A291454 - OEIS

%I #72 Dec 06 2022 01:03:39

%S 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,

%T 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,

%U 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

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

%C 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.

%C From _Robert G. Wilson v_, Aug 25 2017: (Start)

%C First forward difference of A083026.

%C Decimal expansion of 737407/3333333. (End)

%H Muniru A Asiru, <a href="/A291454/b291454.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Scale_(music)">Scale (music)</a>

%F a(n) = floor(12*(n+1)/7) - floor(12*n/7). - _Federico Provvedi_, Oct 18 2018

%F Dirichlet g.f.: 2*zeta(s) - 7^(-s)*(zeta(s,3/7) + zeta(s)). - _Federico Provvedi_, Aug 27 2021

%p 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

%t Table[{2, 2, 1, 2, 2, 2, 1}, 15] // Flatten  (* _Robert G. Wilson v_, Aug 25 2017 *)

%t Table[Floor[12/7 (k + 1)] - Floor[12/7 k], {k, 1, 100}] (* _Federico Provvedi_,Oct 18 2018 *)

%o (PARI) a(n)=[1,2,2,1,2,2,2][n%7+1] \\ _Charles R Greathouse IV_, Aug 26 2017

%o (Magma) [12*(n+1) div 7 - 12*n div 7:  n in [1..80]]; // _Vincenzo Librandi_, Oct 21 2018

%Y Cf. A083026, A286748.

%K nonn,easy,hear

%O 1,1

%A _Halfdan Skjerning_, Aug 24 2017