A366701 - OEIS

A366701

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

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, 16, 17, 18, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

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.

Tall Kite calls n-edo

- 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.

- 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.

- 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 are positive.

- 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.

- 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.

LINKS

Jianing Song, Table of n, a(n) for n = 0..10000

Tall Kite, Notation Guide for EDOs 5-72

Wikipedia, Equal temperament

EXAMPLE

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.

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.

MATHEMATICA

a[n_]:= Round[n * Log[3/2]/Log[2]]; Array[a, 72] (* Stefano Spezia, Apr 12 2024 *)

PROG

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

CROSSREFS