Hindoo Division of the Octave. 



381 



Example. — Order 2, Class 1. 



Third to c : 



4 steps up give e, 



1 unit down \e, which is the required third. 



Third to b : 



4 steps up give //d$, 



1 unit down /c£jf, which is the third. 



Whence, in order 2 class 1, 5, e, a, d (letters of the memoria-technica 

 word) form thirds by one mark up, and all remaining notes by one mark 

 down. 



Similarly, in a system of order r class cc, 5, e, a, d form thirds with 

 r—so marks up, and all the remaining notes with cc marks down. 



Transformations of the Generalized Keyboard. 



It is only necessary to require, in the construction of the generalized 

 keyboard, that all the keys shall equally fit all the bearings, to render it 

 possible to produce any required position system with a sufficient number 

 of the ordinary keys. This requirement has always been attended to in 

 the plans for the sake of simplification ; though the important results 

 which flow from it were not originally foreseen. But it is found that 

 unless the attention of the maker is specially directed to the point, the 

 nature of the finishing processes does not secure the result in question ; 

 there is, however, no difficulty in securing it when it is desired. 



The distance of the end of the key on the plan (projection on a hori- 

 zontal plane) from a line of reference drawn from right to left deter- 

 mines the form of the key completely. 



There are 12 such fundamental positions ; so that we may describe the 

 pattern of any key completely as a function of a series of numbers running 

 from 1 to 12. After 12 the same patterns recur, with reference to a new 

 standard fine, such that the old 12 has the same position as the new 0. 



The ordinary arrangement of a series of 12 fifths may be simply exhi- 

 bited by writing under each note of the series the number of which its 

 pattern is a function. 



Direct Keyboard. 



c g d a e b //# /c% /g% /d% /a% /f /c 

 123456789 10 11 12 1 



Increase of the numbers denotes increased height as well as increased 

 distance from the front ; so that according to this, the original arrange- 

 ment, rise on the keyboard corresponds to rise in the series of fifths. 



Inversion. 



Before the keyboard was originally constructed, it became matter for 

 investigation how far it would be advantageous to make rise in the series 



