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

Author

Peter Woodward, May 09 2015

Keywords

• nonn
• easy

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

Examples

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

Links

• 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).

Crossrefs

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

Programs

Magma

[1+7*(n-1) mod(12): n in [1..80]]; // Vincenzo Librandi, May 10 2015

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 *)

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

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)

Extensions

Extended by Ray Chandler, Aug 27 2015