login
The number of multiples (k) of n where an equal temperament with k*n divisions of the octave's best approximation of the just perfect fifth (3/2) is equal to that of an equal temperament with n divisions.
1

%I #45 May 12 2024 11:37:13

%S 1,2,2,1,6,1,5,1,1,3,1,25,1,2,2,1,8,1,4,1,1,3,1,12,1,2,2,1,13,1,3,1,1,

%T 4,1,8,1,2,2,1,30,1,3,1,1,5,1,6,1,2,2,1,165,1,2,2,1,6,1,5,1,1,3,1,22,

%U 1,2,2,1,9,1,4,1,1,3,1,11,1,2,2,1,15,1,3

%N The number of multiples (k) of n where an equal temperament with k*n divisions of the octave's best approximation of the just perfect fifth (3/2) is equal to that of an equal temperament with n divisions.

%C "Best" is by minimum absolute difference, so the best approximation for n divisions is the rational r = f/n which minimizes d = abs(r - L) where L = log_2(3/2). The value of f for each n is A366701(n).

%C a(n) = k is the largest k for which A366701(k*n) = k * A366701(n); i.e., subdividing by k has not offered a better approximation than r.

%C This k is the largest k for which 1/(k*n) > 2*d, since a step to f +- 1/(k*n) is not in the range (L-d, L+d) and therefore is not closer to L.

%H Soren Allen, <a href="/A371886/b371886.txt">Table of n, a(n) for n = 1..10000</a>

%H Soren Allen, <a href="/A371886/a371886.png">Plot of n, log(a(n)) for n = 1..3000</a>; the asymptotic "peak-like" curves in this plot consist of values of n which differ by 53.

%H Soren Allen, <a href="/A371886/a371886_1.png">Plot of n, log(a(n)) for n = 1..100000</a>; the asymptotic "peak-like" curves in this plot consist of values of n which differ by 665.

%F a(n) = floor(1 / (2 * abs(round(n*log_2(3/2)) - n*log_2(3/2)))).

%e For n = 12, the nearest integer to 12*log_2(3/2) is 7, yielding r = 7/12. Because r is equal for all multiples of 12 (24 yields 14/24, 36 yields 21/36, etc.) through 12*25=300 (175/300), a(12) = 25. 300 is in fact the highest multiple, as the best approximation for 12*26=312 is 183/312, which does not reduce to 7/12.

%t a[n_]:= Floor[1 / (2 * Abs[Round[n*Log2[3/2]] - n*Log2[3/2]])]; Array[a,84] (* _Stefano Spezia_, Apr 12 2024 *)

%Y Cf. A366701.

%K nonn

%O 1,2

%A _Soren Allen_, Apr 10 2024