The Musical Scale. 
167 
1919-20.] 
If, now, we rearrange these symmetric intervals in pitch order, we shall 
have the 
symmetrical chromatic scale : — 
SCALE-NAMES. 
Bb 
B 
C 
Db 
D Eb E F 
RATIOS. 
F# 
G 
Ab 
A 
9 
15 
1 or 
16 
9 6 5 4 
45 
3 
8 
5 
5 
8 
2 
15 
8 5 4 3 
SEMITONE-RATIOS. 
32 
2 
5 
3 
25 16 16 135 16 25 16 135 16 16 25 
24 15 15 128 15 24 15 128 15 15 24 
It will be seen that the semitone-ratios have the same order when 
read from either end, which proves that the scale-ratios themselves are 
symmetrical. The symmetry which thus appears only in our third line is, 
of course, involved equally in both the others, but is effectually obscured 
by making C the centre of measurement, whereas the true centre is the 
interval C G. 
It now remains for us to examine the functional character of the scale- 
notes ; to show the order of the extended scale, and the twelve-note limit 
of the tonal system ; to consider the subjects of tonality and key- 
relationship (with special regard to the relative minor key), and to mention 
very briefly the subject of comparative magnitudes. 
III. Functional Characters of the Scale-notes. 
As we saw in its symmetric order, the chromatic scale is divisible into 
two equal and similar parts, in each of which (so far as the mathematical 
element is concerned) every note has a function corresponding to that 
of its fellow (Diagram V). The twelve notes, therefore, comprise six 
functional pairs. It is not an easy matter to find simple and adequate 
names for these co-functionaries, but their order and character may be 
conveniently intimated as follows : — 
Diagram VI. 
Mathematical Co-functionaries of the Scale. 
Scale-names. Functional Characters. 
C G . . . Dominators. 
E Eb . 1 Mediants. 
B Ab . Leaders. 
F D . Ultra-dominators. 
A Bb . Ultra-mediants. 
F| Db . Ultra-leaders. 
