A258054 -id:A258054 - cp's OEIS frontend

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

Peter Woodward, May 09 2015

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

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

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).

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

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

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 *)

a(n)=7*(n-1)%12+1 \\ Charles R Greathouse IV, Jun 02 2015

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

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)

Extended by Ray Chandler, Aug 27 2015

A358029 Decimal expansion of the ratio between step sizes of the diatonic and chromatic semitones produced by a circle of 12 perfect fifths in Pythagorean tuning.

Original entry on oeis.org

1, 2, 6, 0, 0, 1, 6, 7, 5, 2, 6, 7, 0, 8, 2, 4, 5, 3, 5, 9, 3, 1, 2, 7, 6, 1, 2, 2, 6, 0, 3, 9, 2, 4, 2, 3, 3, 7, 1, 8, 1, 1, 5, 5, 7, 9, 2, 3, 2, 7, 6, 7, 8, 3, 3, 4, 1, 0, 6, 5, 2, 0, 1, 6, 1, 6, 2, 0, 8, 7, 4, 8, 0, 0, 8, 3, 1, 2, 2, 7, 8, 4, 6, 8, 8, 1, 4

Offset: 1

Eliora Ben-Gurion, Oct 25 2022

Pythagorean tuning is a form of tuning produced by repeated stacking of the perfect fifth, which has the frequency ratio of 3:2. A circle of 12 perfect fifths is approximately equal to the tuning system predominantly in use in the world today. If the perfect fifth is stacked 12 times and the resulting sequence is octave-reduced, then this divides the octave into 5 chromatic semitones which are equal to 2187/2048 (A229948), and 7 diatonic semitones which are equal to 256/243 (A229943). Diatonic semitones are those which are derived from a circle of 7 perfect fifths, the diatonic scale, and 5 chromatic semitones are a byproduct of an addition of 5 more perfect fifths, that is, another rotation, to the scale.

			1.2600167526708245359312761226039242337181155792327678334106520161...

Wikipedia, Pythagorean tuning.
Wikipedia, Semitone, section Pythagorean tuning.
Index entries for sequences related to music.

Cf. A131071, A221363, A258054, A257811.

Cf. A229943, A229948.

RealDigits[(7*Log[3] - 11*Log[2])/(8*Log[2] - 5*Log[3]), 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

Equals log(2187/2048) / log(256/243).
Equals log(A229948) / log(A229943).
Equals (7*log(3) - 11*log(2))/(8*log(2) - 5*log(3)).

cp's OEIS Frontend

A257811 Circle of fifths cycle (clockwise).

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