66 
by angular spaces of 30° each, the whole twelve thus amounting 
to 360°, and so representing what may be regarded as the actual 
coincidence to the ear of the tonic and its octave. 
The meaning of this will be seen from the annexed figure. 
With centre O describe a circle, and divide the circum- 
ference into twelve equal parts. 
Join the dividing points with Fig. I 
the centre, and put letters as 
in the figure. 
It will be observed that 
each diameter has the same 
letter at each extremity, the 
letters at the two extremities 
being distinguished by an ac- 
cent affixed to one of them; it 
will also be observed that one 
diameter has an ambiguity, 
being either the diameter BOS’, or the diameter FOF"; this arises 
from the fact that in the natural diatonic scale F is only a 
semitone removed from H, and B only a semitone from C. In 
the figure the angular spaces BOC, HOF have been shaded, 
to indicate at once to the eye that these are the two semitone 
intervals in the diatonic scale. 
For the sake of distinctness, call COC’ the tonic line, and 
GOG the dominant line. Then it will be seen that the tonic 
line and the dominant line are inclined to each other at an 
angle of 30°: they are in fact next to each other in the group of 
note-lines which have been drawn through 0. 
It will be seen also that the shaded spaces corresponding 
to the two semitones are situated symmetrically with respect 
to DOD’, the line corresponding to the second note of the 
scale. 
Now let us see what effect will be produced upon our figure 
' by sharpening the subdominant F. The arrangement will then 
