Unqualified page numbers on the left conform to the original German 1911 edition;
those on the right conform to Carter's 1978 English translation.
[p 21]
[Carter, p 23:]
4 - The Major Mode and the Diatonic Chords
Our major scale,
the series of tones c, d, e, f, g, a, b,
the tones that also provided the basis for the Greek and church modes,
we can explain as having been found through imitation of nature.
Intuition and inference assisted
in translating
the most important characteristic of the tone, the overtone series,
from
the vertical (as we imagine the position of all simultaneous sounds)
into
the horizontal, into
separate, successive tones.
The natural model, the tone, exhibits the following characteristics:
1. A musical sound is a composite,
made up of a series of tones sounding together, the overtones;
hence, it forms a chord.
From a fundamental, C, these overtones are:
|
etc. |
|
|
C |
c |
g |
c 1 |
e 1 |
g 1 |
(b b1) |
c 2 |
d 2 |
e 2 |
f 2 |
g 2 |
|
|
overtone |
Fundamental |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
partial |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
|
prime factor |
n 0 |
n 0 |
31 |
n 0 |
51 |
31 |
71 |
n 0 |
32 |
51 |
111 |
31 |
|
111 |
||||
|
F |
||||
|
71 |
||||
|
B b |
||||
|
(7... n)0 |
E |
|||
|
C |
G |
D |
2. In this series the c is the strongest sound because
3. After the c
the next strongest tone is g, because
If we think of this g as a real tone
(as indeed occurs when the overtone series is realized horizontally,
when, for example, the 5th of a horn tuned in c is played),
it then has overtones itself (as a tone actually played); these are:
|
etc. |
|
|
g |
g 1 |
d 2 |
g 2 |
b 2 |
d 3 |
etc. |
|
|
overtone |
fundamental |
1 |
2 |
3 |
4 |
5 |
|
|
partial |
1 |
2 |
3 |
4 |
5 |
6 |
|
|
prime factor |
31 |
31 |
32 |
31 |
3151 |
32 |
|
(7... n)0 |
E |
B |
||
|
C |
G |
D |
and at the same time
this g, together with its overtones,
presupposes the C (fundamental of the horn).
Thus it happens that
the overtones of the overtones also contribute to the total sound.
dependent upon a tone a 5th below, the C.
The conclusion that follows from the foregoing:
This tone, C, is likewise dependent upon the tone a 5th below it, F.
[p 22]
Now if the C is taken as the midpoint, then
its situation can be described by reference to two forces: