72 
the principle that a semitone is the smallest interval by which 
it is agreeable to the human ear that musical sounds should 
follow each other, and if we further admit that the third and 
fifth must find their place in the scale, then we find our musical 
circle divided into three portions which contain about (not 
accurately) 4, 3, and 5 semitones respectively. 
No other division of these intervals seems possible, except 
that of dividing the first two in- Fig. V. 
tervals into two parts each, and the oh 
last into three. The positions of the 
two semitones, the existence of which 
is manifestly unavoidable, will still 
be undecided, but there will not be 
much difficulty in determining their 
position; for the satisfaction which 
the ear experiences in the sound of 
a leading note, or a note approaching 
within half a note of the tonic, is so great as to leave no 
doubt as to the position of the semitone between G and C; 
it must clearly be immediately contiguous to C; and this being 
so, it will be indifferent where we place the other semitones. 
For suppose we put it next to #, then we have the ordinary 
arrangement of the diatonic scale, the tonic being C: but 
suppose we take the other course and put it next to G, then the . 
result is that the arrangement of semitones in the musical circle 
is exactly the same as before, only the tonic will be G instead 
of C. Hence, granting to the third and fifth their places in the 
scale honoris causd, and allowing the necessity of a leading note, 
it appears that the arrangement of the semitones in the scale 
must be that with which we are familiar. And thus we seem to 
get at a rationale of the ordinary system of notes, which is mm 
some respects more instructive than that which is usually given, 
as for example in the treatise of Sir John Herschel above cited; 
for the reasons there adduced depend upon considerations of the 
