Geometric Sequences and

The Frequencies of the 88 Keys on a Piano

(The Equal Tempered Chromatic Musical Scale)

By Don Cohen- The Mathman

February, 2006

A geometric sequence is found by starting with a number, then multiplying this by a 'certain number'. Whatever one gets, you multiply that answer by that same 'certain number', and continue that.  The multiplying number is always the ratio of two consecutive numbers in the sequence. For example, let's start with 3, and let our multiplying number (that 'certain number' )or ratio, be 2, and write the first 4 terms of this geometric sequence.

3,  3*2,  3*2*2,  3*2*2*2,.. , or using exponents: 3*20 , 3*21 , 3*22 , 3*23 or  3, 6,12, 24, ..    (Notice 20  =1, 21  =2,  22  =4,  23  =8).. What's the ratio of consecutive numbers in the sequence? Well 6/3 = 2,  12/6 = 2,  and 24/12 = 2 ..

Problem 1: If the first number of a geometric sequence is 1 and the fourth number in the geometric sequence is 64, what are the second and third number in the geometric sequence, or what are the two geometric means between 1 and 64?  1,  ?   , ?  , 64

Solution 1: 1= 1*r0, (r being the ratio), the second number is 1*r1, the third term is 1*r2 and the fourth number is 1*r3 = 64, so r3 = 64. What number cubed, or multiplied by itself 3 times = 64 or

what is the cube root of 64 or ?  4. Because 4*4*4 = 64, so r = 4. Our geometric sequence then is 1, 4, 16,  64.

Problem 2: If the first number of a geometric sequence is 1 and the thirteenth number in the geometric sequence is 2, what are the 2nd through 12th numbers in the geometric sequence, or the 11 geometric means between 1 and 2?

Solution 2:  1       2       3       4        5       6       7        8      9      10      11        12         13

                  1= r0 , 1*r1 , 1*r2 , 1*r3 , 1*r4 , 1*r5 , 1*r6 , 1*r7 , 1*r8,  1*r9 , 1*r10  ,  1*r11  ,  1*r12

                      1 ,   ?   ,    ?    ,  ?    ,  ?    ,  ?    ,   ? ,     ?   ,   ?  ,    ?   ,    ?  ,        ?   ,       2                                                                                                       

So  r12  =   2   and  r = = 1.05946'

The subscript 3 below indicates the 3rd octave, if one starts with the 0th octave A0 = 27.5 Hertz (vibrations/sec),

If the first term, instead of 1, is middle C3 = 261.63 Hz, in the equal tempered chromatic scale,

                           the second term is  C3# = 261.63*1.0594 = 261.63* = 277.18 Hz and

                                                            D3 = 261.63*1.05942 = 261.63* = 293.66 Hz and

                                                            D3# = 261.63*1.05943 = 261.63* = 311.12 Hz..

                                                       to  A4  = 261.63*1.05949 = 261.63* = 440.00 Hz .. 

                                                 to  C4   = 261.63*1.059412 = 261.63* = C 3*2 = 523.25 Hz.   C4 is double the frequency of C 3 , and an octave higher than C 3 .

An octave is divided into twelve intervals to form the equal tempered chromatic scale; the Intervals of pitch are described in terms of the ratios of the frequencies ~1.0594 =                                                    

The frequency of each note in the scale can be figured by multiplying each successive note by this number to get the next. The frequency of any note can also be figured from A0 = 27.5 Hz, using the formula:

f(N) = 27.5*2(N/12),  where N is an index into the equal tempered chromatic scale notes starting with N= 0 for A0, the lowest note on the keyboard. N increases by 1 for each note on the keyboard. Note that the actual key on the keyboard is N + 1. The 88th key has a frequency for N=87 in the formula above, so f(87) = 27.5*2(87/12) = 4186.009 Hz, the frequency of the note C7

The table below shows the note, the index number (N) and corresponding frequencies (f) and the Octave for all of the keyboard notes based on A4 = 440 Hz  in the equal tempered chromatic scale.

Oct

 

0

 

1

 

2

 

3

 

4

 

5

 

6

 

7

Note

N

f

N

f

N

f

N

f

N

f

N

f

N

f

N

f

A

0

27.500

12

55.0000

24

110.0000

36

220.0000

48

440.0000

60

880.0000

72

1760.000

84

3520.000

A#

1

29.135

13

58.2705

25

116.5409

37

233.0819

49

466.1638

61

932.3275

73

1864.655

85

3729.310

B

2

30.867

14

61.7354

26

123.4708

38

246.9417

50

493.8833

62

987.7666

74

1975.533

86

3951.066

C

3

32.703

15

65.4064

27

130.8128

39

261.6256

51

523.2511

63

1046.502

75

2093.005

87

4186.009

C#

4

34.647

16

69.2957

28

138.5913

40

277.1826

52

554.3653

64

1108.731

76

2217.461

88

4434.922

D

5

36.708

17

73.4162

29

146.8324

41

293.6648

53

587.3295

65

1174.659

77

2349.318

89

4698.636

D#

6

38.890

18

77.7817

30

155.5635

42

311.1270

54

622.2540

66

1244.508

78

2489.016

90

4978.032

E

7

41.203

19

82.4069

31

164.8138

43

329.6276

55

659.2551

67

1318.510

79

2637.020

91

5274.041

F

8

43.653

20

87.3071

32

174.6141

44

349.2282

56

698.4565

68

1396.913

80

2793.826

92

5587.652

F#

9

46.249

21

92.4986

33

184.9972

45

369.9944

57

739.9888

69

1479.978

81

2959.955

93

5919.911

G

10

48.999

22

97.9989

34

195.9977

46

391.9954

58

783.9909

70

1567.982

82

3135.963

94

6271.927

G#

11

51.913

23

103.8262

35

207.6523

47

415.3047

59

830.6094

71

1661.219

83

3322.438

95

6644.875

The modern piano has 88 keys (seven octaves = 7*12 = 84, then add 4 to get 88. The 4 is a minor third = (1/3)*12= 4; so the 88 piano key notes go from from A0, the lowest note, or 27.5 Hz, to C7,  the highest note on the piano, 4186.009 Hz.  

The following graph done in Mathematica, shows a polar graph of the frequencies for the octave from middle C 3 (261.63) to C4 (523.25). Each point has as its distance from the origin (or radius) the pitch or frequency, and the angle from the starting position is (360/13)*n, starting with n=0, going counterclockwise. The command to do this graph is:  PolarListPlot[Table[261.6256*2^(n/12),{n,0,12}],PlotStyle->PointSize[0.02]].

 

In trying to understand how all this works, Don found these two sources very helpful:

'The Handbook of Chemistry and Physics' (26th edition) edited by Charles D. Hodgman and Harry N. Holmes, Published in 1942 by Chemical Rubber Publishing Co., Cleveland, OH

An internet file 'The Equal Tempered Scale and Some Peculiarities of Piano Tuning' with the table above, by Jim Campbell


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