380 



Mr. R. H. M. Bosanquet on the 



of controlling only systems which form their thirds by either 4 fifths up 

 or 8 fifths down. The systems included by these conditions are all those 

 of the first order, positive and negative, and all systems of any order of 

 class or class r. These embrace all that are likely to be interesting 

 with reference to European harmonious music, with the possible exception 

 of the system of: 34 above alluded to. 



The principles of position on which the keyboard is founded are, how- 

 ever, applicable to all higher systems • and I shall presently investigate 

 its transformations. The keyboard of the second order thus obtained 

 will afford a means of controlling, in a convenient manner, systems of 

 the first class in that order, and dealing with facility with either the 

 Hindoo s} 7 stem of 22, or the system of 34 above mentioned. 



But before proceeding to discuss these arrangements, it is desirable to 

 provide the extension of our notation, which is necessary for dealing with 

 system's of the rth order and classes other than r and 0. 



Generalized Notation. 



The notation which I have hitherto employed has always assumed that 

 the deviation, or departure, due to a circle of 12 fifths is identical with 

 one unit of the system employed. 



Thus c—./c represented both the departure of 12 fifths and the smallest 

 interval, or unit, of the system. In non-cyclical systems, and in systems 

 of the first order, this representation is consistent and satisfactory ; but 

 in systems of higher orders these two conceptions diverge. The departure 

 of 12 fifths and the unit of the system can no longer be represented by 

 the same symbol. 



The choice we will make is, that the symbol of elevation or depression 

 shall represent primarily one unit of the system. Thus c—/c will always 

 represent the unit, but will only represent the departure of 12 fifths in 

 systems of the first order. 



c—//c will be the departure of twelve fifths in systems of the second 

 order ; c — ///c in systems of the third order, and so on. 



It follows that, in a continuous series of fifths, at the point where two 

 consecutive series of the notation join, the difference of the marks, on the 

 two notes which constitute the joining fifth, will be r. 



Thus the following are fifths which join the unmarked series to that 

 next above it : — 



In the 1st order, h—/f%, 

 2nd „ , &-///#, 

 3rd „ , 6 -////J, 



and so on. 



We now require only to find the thirds. Introducing the condition 

 that the system be of class x\ we find the third as follows : — Pass up four 

 steps in the series of fifths, and then x units down. 



