510 Mr. R. H, M. Bosanquet on Temperament, 



For, recalling the definition of rth order (departure of 12 fifths 

 = +r units of system), the proposition follows at once from 

 Theorem 2. 



This proposition, taken with theorem 1, enables us to ascer- 

 tain the number of divisions in the octave in systems of any 

 order. The systems mentioned below are all of some historical 

 or other interest; those of higher orders possess but little 

 interest. 



Primary (1st order) positive. 



Number of units in 



7-fifths semitone 5-fifths semitone octave (Theorem 1), 



=6' units. =/ units. 5s-\-7f. 



2 1 17 



3 2 29 



4 3 41 



5 4 53 



6 5 65 



Secondary (2nd order) , positive. 

 11 9 118 





Primary Negative. 





1 



2 



19 



2 



3 



31 



Secondary Negative, 

 3 5 50 



The formation of scales and the discussion of the properties 

 of these systems are reserved for a future occasion. 



Theorem 4. If a system divide the octave into n equal inter- 

 vals, the condition that it may be of the rth order is that r + 7n 



r + 7n 

 is a multiple of 12 ; and the number , is the number of 



units in the fifth of the system. 



Let m be the number of units in the fifth. 



12 



is the unit in semitones (E. T.); 



n 



12 



.•. m . — is the fifth in semitones, 



n 



m 



and 12 7 the departure of the fifth in semitones. 



n r 



12 (12 7) = departure of 12 fifths in semitones 



=r units of system (Def.) 



12 



= r . — : 



