1919-20.] 
The Musical Scale. 
161 
XVIII. — The Musical Scale. By Joseph Goold. 
Communicated by C. R. Gibson. 
(MS. received January 1, 1920. Read June 21, 1920.) 
(Abstract.) 
I. The Tonal Order of the Chromatic Scale. 
The exact ratios of the chromatic scale were first made known by Claudius 
Ptolemy of Alexandria about a.d. 150. We give them here as he found 
them, unaltered and unalterable. 
Diagram I. 
Ptolemy’s Scale-Patios. 
Scale-names 
C 
Db 
D 
Eb 
E 
F 
F# 
G 
Ab 
A 
Bb 
B 
C 
Scale-ratios 
1 
16 
9 
6 
5 
4 
45 
3 
8 
5 
9 
15 
2 
1 
15 
8 
5 
4 
3 
32 
2 
5 
3 
5 
8 
1 
It is a most remarkable fact that while the ratios of the chromatic 
scale have thus been known for nearly two thousand years, its mathe- 
matical symmetry and origin have remained unrecognised down to the 
present day. 
For the concrete musical faculty is compounded of two distinct elements, 
physical and mathematical. 
In its physical aspect (with which we are familiar), the scale is a series 
of notes arranged in order of pitch upwards from the tonic (or keynote). 
In its mathematical aspect, the scale is a system of intervals measured 
symmetrically upwards and downwards from tonic and dominant as 
central interval (or tonal centre). 
The physical order of the scale is that which we most easily appreciate, 
but the mathematical order is that in which it originates. The chromatic 
scale therefore is, 'primarily , not a series of notes but a system of intervals 
(or tonal system). 
Diagram I shows the scale in its familiar order of pitch, and it is 
remarkable how effectually this physical order conceals the mathematical 
from view. 
VOL. XL. 
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