68 
After this there is an apparent discontinuity, but not a real 
one, as the next key would be that of FZ, and this is in fact 
the key indicated by the line OC’, being intermediate to OF 
and OG. ; 
Thus the passage from one natural key to another is repre- 
sented by the orderly revolution of a radius of our musical 
circle through angular spaces of 30°. 
It is hardly necessary to say that the same method is appli- 
cable to the representation of the succession of flat keys: but it 
may be interesting to exhibit the method to the eye. 
The successive flat keys are produced by flattening the leading 
Fig. II. 
note, or seventh of the scale. If we 
perform this process upon Fig. I. we 
have the annexed figure. Here the 
shaded semitonal spaces are sym- 
metrical with respect to GOG' ex- 
actly as in Fig. 1. they were sym- >, 
metrical with respect to DOD’, and in : 
Fig. 11. with respect to AOA’. Con- 
sequently G is the second note of the a 
scale, or F is the tonic. 
Hence for the flat keys our tonic line revolves through one 
angular space of 30° in the sense contrary to the motion of the 
hands of a clock, or contrary to that in which it revolves for 
the sharp keys; and therefore Fig. mr. will give us for the 
key-notes corresponding to the signatures one, two, three, four, 
five flats, the following 
PR AE eA aD Gas 
or, as will be seen by inspecting the figures, 
Ih Bins Dy Jay) DID, 
Il. I now pass on to the second subject which I proposed 
to discuss in this paper; and I shall endeavour to exhibit by 
the machinery already introduced the great convenience of the 
