164 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The Mathematical System. 
The scale-notes being derived by measurements of fifths and thirds 
from the tonal centre, their position on the diagram (No. II) enables us 
to calculate the ratio of each pair. 
For example, taking C as the base of measurement, the note B is one- 
fifth and one-third above ; its ratio therefore is . 
3 5 _ 15 
2 X 4 “ 8 
In this way we can prove all our scale-ratios to be identical with those 
of the Ptolemaic series (Diagram I). 
In doing this we must realise that the primary measurements of these 
intervals actually extend over a compass of more than nine octaves ; as, 
however, the character of an interval remains the same regardless of 
octave-measurement, we reduce the primary ratios to scale-limit, multiply- 
ing or dividing them by powers of 2, according to the number of octaves 
we require to raise or lower them. 
This arithmetical process applied to Diagram II gives Diagram III. 
Diagram III. 
1 1 
5 
5 
15 
45 
3 
4 
8 
32 
4 
. ( 
1 
3 
9 
3 
Ilia 
2 
8 
16 
8 
6 
9 
15 
5 
5 
5 
Scale-ratios of the tonal system identical with those of the Ptolemaic series. 
II. The Harmonic Order of the Scale. 
We have said that the mathematical order of the scale may be demon- 
strated in several different ways, and we have already viewed it as a 
symmetrical system based upon a central interval ; that is, its tonal order : 
we are now to see that it has another symmetry, equally perfect, but based 
upon its harmonic order, and measured from its harmonic extremes. 
