A206788 -id:A206788 - cp's OEIS frontend

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.

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

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

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

Cf. A054540, A060525, A060526, A060527, A254351.

A005664 is a subsequence, A206788 is a supersequence.

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 */

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

Incorrect term 571611 removed by K. G. Stier, Jan 29 2015

More terms from Jon E. Schoenfield, Feb 06 2015

A254351 Numerators of increasingly better rational approximations to log(3)/log(2) with increasing denominators.

Original entry on oeis.org

2, 3, 5, 8, 11, 19, 46, 65, 84, 317, 401, 485, 569, 1054, 13133, 14187, 15241, 16295, 17349, 18403, 19457, 20511, 21565, 22619, 23673, 24727, 50508, 125743, 176251, 301994, 8632083, 8934077, 9236071, 9538065, 9840059, 10142053, 10444047, 10746041, 11048035

Offset: 1

K. G. Stier, Jan 29 2015

log(3)/log(2) = 1.5849625... (see A020857) is an irrational number. The fractions (2/1, 3/2, 5/3, 8/5, 11/7, 19/12, 46/29, 65/41, 84/53, 317/200, 401/253, 485/306, 569/359, 1054/665, ...) are a sequence of approximations to log(3)/log(2), where each is an improvement on its predecessors.

Numerators are shown here, the respective denominators are A060528 (and can also be found among the terms of A206788), both of which refer to equal divisions of the octave and good approximations to musical harmonics.

Cf. A060528 (denominators), A020857, A206788.

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(numerator)));

cp's OEIS Frontend

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.

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