378 



Mr. R. H. M. Bosanquet on the 



Diagram L 



Characteristic numbers of system of 22 in order of fifths. 



c g d a e b /# c# g% e\) b\> f c 

 2 2 



11 



20 



7 



16 



3 



1 12 1 



21 



8 



17 



4 



13 







Formation of Thirds. 



Thirds may be formed either by the notes of the circle of fifths with 

 which we start, or by the notes of another circle any number of units 

 above or more generally below the first. 



In the system of 22 we have seen that the third is 7 units. Looking 

 at the circle of fifths, the third by 4 fifths up is 8 units. We may form 

 the third to any note therefore by ascending through 4 fifths of the series 

 and then descending one unit ; i. e. the third is formed in the circle of 

 fifths one unit below that which contains the fundamental. 



This mode of formation has not been previously considered. It leads 

 to the following observation, which is important in the practical employ- 

 ment of the systems : — 



Modulation through a third, in systems of this character, cannot be 

 generally treated as equivalent to modulation through any number of 

 fifths. 



"We proceed to a further classification of the higher systems, based on 

 this property. 



By definition, the interval between the two ends of the circle of fifths 

 is r units. Let r circles of fifths be placed in juxtaposition, so that cor- 

 responding pairs of notes are all one unit apart, and consider the third 

 formed with the starting point of the uppermost series. 



Then we shall define a system as being of class x, when the third lies 

 in the ccth series below the upper one. 



In the system of 22, the third (7) to c (0) lies one series below that in 

 which c is, so that we may define the properties of the system of 22 by 

 saying that it is of rrder 2 and class 1. 



The simplest systems of higher orders are those which form their thirds 



