A291454 Number of half tones between successive pitches in a major scale.

2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1

Offset: 1

Author

Halfdan Skjerning, Aug 24 2017

Keywords

• nonn
• easy
• hear

Comments

In music theory the repeating sequence '2,2,1,2,2,2,1' is the number of steps of half tones in pitch between the tones of a major scale. Starting at, for example, the tone 'C' that is the first tone of the C major scale, 2 half tones up leads to 'D', which is the second tone in the scale. The scale then is: C,D,E,F,G,A,B and C. Starting at another term in the sequence will produce a different scale; for example, '2,1,2,2,1,2,2' will produce a minor scale.

From Robert G. Wilson v, Aug 25 2017: (Start)

First forward difference of A083026.

Decimal expansion of 737407/3333333. (End)

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..1000
• Wikipedia, Scale (music)

Crossrefs

Cf. A083026, A286748.

Programs

Magma

[12*(n+1) div 7 - 12*n div 7:  n in [1..80]]; // Vincenzo Librandi, Oct 21 2018

Maple

a:=proc(n) floor(12*(n+1)/7-floor(12*n/7)) end: seq(a(n),n=1..110); # Muniru A Asiru, Oct 19 2018

Mathematica

Table[{2, 2, 1, 2, 2, 2, 1}, 15] // Flatten  (* Robert G. Wilson v, Aug 25 2017 *)
Table[Floor[12/7 (k + 1)] - Floor[12/7 k], {k, 1, 100}] (* Federico Provvedi,Oct 18 2018 *)

PARI

a(n)=[1,2,2,1,2,2,2][n%7+1] \\ Charles R Greathouse IV, Aug 26 2017

Formula

a(n) = floor(12*(n+1)/7) - floor(12*n/7). - Federico Provvedi, Oct 18 2018

Dirichlet g.f.: 2*zeta(s) - 7^(-s)*(zeta(s,3/7) + zeta(s)). - Federico Provvedi, Aug 27 2021