Return to Jody Nagel's Writings 
Return to Jody Nagel's Jomar Page 
Return to JOMAR Home Page 

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 equaltempered 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 equaltempered 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. 

Return to Jody Nagel's Writings 
Return to Jody Nagel's Jomar Page 
Return to JOMAR Home Page 

Copyright © 2003 by Jody Nagel. All rights reserved. 
Call 17657591013 or
Email for additional information. 