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a(n) is the nearest integer to n*log_2(3/2).
2

%I #21 Apr 12 2024 07:56:37

%S 0,1,1,2,2,3,4,4,5,5,6,6,7,8,8,9,9,10,11,11,12,12,13,13,14,15,15,16,

%T 16,17,18,18,19,19,20,20,21,22,22,23,23,24,25,25,26,26,27,27,28,29,29,

%U 30,30,31,32,32,33,33,34,35,35,36,36,37,37,38,39,39,40,40,41,42,42

%N a(n) is the nearest integer to n*log_2(3/2).

%C For n > 0, a(n) is the number of steps of a perfect fifth in n-edo (equal divison of octaves).

%C In n-edo, a chromatic semitone is 7*a(n)-4*n steps wide, and a diatonic semitone is 3*n-5*a(n) steps wide.

%C Tall Kite calls n-edo

%C - superflat if a(n)/n < 4/7, which occurs at n = 2, 4, 9, 11, 16, 23. In these tuning systems a chromatic semitone has negative width, so the sharp sign lowers the pitch and the flat sign raises it, and major intervals are narrower than minor intervals.

%C - perfect if a(n)/n = 4/7, which occurs at n = 7, 14, 21, 28, 35. In these tuning systems a chromatic semitone has zero width, so there are no sharps or flats, and there is no difference between major and minor keys.

%C - diatonic if 4/7 < a(n)/n < 3/5, which occurs at n = 12, 17, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36 or more. In these tuning systems both the width of a chromatic semitone and the width of a diatonic semitone is positive.

%C - pentatonic if a(n)/n = 3/5, which occurs at n = 5, 10, 15, 25, 25, 30. In these tuning systems a diatonic semitone has zero width, so we have B being enharmonic to the adjacent C and F being enharmonic to the adjacent E.

%C - supersharp if a(n)/n > 3/5, which occurs at n = 1, 3, 6, 8, 13, 18. In these tuning systems a diatonic semitone has negative width, so we have B higher the adjacent C and F lower than the adjacent E.

%H Jianing Song, <a href="/A366701/b366701.txt">Table of n, a(n) for n = 0..10000</a>

%H Tall Kite, <a href="https://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf">Notation Guide for EDOs 5-72</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Equal_temperament">Equal temperament</a>

%e a(12) = 7 since 12*log_2(3/2) = 7.01955... This corresponds to that a perfect fifth is 7 steps wide in 12-edo.

%e a(19) = 11 since 19*log_2(3/2) = 11.11428... This corresponds to that a perfect fifth is 11 steps wide in 19-edo.

%t a[n_]:= Round[n * Log[3/2]/Log[2]]; Array[a,72] (* _Stefano Spezia_, Apr 12 2024 *)

%o (PARI) a(n) = round(n * log(3/2)/log(2))

%Y Cf. A020857 (log_2(3) = 1 + log_2(3/2)).

%K nonn,easy

%O 0,4

%A _Jianing Song_, Oct 16 2023