A071832 -id:A071832 - cp's OEIS frontend

A071831 Frequency ratios for notes of C-major scale starting at c = 1 (numerators).

1, 9, 5, 4, 3, 5, 15, 2, 9, 5, 8, 3, 10, 15, 4, 9, 5, 16, 6, 20, 15, 8, 9, 10, 32, 12, 40, 15, 16, 18, 20, 64, 24, 80, 30, 32, 36, 40, 128, 48, 160, 60, 64, 72, 80, 256, 96, 320, 120, 128, 144, 160, 512, 192, 640, 240, 256, 288, 320, 1024, 384, 1280, 480, 512, 576, 640, 2048

Offset: 0

N. J. A. Sloane, Jun 10 2002

			The ratios are 1 (c), 9/8 (d), 5/4 (e), 4/3 (f), 3/2 (g), 5/3 (a), 15/8 (b); followed by these 7 numbers multiplied by successive powers of 2.

OEIS index entry for music.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2).

Cf. A071832, A071833.

Numerator[2^Floor[#/7]Rationalize[2^((-1+Floor[12(1+Mod[#,7])/7])/12),2^-6]]&/@Range[0,70] (* Federico Provvedi, Feb 14 2014 *)

r=[1,9/8,5/4,4/3,3/2,5/3,15/8]; for(n=0,10, a=2^n*r; for(m=1,7, print1(numerator(a[m])", "))) \\ Rick L. Shepherd, Apr 06 2006

A071831(n)=numerator([1,9/8,5/4,4/3,3/2,5/3,15/8][n%7+1]*2^(n\7))  \\ M. F. Hasler, Jun 13 2012

a(n+7) = 2*a(n) for n >= 21. - Rick L. Shepherd, Apr 06 2006

G.f.: (15*x^27 + 9*x^22 + 15*x^20 + 5*x^16 + 9*x^15 + 15*x^13 + 3*x^11 + 5*x^9 + 9*x^8 - 15*x^6 - 5*x^5 - 3*x^4 - 4*x^3 - 5*x^2 - 9*x - 1) / (2*x^7 - 1). - Colin Barker, Feb 14 2014

More terms from Rick L. Shepherd, Apr 06 2006

A071833 Frequency ratios for notes of C-major scale starting at c = 24 and using Ptolemy's intense diatonic scale.

Original entry on oeis.org

24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 64, 72, 80, 90, 96, 108, 120, 128, 144, 160, 180, 192, 216, 240, 256, 288, 320, 360, 384, 432, 480, 512, 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1440, 1536, 1728, 1920, 2048, 2304, 2560, 2880

Offset: 0

N. J. A. Sloane, Jun 10 2002

All terms are 5-smooth numbers due to the 5-limit-tuning of the natural major scale, where all the ratios prime factors are all less than or equal to 5. - Federico Provvedi, Sep 09 2022
From Federico Provvedi, Apr 19 2024: (Start)
This natural scale has interesting musical and mathematical Diophantine relations between the sum of distinct interval ratios a(n)/a(0) and their own indices: with indices i(k) != j(k), Sum_{k=1..n} a(i(k)) = Sum_{k=1..n} a(j(k)) and
  Sum_{k=1..n} i(k)  = Sum_{k=1..n} j(k), for n=4 a solution is:
     1 + 4/3 + 5/3 + 15/8  = 9/8 + 5/4 + 3/2 +   2 ,
     I +  IV +  VI +  VII  =  II + III +  V  + VIII,
     1 +   4 +   6 +    7  =   2 +  3  +  5  +   8 ,
  a(0) + a(3) + a(5) + a(6) = a(1) + a(2) + a(4) + a(7). (End)
In the terminology of classical music theory, a(0) to a(7) are the frequencies of the diatonic C-major scale (C,D,E,F,G,A,B,C) as tuned in "Just Intonation", starting with frequency C=24=a(0). On keyboard instruments, these are the "white notes". Each higher octave of 8 notes doubles the frequencies of the prior octave, hence, a(n+7) = 2*a(n). The a(n) frequencies of Just Intonation are uniquely determined by requiring that the notes in each of the three principal major triads, namely, the tonic triad (C:E:G), the dominant triad (G:B:D), and the subdominant triad (F:A:C), all have frequencies with exact ratios of 4:5:6. The base frequency of C=24=a(0) is the lowest frequency of C for which all a(n) are integers. (In actual practice, keyboard notes are usually tuned to non-integer frequencies, are based on a "middle C" frequency around 261.62 Hz, and have irrational frequency ratios due to "equal temperament" - see A010774.) - Robert B Fowler, Aug 21 2024

			The ratios are 24 times 1 (c), 9/8 (d), 5/4 (e), 4/3 (f), 3/2 (g), 5/3 (a), 15/8 (b), followed by these 7 numbers multiplied by successive powers of 2.

Wikipedia, Ptolemy's intense diatonic scale.
Wikipedia, Five-limit tuning. Diatonic scale.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2).

Cf. A071831, A071832, subset of A051037, A010774.

Table[ 2^Floor[n/7] ( 3*(91 + (-1)^Mod[n, 7] ) + 42 Mod[n, 7] + 8 Sqrt[3] Sin[Pi(1 + Mod[n, 7])/3] ) / 12,  {n, 0, 70}] (* Federico Provvedi, Aug 28 2012 *)
3*2^(3+Floor[#/7])*Rationalize[2^((-1+Floor[12(1+Mod[#,7])/7])/12),2^-6]&/@Range[0,70] (* Federico Provvedi, Oct 13 2013 *)
LinearRecurrence[{0,0,0,0,0,0,2},{24,27,30,32,36,40,45},50] (* Harvey P. Dale, May 23 2016 *)

def a(n): return [24, 27, 30, 32, 36, 40, 45][n % 7] << (n // 7) # Peter Luschny, Aug 22 2024

a(n) = 2^floor(n/7) * (3*(91 + (-1)^(n mod 7)) + 42*(n mod 7) + 8*sqrt(3) * sin(Pi*(1+(n mod 7))/3))/12. - Federico Provvedi, Aug 28 2012
G.f.: -(45*x^6 + 40*x^5 + 36*x^4 + 32*x^3 + 30*x^2 + 27*x + 24) / (2*x^7 - 1). - Colin Barker, Feb 14 2014
a(b(n)) - a(b(n)+1) - a(b(n)+2) + a(b(n)+3) - a(b(n)+4) + a(b(n)+5) + a(b(n)+6) - a(b(n)+7) = 0, where b(n) = A047274(n). - Federico Provvedi, Apr 19 2024
a(n) = 2^floor(n/7) * round(24 * 2^(floor( (12*(n mod 7)+5)/7) / 12)). - Robert B Fowler, Aug 22 2024

More terms from Kerri Sullivan (ksulliva(AT)ashland.edu), Oct 31 2005

Name made more specific by Jon E. Schoenfield, Sep 12 2022

A131062 Rounded frequencies of notes in a Pythagorean scale, starting with 260.7 Hertz for a C.

Original entry on oeis.org

261, 293, 330, 348, 391, 440, 495, 521, 587, 660, 695, 782, 880, 990, 1043, 1173, 1320, 1391, 1564, 1760, 1980, 2086

Offset: 1

Hans Isdahl, Sep 24 2007

nonn

The approximate value of 260.7 Hz for the C corresponds to 16/27 * 440 Hz. The frequencies correspond to the ratios [1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1].

Wikipedia, Just intonation - Pythagorean tuning.
Wikipedia, Pythagorean tuning.
Index entries for sequences related to music.

Cf. A131071 for the same scale including half-tones.
Cf. A071831/A071832 = A071833/24. - M. F. Hasler, Jun 14 2012
Cf. A101285.

Value of a(8) corrected, sequence extended to 3 octaves and comments added by M. F. Hasler (following suggestions by Franklin T. Adams-Watters and Charles R Greathouse IV), Oct 05 2011

A101285 Rounded frequencies in Hertz of the notes of the C major music scale beginning at A (A Minor equal-tempered).

Original entry on oeis.org

55, 62, 65, 73, 82, 87, 98, 110, 123, 131, 147, 165, 175, 196, 220, 247, 262, 294, 330, 349, 392, 440, 494, 523, 587, 659, 698, 784, 880, 988, 1047, 1175, 1319, 1397, 1568, 1760, 1976, 2093, 2349, 2637, 2794, 3136, 3520, 3951, 4186, 4699, 5274, 5588, 6272

Offset: 1

Angela Johansson (angvi798(AT)student.liu.se), Dec 20 2004

nonn

The scale is equal-tempered ("Wohltemperiert"), introduced by Johann Sebastian Bach.

Subsequence of A101286, obtained by removal of the 5 black keys' frequencies in each block of 12 keys. - R. J. Mathar, Mar 12 2008

T. Yahaya Abdullah, Music Scales, part of Synthesizers, Music and Television.
Wikipedia, Equal temperament
Wikipedia, Aeolian scale

Cf. A071831, A071832, A071833.

A101286x := proc(n) 55*2.0^((n-1)/12.0) ; end: A101285x := proc(n) if n >= 8 then 2*A101285x(n-7) ; else A101286x(op(n,[1,3,4,6,8,9,11])) ; fi ; end: A101285 := proc(n) round(A101285x(n)) ; end: seq(A101285(n),n=1..80) ; # R. J. Mathar, Mar 12 2008

Table[Round[55*2^((Floor[3(4k-1)/7]-1)/12)],{k,1,49}] (* Federico Provvedi, Feb 14 2014 *)

forstep(i = 0, 100, [2, 1, 2, 2, 1, 2, 2], print(round(55*2^(i/12)))) \\ David Wasserman, Mar 17 2008

From David Wasserman, Mar 17 2008: (Start)
a(7n) = round(55*2^(n-1/6));
a(7n+1) = 55*2^n;
a(7n+2) = round(55*2^(n+1/6));
a(7n+3) = round(55*2^(n+1/4));
a(7n+4) = round(55*2^(n+5/12));
a(7n+5) = round(110*2^(n-5/12));
a(7n+6) = round(110*2^(n-1/3)). (End)
a(n) = round(55*2^(int(3*(4*k-1)/7-1)/12)). - Federico Provvedi, Feb 14 2014

More terms from Jonathan R. Love (japanada11(AT)yahoo.ca) and R. J. Mathar, Mar 08 2007

A101286 Rounded frequencies in Hz of the notes of the chromatic music scale beginning at A.

Original entry on oeis.org

55, 58, 62, 65, 69, 73, 78, 82, 87, 92, 98, 104, 110, 117, 123, 131, 139, 147, 156, 165, 175, 185, 196, 208, 220, 233, 247, 262, 277, 294, 311, 330, 349, 370, 392, 415, 440, 466, 494, 523, 554, 587, 622, 659, 698, 740, 784, 831, 880, 932, 988, 1047, 1109, 1175

Offset: 1

Angela Johansson (angvi798(AT)student.liu.se), Dec 20 2004

nonn

The scale is equal-tempered. ("Wohltemperiert", introduced by Johann Sebastian Bach.)

T. Yahaya Abdullah, Music Scales, part of Synthesizers, Music and Television.
Wikipedia, Equal temperament

Cf. A071831, A071832, A071833.

A101286 := proc(n) round( 55*2.0^((n-1)/12.0)) ; end: seq(A101286(n),n=1..80) ; # R. J. Mathar, Mar 12 2008

a(1+n) = round(55*2^(n/12)). - R. J. Mathar, Mar 12 2008

Corrected by T. D. Noe, Nov 02 2006

More terms from R. J. Mathar, Mar 12 2008

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A071831 Frequency ratios for notes of C-major scale starting at c = 1 (numerators).

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A071833 Frequency ratios for notes of C-major scale starting at c = 24 and using Ptolemy's intense diatonic scale.

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A131062 Rounded frequencies of notes in a Pythagorean scale, starting with 260.7 Hertz for a C.

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A101285 Rounded frequencies in Hertz of the notes of the C major music scale beginning at A (A Minor equal-tempered).

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A101286 Rounded frequencies in Hz of the notes of the chromatic music scale beginning at A.

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Maple

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A101287 Rounded frequencies in Hz of the notes of the pentatonic music scale beginning at F#.

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