154 Scientific Proceedings, Royal Dublin Society. 



tiplied out has 179 digits in the numerator; and yet, when reduced to 

 its lowest terms, it is exactly equal to 2. Consequently, it accurately 

 represents the interval of the octave. 



which proves that the differences for the dominant and the mediant are commensurable. 

 But the diatonic scale is made up of combinations of the major fifth and major third ; 

 consequently, if the dominant and mediant have differences which are multiples of a 

 constant, then the other degrees of the scale wiU have the same property. 



Again, if the relation holds good for one diatonic scale, it will also hold good for any 

 others that are built up on it ; for the difference of any degree in a scale built on to the 

 primary scale must be equal to the difference due to that degree plus the difference 

 belonging to the tonic on which the scale is formed. For example : — F2n is the super- 

 tonic of E2. Now the difference for the supertonic is - 2a, and the difference for Eo 

 is + 7«. Consequently the difference for F2Ji is + Sc. N. E. — The equality of the 

 above last equation is not exact, the left side being 7 (-000490474126), and the right 

 7 (-000490428252). 



