or the Division of the Octave. 5 L 1 



whence 



r + 7n 



m = 



12 ' 



and m is an integer by hypothesis. Whence the proposition. 



This formula is useful for finding corresponding values of r 

 and n. The results are the same as those of Theorem 3. 



Theorem 5. If a system divide the octave into n equal in- 

 tervals, the total departure of all the n fifths of the system = +r 

 E. T. semitones, where +r is the order of the system. 



For the departure of 12 fifths = ±r units 



, 12 



= +r — semitones; . . («) 

 n 



.'. departure of n fifths = ±r semitones. 



Cor. Dividing equation (a) by 12, we have 



r 

 Departure of 1 fifth = + -, 



a proposition which will be useful hereafter. 



This theorem gives rise to a curious method of deriving the 

 various systems. 



Suppose the notes of an equal >temperament series arranged 

 in order of fifths and proceeding onwards indefinitely, thus, 

 c, g, d, a, e, 6,/fc c#, g%, d#, a#,f,c, g, . . ., and so on. 



Let a system of fifths, say positive, start from c; then at 

 each step the system rises further from the E. T. It can only 

 return to c by sharpening an E. T. note. 



Suppose that b is sharpened one E. T. semitone so as to 

 become c; then the return may be effected in 5 fifths, or at 

 the next b in 17 fifths, or at the next in 29, or in 41, or 

 in 53, and so on. These are the primary positive systems. 

 Secondary positive systems may be got by sharpening b \) by 

 two semitones, and so on. So with negative fifths : to return 

 to c, c% may be depressed a semitone; these returns give 7, 

 19, 31, . .. fifths; or d may be depressed two semitones, and 

 so on. There remain to be considered in connexion with this 

 subject : — 



Points of historical interest. 



Formation of scales and properties of systems. 



Instrumental means of control. 



