguity.

Original entry on oeis.org

1, 3, 12, 19, 34, 53, 118, 441, 612, 730, 1171, 1783, 2513, 4296, 25164, 52841, 73709, 78005
Offset: 0

Views

Author

Gene Ward Smith, Mar 28 2006

Keywords

Comments

We may define the p-limit Pepper ambiguity, for any odd prime p, as the maximum of the ratios of the errors of the nearest approximation to the members of the p-limit tonality diamond to the next nearest. In the 5-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4 and 5/3 to the next nearest.
The 3-limit version of this is A005664, so in some sense this is a generalization of that. However it is also very closely related to A054540.

Links

Crossrefs

Cf. A117555, A117556, A117557, A117558, A117559, A005664, A054540.

A117555 Equal divisions of the octave of decreasing seven-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 3, 4, 12, 22, 27, 31, 99, 171, 3125, 6691, 11664, 18355, 84814, 103169
Offset: 0

Views

Author

Gene Ward Smith, Mar 28 2006

Keywords

Comments

We may define the n-limit Pepper ambiguity, for any odd number greater than one n, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest. In the 7-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4, 7/5 and 7/6 to the next nearest.

Links

Crossrefs

Cf. A117554, A117556, A117557, A117558, A117559, A005664, A117536, A117538.

A117556 Equal divisions of the octave of decreasing nine-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 4, 5, 12, 19, 31, 41, 99, 171, 3125, 11664, 18355, 84814, 103169
Offset: 0

Views

Author

Gene Ward Smith, Mar 28 2006

Keywords

Comments

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest. In the 9-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4, 7/5, 7/6, 9/8, 9/7 and 9/5 to the next nearest.

Links

Crossrefs

Cf. A117554, A117555, A117557, A117558, A117559, A005664, A117536, A117538.

A355512 Sum of numerator and denominator in the convergents of the approximation of log(2)/log(3) by a continued fraction.

Original entry on oeis.org

2, 3, 5, 13, 31, 106, 137, 791, 1719, 40328, 82375, 205078, 287453, 492531, 27376658, 27869189, 138853414, 444429431, 583282845, 1027712276, 15998966985, 17026679261, 169239080334, 355504839929, 1946763279979, 13982847799782, 15929611079761, 29912458879543, 135579446597933
Offset: 1

Views

Author

Hugo Pfoertner, Jul 05 2022

Keywords

Crossrefs

Cf. A005663, A005664, A102525, A355513, A355515.
Cf. A355514 for the relation to potential cycle lengths of the 3x+1 problem.

Programs

  • PARI
    a355512(upto) = {my(q=log(2)/log(3), fa=oo); for (denmax=1, upto, my(f=bestappr(q,denmax)); if (fa!=f, print1(numerator(f)+denominator(f),", "); fa=f))};
\\ needs increased precision for larger terms
a355512(10^6)
  • PARI
    \\ See also A005663 and A005664 for efficient code.

A117558 Equal divisions of the octave of decreasing thirteen-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 7, 8, 24, 37, 46, 58, 130, 198, 224, 270, 494, 1506, 2684, 5585, 6079, 14618, 20203, 81860, 87939, 96478
Offset: 0

Views

Author

Gene Ward Smith, Mar 28 2006

Keywords

Comments

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest.

Links

Crossrefs

Cf. A117554, A117555, A117556, A117557, A117559, A005664, A117536, A117538.
Showing 1-10 of 14 results. Next

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

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