A257811 Circle of fifths cycle (clockwise).

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

Offset

1,2

Comments

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

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

Links

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

Alonso del Arte, Alvin Hoover Belt, Daniel Forgues, and Charles R Greathouse IV, The multi-faceted reach of the OEIS: Music

Wikipedia, Circle of fifths

Wikipedia, Pythagorean comma

OEIS index entry for music

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

Formula

Periodic with period 12: a(n) = 1 + 7*(n-1) mod 12.

From Colin Barker, Nov 15 2019: (Start)

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

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

(End)

Example

For a(3), 1+7+7 == 3 (mod 12).
For a(4), 1+7+7+7 == 10 (mod 12).

Mathematica

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

Prog

(Magma) [1+7*(n-1) mod(12): n in [1..80]]; // Vincenzo Librandi, May 10 2015
(PARI) a(n)=7*(n-1)%12+1 \\ Charles R Greathouse IV, Jun 02 2015
(PARI) Vec(x*(1 + 8*x + 3*x^2 + 10*x^3 + 5*x^4 + 12*x^5 + 7*x^6 + 2*x^7 + 9*x^8 + 4*x^9 + 11*x^10 + 6*x^11) / (1 - x^12) + O(x^80)) \\ Colin Barker, Nov 15 2019

Crossrefs

Cf. A194835 (Contains this circle of fifths sequence), A007337 (sqrt(3) sequence), A258054 (counterclockwise circle of fifths cycle).

Sequence in context: A011467 A246671 A069610 * A122159 A365445 A069200

Adjacent sequences: A257808 A257809 A257810 * A257812 A257813 A257814

Keyword

nonn,easy

Author

Peter Woodward, May 09 2015

Extensions

Extended by Ray Chandler, Aug 27 2015

Status

approved