The Musical Scale. 
165 
1919-20.] 
It is a well-known fact that the natural harmonic series is included in 
a series of whole multiples of 1, only a definite selection of these multiples 
having any place in the scale. We shall see, for example, that in the 
chromatic scale we have only 12 out of 675 possible harmonics; and 
although in the extended scale there is an indefinite number, they are all 
of them multiple powers of the two prime numbers 3 and 5. In fact, all 
ratios of notes and intervals that are used in music are products of the 
powers of the three simplest prime numbers 2, 3, and 5. 
We will demonstrate this harmonic scale-order mathematically. 
Let the indices of the simplest positive powers of 3 and 5 be arranged 
in a series of twelve pairs, in order of magnitude : — 
Powers of 5 . .001012012122 
3 . .010210321323 
Under these pairs of indices we now write the corresponding product 
in whole numbers : — 
Harmonics . 1 3 5 9 15 25 27 45 75 135 225 675 
Here we have the select series of harmonics constituting the chromatic 
scale, viz. the twelve simplest available powers of the two simplest prime 
numbers which can divide the octave. These are the root-ratios of the C 
scale, expressed as harmonics of Db. 
Ratios 
1 
3 
5 
9 
15 
25 
27 
45 
75 
135 
225 675 
• 1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 1 
Scale-names 
. Db 
Ab 
F 
Eb 
C 
A 
Bb 
G 
E 
D 
B Ft 
If, now, we divide this series by 15 (to make C the unit of measure- 
ment), we shall at once recognise the primary ratios of the C scale, though 
still in their harmonic order : — 
111 i 159359 15 45 
15 535135111 11 
Db Ab F Eb C A Bb G E D B F# 
In this form these ratios occupy a compass of nine octaves and one 
augmented third : reduced to scale-limit and arranged in scale-order, we 
o o 
recognise them as Ptolemy’s ratios ! 
1 
C 
16 9654 45 3859 
15 8543 32 2535 
Db D Eb E F F# 
15 
8 
2 
G Ab A Bb B 
C 
