A010774 -id:A010774 - cp's OEIS frontend

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

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

A103922 Continued fraction expansion of the twelfth root of two, 2^(1/12).

Original entry on oeis.org

1, 16, 1, 4, 2, 7, 1, 1, 2, 2, 7, 4, 1, 2, 1, 60, 1, 3, 1, 2, 8, 5, 1, 2, 1, 1, 1, 5, 1, 1, 3, 5, 1, 1, 1, 2, 3, 1, 1, 1, 73, 3, 5, 1, 1, 1, 2, 26, 2, 1, 1, 2, 7, 2, 1, 1, 2, 3, 1, 1, 1, 14, 1, 2, 1, 4, 1, 19, 8, 3, 6, 5, 1, 2, 1, 1, 1, 5, 1, 4, 1, 1, 3, 10, 1, 1, 4, 4, 9, 4, 4, 1, 1, 1, 4, 1, 19, 16, 13

Offset: 0

Nicholas A Kooij (nickkooij(AT)hotmail.com), May 02 2005

Successive note frequencies in the twelve-tone equal temperament chromatic scale are 2^(1/12) higher than their predecessor.

Michael Rubinstein, University of Waterloo, The Conspiracy of Equal Temperament
Eric Weisstein's World of Mathematics, Continued Fractions
Wikipedia, Equal Temperament

Cf. A010774, A341113.

ContinuedFraction[2^(1/12), 100]
ContinuedFraction[Surd[2,12],120] (* Harvey P. Dale, Jan 02 2019 *)

default(realprecision, 120); contfrac(2^(1/12)) \\ Seiichi Manyama, Feb 05 2021

A221363 Decimal expansion of the Pythagorean comma.

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Jan 19 2013

In musical tuning, the Pythagorean comma is 12 fifths / 7 octaves = (3/2)^12 / 2^7.

			3^12 / 2^19 = 531441/524288 = 1.0136432647705078125

Larry Baggett, In the Dark on the Sunny Side: A Memoir of an Out-of-Sight Mathematician, Mathematical Association of America, 2012, p. 78.
Dave Benson, Music: A Mathematical Offering. Cambridge: Cambridge University Press (2006): 164.
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 257.

Peter A. Frazer, Temperament, Midicode, January 2010.
Eric Weisstein's World of Music, Comma of Pythagoras
Wikipedia, Pythagorean comma
OEIS index entry for music
The multi-faceted reach of the OEIS: Music

Cf. A005663, A005664, A010774, A046102.

Mathematica
```
RealDigits[N[31441/524288, 50]][[1]]
```

A229948/A229943 - Omar E. Pol, Oct 25 2013

A214832 Integer part of A440 piano key frequencies, start with A0 = the 1st key.

Original entry on oeis.org

27, 29, 30, 32, 34, 36, 38, 41, 43, 46, 48, 51, 55, 58, 61, 65, 69, 73, 77, 82, 87, 92, 97, 103, 110, 116, 123, 130, 138, 146, 155, 164, 174, 184, 195, 207, 220, 233, 246, 261, 277, 293, 311, 329, 349, 369, 391, 415, 440, 466, 493, 523, 554, 587, 622, 659, 698, 739, 783, 830, 880, 932, 987, 1046, 1108, 1174, 1244, 1318, 1396, 1479, 1567, 1661, 1760, 1864, 1975, 2093, 2217, 2349, 2489, 2637, 2793, 2959, 3135, 3322, 3520, 3729, 3951, 4186

Offset: 1

Jon Perry, Mar 07 2013

A254531(a(k)) = k, k = 1..88. - Reinhard Zumkeller, Feb 04 2015

			Middle C is 261.626 Hz so a(40) = 261.

BGfL, A Virtual Keyboard
Wikipedia, Piano Key Frequencies
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A059620, A079731, A131062, A101285, A131071.

Cf. A254531, A010774.

a214832 = floor . (* 440) . (2 **) . (/ 12) . fromIntegral . subtract 49
-- Reinhard Zumkeller, Nov 23 2014

for (i=1;i<=88;i++) document.write(Math.floor(Math.pow(2,(i-49)/12)*440)+", ");

PARI
```
a(n)=floor(440*2^((n-49)/12));
```

a(n) = floor[2^((n-49)/12)*440] (Hz) for 1 <= n <= 88.

A229943 Decimal expansion of 256/243, the Pythagorean semitone.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 25 2013

The Pythagorean diatonic semitone is one of the musical intervals. Has a ratio of 256/243, and is often called the Pythagorean limma. It is also sometimes called the Pythagorean minor semitone.

After the initial term the sequence has period 27, repeat: 0, 5, 3, 4, 9, 7, 9, 4, 2, 3, 8, 6, 8, 3, 1, 2, 7, 5, 7, 2, 0, 1, 6, 4, 6, 0, 9.

			1.053497942386831275720164609...

J. M. Merino de la Fuente, Las vibraciones de la música, Editorial Club Universitario (2006), 133.
Alberto Rojo, La física en la vida cotidiana, Siglo Veintiuno Editores (2011), 137.

Wikipedia, Semitone
OEIS index entry for music
The multi-faceted reach of the OEIS: Music

Cf. A010774, A221363, A229948, A230437.

RealDigits[256/243,10,120][[1]] (* Harvey P. Dale, Jul 17 2019 *)

A229948/A221363 = (3^7/2^11)/(3^12/2^19) = 2^8/3^5 = 256/243.

A229948 Decimal expansion of 2187/2048, the Pythagorean apotome.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 25 2013

The Pythagorean apotome 2187/2048 (also called the apotome Pythagorica) is one of the musical intervals.

			1.06787109375

Wikipedia, Semitone
OEIS index entry for music
The multi-faceted reach of the OEIS: Music

Cf. A010774, A221363, A229943, A230437.

A221363 * A229943 = (3^12/2^19)*(2^8/3^5) = 3^7/2^11 = 2187/2048.

A254531 a(n) is the position of the piano key whose frequency is closest to n Hz, start with A0 = the 1st key.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22

Offset: 27

Reinhard Zumkeller, Feb 01 2015

			.     | Frequency [Hz] | Piano key | Pitch
.   i | f = A079731(i) |      a(f) |
.  ---+----------------+-----------+------
.   0 |             28 |         1 |  A0
.   1 |             55 |        13 |  A1
.   2 |            110 |        25 |  A2
.   3 |            220 |        37 |  A3
.   4 |            440 |        49 |  A4    A440
.   5 |            880 |        61 |  A5
.   6 |           1760 |        73 |  A6
.   7 |           3520 |        85 |  A7 .

Reinhard Zumkeller, Table of n, a(n) for n = 27..4308
Wikipedia, Piano Key Frequencies
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A214832, A079731, A010774.

a254531 = (+ 49) . round . (* 12) . logBase 2 . (/ 440) . fromIntegral

a(n) = round(12*log(n/440)/log(2))+49 \\ Jianing Song, Oct 14 2019

a(n) = round(12*log_2(n/440)) + 49, 27 <= n <= 4308.

a(A214832(k)) = k for k = 1..88.

Corrected by Jianing Song, Oct 14 2019

A328229 Decimal expansion of 2^(7/12).

Original entry on oeis.org

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

Offset: 1

Daniel Hoyt, Oct 08 2019

2^(7/12) is the multiplier with respect to a base frequency to produce a perfect fifth interval in an equal tempered chromatic scale.

			1.498307076876681498...

Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A010774, A103922, A328228.

First@ RealDigits[2^(7/12), 10, 105] (* Michael De Vlieger, Oct 25 2019 *)

sqrtn(2^7, 12) \\ Michel Marcus, Oct 08 2019

Equals A010774^7. - Michel Marcus, Oct 08 2019

a(17) ff. corrected by Georg Fischer, Apr 04 2020

A329219 Decimal expansion of 2^(10/12) = 2^(5/6).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(10/12) is the ratio of the frequencies of the pitches in a minor seventh (e.g., D4-C5) in 12-tone equal temperament.

			1.78179743...

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (this sequence)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/6), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PARI
```
default(realprecision, 100); 2^(10/12)
```

Equals 2/A010768.

Equals Product_{k>=0} (1 + (-1)^k/(6*k + 1)). - Amiram Eldar, Jul 25 2020

A230437 Decimal expansion of (2/(3 - 2^(1/2)))^(1/4).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 25 2013

Approximation to the frequency ratio of a semitone in equal temperament, suggested by Marin Mersenne.

This constant is close to A010774, the frequency ratio of a semitone in equal temperament. The difference is 0.0002695778428445...

			1.059732672202139808746969376992583251211677...

Wikipedia, Marin Mersenne.
Wikipedia, Semitone.
OEIS index entry for music.
The multi-faceted reach of the OEIS: Music.

Cf. A010774, A221363, A229943, A229948.

RealDigits[Surd[2/(3 - 2^(1/2)), 4], 10, 120][[1]] (* Amiram Eldar, May 16 2023 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A103922 Continued fraction expansion of the twelfth root of two, 2^(1/12).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A221363 Decimal expansion of the Pythagorean comma.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A214832 Integer part of A440 piano key frequencies, start with A0 = the 1st key.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

JavaScript

PARI

Formula

A229943 Decimal expansion of 256/243, the Pythagorean semitone.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A229948 Decimal expansion of 2187/2048, the Pythagorean apotome.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A254531 a(n) is the position of the piano key whose frequency is closest to n Hz, start with A0 = the 1st key.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

PARI