A060106 -id:A060106 - cp's OEIS frontend

A060107 Numbers that are congruent to {0, 2, 3, 5, 7, 8, 10} mod 12. The ivory keys on a piano, start with A0 = the 0th key.

0, 2, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 44, 46, 48, 50, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 80, 82, 84, 86, 87, 89, 91, 92, 94, 96, 98, 99, 101, 103, 104, 106, 108, 110, 111, 113, 115

Offset: 1

Henry Bottomley, Feb 27 2001

More precisely, the key-numbers of the pitches of a minor scale on a standard chromatic keyboard, with root = 0 and flat seventh.
Also key-numbers of the pitches of an Aeolian mode scale on a standard chromatic keyboard, with root = 0. An Aeolian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone A.
A piano sequence since if a(n) < 88 then A059620(a(n)) = 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A059620, A081031. Complement of A060106.
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): this sequence (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032

[n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 8, 10]]; // Wesley Ivan Hurt, Jul 20 2016

A060107:=n->12*floor(n/7)+[0, 2, 3, 5, 7, 8, 10][(n mod 7)+1]: seq(A060107(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Select[Range[0,120], MemberQ[{0,2,3,5,7,8,10}, Mod[#,12]]&] (* or *) LinearRecurrence[{1,0,0,0,0,0,1,-1}, {0,2,3,5,7,8,10,12}, 70] (* Harvey P. Dale, Nov 10 2011 *)

x='x+O('x^99); concat(0, Vec(x^2*(2+x+2*x^2+2*x^3+x^4+2*x^5+2*x^6)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ Jianing Song, Sep 22 2018

a(n) = a(n-7) + 12 for n > 7.
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f.: x^2*(2 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6)/((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) = (84*n - 91 - 2*(n mod 7) + 5*((n + 1) mod 7) - 2*((n + 2) mod 7) - 2*((n + 3) mod 7) + 5*((n + 4) mod 7) - 2*((n + 5) mod 7) - 2*((n + 6) mod 7))/49.
a(7k) = 12k - 2, a(7k-1) = 12k - 4, a(7k-2) = 12k - 5, a(7k-3) = 12k - 7, a(7k-4) = 12k - 9, a(7k-5) = 12k - 10, a(7k-6) = 12k - 12. (End)
a(n) = A081031(n) - 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019

A059620 Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, start with A0 = the 0th key.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Naohiro Nomoto, Feb 19 2001

			.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1
0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00

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

Cf. A060106, A060107, A081031, A081032.

Flatten[Table[{0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1}, {8}]][[1 ;; 88]] (* Jean-François Alcover, Apr 24 2017 *)
PadRight[{},88,{0,1,0,0,1,0,1,0,0,1,0,1}] (* Harvey P. Dale, Sep 14 2020 *)

a(n) = floor((5n+7)/12) - floor((5n+2)/12). - David W. Wilson, Mar 02 2003
G.f.: -x*(1+x^3+x^5+x^8+x^10) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) ). - R. J. Mathar, Dec 16 2016
a(n) = 0 if n is in A060107, a(n) = 1 if n is in A060106. - Jianing Song, Oct 14 2019

A081032 Positions of black keys on piano keyboard, starting with A0 = the 1st key.

Original entry on oeis.org

2, 5, 7, 10, 12, 14, 17, 19, 22, 24, 26, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 55, 58, 60, 62, 65, 67, 70, 72, 74, 77, 79, 82, 84, 86

Offset: 1

David W. Wilson, Mar 02 2003

			2nd, 5th, 7th, 10th, etc. keys of piano keyboard are black.

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

Cf. A059620, A081031, A060106.

LinearRecurrence[{1,0,0,0,1,-1},{2,5,7,10,12,14},36] (* Harvey P. Dale, Sep 15 2018 *)

a(n) = floor((12n+2)/5).
From Chai Wah Wu, Sep 11 2018: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
G.f. for a keyboard with an infinite number of keys: x*(2*x^4 + 3*x^3 + 2*x^2 + 3*x + 2)/(x^6 - x^5 - x + 1). (End)
a(n) = A060106(n) + 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019

A356464 Number of black keys in each group of black keys on a standard 88-key piano (left to right).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Peter Woodward, Aug 08 2022

On a standard piano keyboard, the black keys appear in groups of two and three, with each group separated from adjacent groups by the presence of two white keys that have no black key between them.
The black keys in a group of two are C#/Db and D#/Eb; the black keys in a group of three are F#/Gb, G#/Ab, and A#/Bb.
The A#/Bb key near the left end of the keyboard is a special case; it is the only black key in its group because the white A key to its left is the leftmost key on the keyboard.

			From _Jon E. Schoenfield_, Aug 12 2022: (Start)
In the diagram below, five octaves (i.e., sets of 12 consecutive keys) have been omitted (as represented by the ellipses):
.
    n |  1       2         3       ...     14        15
  ----+---------------------------------------------------------
  a(n)|  1       2         3       ...      2         3
    ______________________________ ... _________________________
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |_| | |_||_| | |_||_||_| |     | |_||_| | |_||_||_| |  |
      |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |
      |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |
      |__|__|__|__|__|__|__|__|__|     |__|__|__|__|__|__|__|__|
       A  B  C  D  E  F  G  A  B   ...  C  D  E  F  G  A  B  C
(End)

Cf. A059620, A060107, A060106, A081031, A081032.

Cf. A329207.

cp's OEIS Frontend

A060107 Numbers that are congruent to {0, 2, 3, 5, 7, 8, 10} mod 12. The ivory keys on a piano, start with A0 = the 0th key.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A059620 Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, start with A0 = the 0th key.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A081032 Positions of black keys on piano keyboard, starting with A0 = the 1st key.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A356464 Number of black keys in each group of black keys on a standard 88-key piano (left to right).

Views

Author

Keywords

Comments

Examples

Crossrefs