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The "Pythagorean Comma"
 
 
         
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The "Pythagorean Comma"
Explanation by Dr. Jody Nagel
 
 

There is found to be a small intervallic difference when determining (A) a pitch seven octaves higher than a given starting pitch, or (B) a pitch twelve perfect fifths higher than the same given starting pitch, when respectively using the pure 2:1 ratio for finding octaves and when using the pure 3:2 ratio for finding perfect fifths. This generally seems "weird" to the beginner used to playing an equal-tempered piano, where "seven octaves higher" or  "twelve perfect fifths higher" (than some given starting pitch) produces the exact same result: a pitch that is 84 equal-tempered semitones higher than the given starting pitch. Here is the math that explains this phenomenon, which was first explored by the school of Pythagorus in ancient Greece.
 
 

  7 perfect octaves (84 semitones)
(8ve = 12 semitones)
  12 perfect fifths (84 semitones)
(5th = 7 semitones)
 
 
In equal temperament, 7 x 12 = 12 x 7, however . . .
 

  7 perfect octaves
(8ve = 2:1 frequency ratio)
  12 perfect fifths
(5th = 3:2 frequency ratio)
 

If the fundamental frequency is 100 Hz, then:
 
 
  (in Hertz)   (in Hertz)  
0.      100 0.      100.  
1.      200 1.      150.  
2.      400 2.      225.  
3.      800 3.      337.5  
4.      1600 4.      506.25  
5.      3200 5.      759.375  
6.      6400 6.      1139.0625  
7.      12800 7.      1708.59375  
    8.      2562.890625  
    9.      3844.3359375  
    10.      5766.50390625  
    11.      8649.755859375  
 
 
 
 
12.     
 
12974.6337890625
 
 

The difference between 12800 and 12974.6337890625 is 174.6337890625.
The frequency 12974.6337890625Hz overshoots 12800Hz (or is "sharp") by 174.6337890625Hz.
 
The interval formed by the ratio 12974.6337890625 : 12800 (or 1.013643265 : 1) is called the "Pythagorean comma."
This interval is equal to about 23.46 cents (where equal-tempered semitones differ by 100 cents.)
 
The two pitches of an equal-tempered semitone have a 1:1.05946 ratio (that is 1:21/12).
An equal-tempered semitone above 12800Hz would be 12800Hz x 1.05946 = 13561.088Hz.
The frequency differential of the equal-tempered semitone, in this range, is 13561.088 - 12800 = 761.088 Hz.
 
The frequency ratio of two pitches that differ by one cent is: 1:2(1/1200) (or 1:1.0005778).
The frequency ratio (Y) of two pitches that differ by X cents is: Y = 1:2(X/1200).
If one knows the frequency ratio, Y, then the number of cents is: X = log2(Y)
x 1200.
 
Note: log2(Y) = loge(Y)
x log2(e).   [log2(e) = 1.4426950].
So, on a calculator that contains the natural logarithm function,
the number of cents, X, of a given frequency ratio, Y, can be determined by:
X = loge(Y)
x 1.4426950 x 1200,     or:     X = loge(Y) x 1731.234.
Thus, the Pythagorean comma, with interval ratio 1.013643265, is obtained by:
loge(1.013643265)
x 1731.234    =   log2(1.013643265) x 1200   =   23.46 cents.
 
 
Dr. Jody Nagel
March 13, 2003
 
 
 

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