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%I #68 Oct 25 2022 20:16:31
%S 0,2,4,5,7,9,11,12,14,16,17,19,21,23,24,26,28,29,31,33,35,36,38,40,41,
%T 43,45,47,48,50,52,53,55,57,59,60,62,64,65,67,69,71,72,74,76,77,79,81,
%U 83,84,86,88,89,91,93,95,96,98,100,101,103,105,107,108,110
%N Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.
%C Key-numbers of the pitches of a major scale on a standard chromatic keyboard, with root = 0.
%C Also key-numbers of the pitches of an Ionian mode scale on a standard chromatic keyboard, with root = 0. An Ionian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone C.
%C Cumulative sum of A291454. - _Halfdan Skjerning_, Aug 30 2017
%H G. C. Greubel, <a href="/A083026/b083026.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>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).
%F G.f.: x^2*(x + 1)*(x^5 + x^4 + x^3 + x^2 + 2)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - _R. J. Mathar_, Oct 08 2011
%F From _Wesley Ivan Hurt_, Jul 20 2016: (Start)
%F a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
%F a(n) = (84*n - 70 - 2*(n mod 7) - 2*((n + 1) mod 7) - 2*((n + 2) mod 7) + 5*((n + 3) mod 7) - 2*((n + 4) mod 7) - 2*((n + 5) mod 7) + 5*((n + 6) mod 7))/49.
%F a(7k) = 12k - 1, a(7k-1) = 12k - 3, a(7k-2) = 12k - 5, a(7k-3) = 12k - 7, a(7k-4) = 12k - 8, a(7k-5) = 12k - 10, a(7k-6) = 12k - 12. (End)
%F a(n) = a(n-7) + 12 for n > 7. - _Jianing Song_, Sep 22 2018
%p A083026:=n->12*floor(n/7)+[0, 2, 4, 5, 7, 9, 11][(n mod 7)+1]: seq(A083026(n), n=0..100); # _Wesley Ivan Hurt_, Jul 20 2016
%t Select[Range[0, 150], MemberQ[{0, 2, 4, 5, 7, 9, 11}, Mod[#, 12]] &] (* _Wesley Ivan Hurt_, Jul 20 2016 *)
%t LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 4, 5, 7, 9, 11, 12}, 70] (* _Jianing Song_, Sep 22 2018 *)
%t Quotient[12*Range[60], 7] - 1 (* _Federico Provvedi_, Sep 10 2022 *)
%o (Magma) [n : n in [0..150] | n mod 12 in [0, 2, 4, 5, 7, 9, 11]]; // _Wesley Ivan Hurt_, Jul 20 2016
%o (PARI) a(n)=[-1, 0, 2, 4, 5, 7, 9][n%7+1] + n\7*12 \\ _Charles R Greathouse IV_, Jul 20 2016
%o (PARI) x='x+O('x^99); concat(0, Vec(x^2*(x+1)*(x^5+x^4+x^3+x^2+2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ _Jianing Song_, Sep 22 2018
%Y Cf. A291454.
%Y A guide for some sequences related to modes and chords:
%Y Modes:
%Y Lydian mode (F): A083089
%Y Ionian mode (C): this sequence
%Y Mixolydian mode (G): A083120
%Y Dorian mode (D): A083033
%Y Aeolian mode (A): A060107 (raised seventh: A083028)
%Y Phrygian mode (E): A083034
%Y Locrian mode (B): A082977
%Y Chords:
%Y Major chord: A083030
%Y Minor chord: A083031
%Y Dominant seventh chord: A083032
%K nonn,easy
%O 1,2
%A James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
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