A258054 Circle of fifths cycle (counterclockwise).

1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8

Offset

1,2

Comments

The twelve notes dividing the octave are numbered 1 through 12 sequentially. This sequence begins at a certain note, travels down a perfect fifth twelve times (seven semitones), and arrives back at the same note. If justly tuned fifths are used, the final note will be flat by the Pythagorean comma (roughly 23.46 cents or about a quarter of a semitone).

Period 12: repeat [1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8]. - Omar E. Pol, May 18 2015

The string [1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8] is also in both A023127 and A054073. - Omar E. Pol, May 19 2015

Links

Table of n, a(n) for n=1..108.

OEIS Wiki, The multi-faceted reach of the OEIS: Music

Wikipedia, Circle of fifths

Wikipedia, Pythagorean comma

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

OEIS index entry for music

Formula

Periodic with period 12: a(n) = 1 + (5(n-1) mod 12).

From Colin Barker, Nov 15 2019: (Start)

G.f.: x*(1 + 6*x + 11*x^2 + 4*x^3 + 9*x^4 + 2*x^5 + 7*x^6 + 12*x^7 + 5*x^8 + 10*x^9 + 3*x^10 + 8*x^11) / (1 - x^12).

a(n) = a(n-12) for n>12.

(End)

Example

For a(3), 1+5+5 = 11 (mod 12).
For a(4), 1+5+5+5 = 4 (mod 12).

Maple

A258054:=n->1+(5*(n-1) mod 12): seq(A258054(n), n=1..100); # Wesley Ivan Hurt, May 22 2015

Mathematica

PadRight[{}, 100, {1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8}] (* Vincenzo Librandi, May 19 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8}, 108] (* Ray Chandler, Aug 27 2015 *)

Prog

(Magma) [1+5*(n-1) mod 12: n in [1..80]]; // Vincenzo Librandi, May 19 2015
(PARI) a(n)=1+5*(n-1) \\ Charles R Greathouse IV, May 22 2015
(PARI) Vec(x*(1 + 6*x + 11*x^2 + 4*x^3 + 9*x^4 + 2*x^5 + 7*x^6 + 12*x^7 + 5*x^8 + 10*x^9 + 3*x^10 + 8*x^11) / (1 - x^12) + O(x^80)) \\ Colin Barker, Nov 15 2019

Crossrefs

Cf. A221363 (Pythagorean comma), A257811 (clockwise circle of fifths cycle).

Sequence in context: A284738 A100129 A009443 * A106540 A212208 A334280

Adjacent sequences: A258051 A258052 A258053 * A258055 A258056 A258057

Keyword

nonn,easy

Author

Peter Woodward, May 17 2015

Extensions

Extended by Ray Chandler, Aug 27 2015

Status

approved