162 Proceedings of the Koyal Society of Edinburgh. [Sess. 
The mathematical order of the scale may be demonstrated in several 
different ways, but in each and all alike we shall find the principles of 
symmetry and simplicity dominant throughout. 
An octave is the simplest of all possible intervals, and therefore it is 
universally recognised as the fundamental musical unit. The chromatic 
scale (or tonal system) is the simplest possible division of the octave (or of 
multiple octaves). 
We have already said that the scale is primarily a system of intervals 
rather than a series of notes ; for though its divisions are marked by notes 
(just as the divisions of a ruler are marked by lines), the divisions them- 
selves are not notes but intervals. These intervals, moreover, are perfect 
fifths and major thirds, and this is so because they are the simplest natural 
intervals by which the octave can be divided. 
It might appear to some that the simplest procedure would be to divide 
the octave by the interval of a semitone ; this, of course, is the principle of 
equal temperament, adopted in our pianos and organs, which, though of 
great practical value, is entirely inadmissible for scientific purposes, being 
both artificial and inaccurate. 
Again, it might be supposed that the perfect fifth alone would suffice 
as a divisor for the octave, and indeed such a suggestion was made by 
Pythagoras, but the result was unsatisfactory : the fifth with the octave 
alone will not yield the notes of the perfect scale, whereas the natural 
scale may be obtained symmetrically by fifths and thirds from a central 
interval. 
The Tonal System. 
Let us now proceed to build up the chromatic scale on these lines. 
Take any convenient musical sound to begin with. We will call it C. 
Before the unprepared ear can recognise it as C (or as any note what- 
ever), some other musical sound must be associated with it. 
Our scale, however, is bound to start with an interval, and the simplest 
interval (next to the octave) is the perfect fifth, C G, and this we take as 
the central interval (or tonal centre). 
If we now add to this central interval the two simplest and most 
symmetrical measurements possible (fifths above and below), we get the 
series of perfect fifths : — 
F, C, G, D. 
These are the four fundamental notes of the scale, belonging equally to 
the major and minor systems. 
