Hindoo Division of the Octave. 



379 



either by 4 fifths up or 8 fifths down in the same series ; these may be 

 spoken of as of order r class 0, and order r class r respectively. Both 

 haye been considered in my paper already referred to. 



I proceed to indicate shortly the general expressions by means of which 

 systems can be discussed. 



The departure of the third formed by 4 fifths up is 



n 



In a system of class x, the third is x units lower, and its departure is 



,r 12 A Sx-.r r , 

 4— x =—4 , (i) 



n n n 



And this has to be compared with the departure of the perfect third, 

 = -•13686, 



= -^3 nearly. 



So that for a determination of the class of any system n of the rth 

 order, we have the approximate condition 



^-^=^2 nearly (ii) 



The formulae (i) and (ii) are sufficient for any required discussion ; 

 they present no difficulty, and I confine myself to a statement of a few 

 of the principal results. 



The departure of the third of all systems of order 2 class 1 is repre- 

 sented by 



_4 



n 



The system of 34, of order 2 class 1, presents both fifths and thirds of 

 exceptional excellence. This system may be of interest for modern pur- 

 poses. 



Systems of the third order and first class have equal-temperament 



thirds ; for (i) vanishes when#=^: or, more generally, a system has 



o 



E.T. thirds when the number of the class is J- that of the order. 



Systems of order r class x which make 3x—r negative need not be 

 considered, as their thirds are sharper than E.T. thirds. 



In the third order, class 2, there is a good system of 87. 



In the fourth order, class 2, there is a good system of 56. 



Neither of these are likely to be of practical interest. 



Practiced Applications. 

 In the light of the foregoing investigation we see that the generalized 

 keyboard, as hitherto constructed, is of limited application ; it is capable 



