Hindoo Division of the Octave. 



377 



I shall presently indicate the mode in which the principles of the gene- 

 ralized keyboard permit us to construct an instrument that will deal prac- 

 tically with this system of 22, and exhibit in a graphical manner the sin- 

 gular laws of harmony to which its notes are subject. 



Theory of the Higher Systems. 



Let us recall what is meant by the order of a system. 



(The letters E.T. are used as an abbreviation for "equal temperament.") 



The E.T. fifth is 7 semitones ; 



the octaTe is 12 semitones. 



.-. 12 E.T. fifths =7 octaves =84 semitones. 



The perfect fifth,- on the other hand, is (very nearly) 7~j ; so that 12 

 12 



perfect fifths = 84gp 



And in other systems there is always a small difference between 12 fifths 

 and 7 octaves. Now the simplest way in which this can be treated is to 

 make this small difference the unit of the system. When this is done 

 the system is said to be of the first order. 



But sometimes this small difference is more than one unit : if it is 

 divided into two units, we say that the system is of the second order ; if 

 into three, of the third, and so on. 



The forms of arrangement into scales and laws connecting the harmony 

 of fifths and thirds depend primarily upon the orders of systems. 



Uef erring back for the details of the investigation to my previous com- 

 munication already cited, I recall only that the systems of each order 

 proceed by differences of 12, and that for the first three orders they are 

 as follows : — 



Order. 











1. 



17 



29 



41 



53 



2. 



22 



34 





118 



3. 



15 



27 



39 





The accompanying illustration (Diagram I.) will make clearer what is 

 meant by saying that the system of 22 is a system of the second order. 

 The numbers are the characteristic numbers of the system ; they are 

 arranged in order of fifths, i. e. they proceed by differences of 13, 22 

 being always cast out. The departure of the sharp fifths from E. T. is 

 represented by displacement in a vertical direction. 



Then the circle of 12 fifths has its terminal points 2 units apart. 



Similarly in systems of the rth order, the circle of 12 fifths has its 

 terminal points r units apart. 



In the illustration we see how the notes may be introduced which form 

 the intervals intermediate between the terminal points ; thus the note 1 

 is introduced midway between and 2, 



