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from one key-note to another. The natural interval of 

 modulation is always a fifth. Now the fifth, the third, 

 and the octave are incommensurable. To quote from 

 Mr. Ellis' translation of Helmholtz's work : "It is impos- 

 sible to form octaves by just 5ths or just 3ds or of both 

 combined, or to form just 3ds by just 5ths, because it is 

 impossible by multiplying any one of the numbers f or 

 I by two, or either by itself or one by the other any 

 number of times, to produce the same result as by multi- 

 plying any other of those numbers by itself any number 

 of times." 



Whenever, therefore, by successive modulations through 

 fifths and transpositions through octaves we arrive at a 

 key-note called by the same name as one of the tones of 

 the diatonic scale deduced from thirds, it will not have 

 the same number of vibrations as that tone. In other 

 words, E, A, and B, the key-notes, are not identical with 

 e, a, and b, the tones of the diatonic scale in C. The 

 proportional difference between any key-note and the 

 corresponding third denoted by the same letter is con- 

 stant, depending upon what may be called the incommen- 

 surable element between f, |, and 2. Suppose the vibra- 

 tion number of C to be 1, the vibration number of E, 

 deduced by fifths and octaves, is 



fXfX^XfXfX^ = fi; 

 the vibration number of e = | = f f . 



Since A bears the same relation to F which E bears to 

 C, and a is the third of F, 



A = 1^ a, and similarly 



B = 1^ b, 

 and in general the vibration number of any key-note is 

 greater than that of the third denoted by the correspond- 

 ing lower case letter by ^V of the latter. This difference 

 is called the comma of Pythagoras. 



