The Musical Scale. 
169 
1919-20.] 
down to the dominant of the new flat key. Though not included in the 
major and minor systems (as familiarly used), they are capable of diatonic 
employment ; indeed, this is implied by their appearance in “ chromatic ” 
chords. 
Before leaving this subject, we must mention [a fact of special interest 
involved in the mathematical symmetry of the scale, viz. that for every 
possible melody or melodic phrase the scale affords a corresponding anti- 
thetic melody or phrase which is a mathematical inversion of the first. 
We must remember, however, both in regard to this and all we have 
been saying of functional characters, that in practice the physical element 
asserts itself and modifies the effect of these mathematical relationships, 
making them difficult to appreciate or even obscuring them altogether. 
IV. The Unlimited (or Extended) Scale.* 
Hitherto we have found it convenient to discuss the principles of scale- 
order only in connection with the key of C. These principles, of course, 
apply equally to all keys. 
For example, any notef in the chromatic scale of C may become the 
keynote of another scale, identical in order, symmetry, and ratio. And 
this process might be repeated to any extent, for the musical scale is quite 
unlimited, extending in fifths and thirds indefinitely, its universal order 
and symmetry transcending all tonal systems. 
The nucleus of this unlimited scale is the twelve-note chromatic scale 
(or tonal system), the boundary of which (as we shall see) is very strictly 
determined. Its measurements of filths and thirds, indefinitely extended, 
create innumerable tonal systems which interpenetrate one another, and 
therefore every tonal system is surrounded by a field of notes, extraneous 
to itself, but having a definite relationship and affording unlimited scope 
for modulation. 
* I.e. “unlimited” mathematically, though in practice merely “extended.” 
t Excepting, of course, the keynote C. 
[Diagram 
