166 Proceedings of the Royal Society of Edinburgh. [Sess. 
The fact of outstanding significance is the mathematical identity 
which is thus revealed between the harmonic and the tonal order of the 
chromatic scale. 
A diagram will illustrate this. 
Diagram IV. 
Chromatic Scale of C. 
25 
75 
225 
675 
5 
15 
45 
135 
1 
3 
9 
27 
Harmonic order measured upwards from Djj. Identical with tonal order, measured 
upwards and downwards from central interval. 
It should be observed that these harmonic measurements bear exactly the 
same relation to one another, upwards or downwards from either extreme. 
The Symmetric Scale. 
It is therefore evident, both from the tonal and the harmonic order of 
the scale, that its twelve notes occupy positions mathematically opposed to 
one another, the whole series comprising six symmetric intervals. 
Diagram V. 
Symmetric Intervals in Harmonic Order. F$ 675 
B 225 
D 135 
9 Eb E 75 
15 C G 45 
25 A Bb 27 
Diagram V shows the mathematical symmetry of the scale-notes very 
clearly, and its importance will soon appear in regard to their functional 
characters, and later in regard to their tonality. 
1 Db 
3 Ab 
5 F 
