A083026 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.

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, 43, 45, 47, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 89, 91, 93, 95, 96, 98, 100, 101, 103, 105, 107, 108, 110

Offset: 1

Author

James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

Keywords

• nonn
• easy

Comments

Key-numbers of the pitches of a major scale on a standard chromatic keyboard, with root = 0.

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.

Cumulative sum of A291454. - Halfdan Skjerning, Aug 30 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Wikipedia, Scale Music
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Cf. A291454.

A guide for some sequences related to modes and chords:

Modes:

Lydian mode (F): A083089

Ionian mode (C): this sequence

Mixolydian mode (G): A083120

Dorian mode (D): A083033

Aeolian mode (A): A060107 (raised seventh: A083028)

Phrygian mode (E): A083034

Locrian mode (B): A082977

Chords:

Major chord: A083030

Minor chord: A083031

Dominant seventh chord: A083032

Programs

Magma

[n : n in [0..150] | n mod 12 in [0, 2, 4, 5, 7, 9, 11]]; // Wesley Ivan Hurt, Jul 20 2016

Maple

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

Mathematica

Select[Range[0, 150], MemberQ[{0, 2, 4, 5, 7, 9, 11}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jul 20 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 4, 5, 7, 9, 11, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Quotient[12*Range[60], 7] - 1 (* Federico Provvedi, Sep 10 2022 *)

PARI

a(n)=[-1, 0, 2, 4, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, Jul 20 2016

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

Formula

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

From Wesley Ivan Hurt, Jul 20 2016: (Start)

a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.

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.

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)

a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018