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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