Hindoo Division of the Octave. 



383 



of fifths correspond to fall on the keyboard, and vice versa. It is a question 

 of manipulation ; the advantages are in some cases rather evenly balanced, 

 and it is very desirable to examine this arrangement practically. 



The first example of Transformation will bear upon this problem : — 

 It is possible to convert a generalized keyboard of the " direct arrange- 

 ment" above described into an "inverted one " by rearranging the keys 



Inverted Keyboard, 

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

 12 11 10 9 8 7 6 5 4 3 2 1 12 



To complete this transformation in an extremely practical manner, we 

 have only to determine the condition that white and black notes shall 

 remain the same. 



Looking at the keyboard of an ordinary piano, which presents the same 

 order of white and black, we see that, as far as colour is concerned, it is 

 symmetrical about two points, d and «[?. Portions of a keyboard, there- 

 fore, which terminate in these points, or in points equidistant on oppo- 

 site sides from either, present, when inverted from right to left, the same 

 sequence of black and white as before. 



The most convenient arrangement for this purpose consists of a compass 

 of keys from c to e, any number of octaves included. 



When inverted, i. e. when the note on the extreme right is placed in 

 the same row on the extreme left, and so on, such an arrangement presents 

 the same sequence of black and white as before. 



The e becomes a c, and the sequence of patterns is that of an inverted 

 series. 



General transformation of the vth order. 



Systems of the rth order were defined as those in which the ends of the 

 circle of 12 fifths include r units of the system. Similarly the keyboard 

 of the rth order may be defined as that which has r unit intervals (r— 1 

 notes) in the vertical line between the ends of a circle of 12 fifths. 



It is easy to obtain the condition of arrangement in the general case. 

 The difference of level of the ends of the series of 12 fifths must amount 

 to 12 steps by course of fifths, and to r steps by course of units. Con- 

 sequently the whole difference of level of the ends of the series of fifths 

 must be made up of 12r primary steps, or steps made by the patterns : 

 each step in course of fifths must be made up of r primary steps, and 

 each step in course of units must be made up of 12 primary steps*. In 

 this manner, with a sufficient supply of notes of the 12 given patterns, a 

 generalized keyboard of any order can be at once arranged. 



Although systems of any order can always be constructed in this 

 manner, it will not generally be the case that they can be played upon 



* Any coinnion factor of r and 12 may be divided out, since it is only necessary that 

 the two classes of steps should be to each other as 12 : r. 



VOL. XXVI. 2 D 



