1877.] 



Hindoo Division of the Octave. 



541 



then remarked that our best waj to a real analysis o£ this music would 

 be to study the system of 22 and compare the results with those actually 

 obtained by Hindoo musicians. The methods which have been employed 

 in the writer's former paper on the subject* are then extended to the 

 higher orders, which have not been before thoroughly discussed. The 

 system of 22 is a system of the second order ; and the nature and pecu- 

 liarities of such systems, and of the system of 22 in particular, are 

 discussed. 



A classification of systems of the higher orders according to their mode 

 of forming thirds is advanced. If the system be arranged in successive 

 series of fifths, differing by one unit in pitch, then the system is said to 

 be of class oo^ if the third of any note is in the series oo units below that 

 which contains the note itself. 



The system of 22 is shown to be of the second order and first class. 



A system of 34, also of the second order and first class, is pointed out 

 as being of considerable excellence, even from a modern practical point 

 of view. 



It is shown that in systems of the second order and first class, modu- 

 lation through a third cannot be regarded as equivalent to modulation 

 through any number of fifths. 



The notation is extended to systems of the rth order. 



The subject of the transformations of the generalized key-board is then 

 entered upon. It is remarked in the first instance that any form of ar- 

 rangement whatever can be constructed by rearranging a supply of keys 

 of the ordinary patterns. 



The problem of inversion is then solved, and it is shown under what 

 circumstances, by simply inverting the succession from end to end, a key- 

 board can be obtained in which rise corresponds to fall of pitch, and vice 

 versa. 



The general transformation of the rth order is then investigated, and 

 a rule is given by which the key-board of the rth. order can be arranged 

 with the ordinary keys. 



This rule is then applied to the construction of the key-board of the 

 second order, and a diagram is given of a portion of a key-board so ar- 

 ranged. Systems of the second order and first class, such as the systems 

 of 22 and 34 above mentioned, can be controlled with facihty by means 

 of this arrangement. 



* Proc. Roy. Soc. 1875, vol. xxiii. p. 390, and 'An Elementary Treatise on Musical 

 Intervals and Temperament' (Macmillan, 1876). 



