The Origin and Development of Musical Scales [pp. 324-343]

THE ORIGIN AND DEVELOPMENT OF MUSICAL SCALES. 337
Helmholtz adduces examples of five of the possible scales of
five tones, as follows:
(i) c-d —f-g-a —ac',
I 4 X X 2
(2) c —eb -f-g —bb-c',
I f I I V 2
(3) c-d — f-g-b-c',
I * I I M 2
(4) c-d-e —g-a —c',
I X I i 1 2
(5) c —eb- f —a — b-c',
I 11 V 2
Turning now to systems admitting shorter intervals than x_,
and especially to that of the Greeks, we find that they began by
admitting the interval f, but soon restricted themselves to A.
As they commenced with "octaves," so to speak, only a fourth
long, we must first look at the combinations within that short
compass.
Pythagoras, it was said, is supposed to have succeeded in
applying his method of tuning by perfect fifths and octaves to
the Dorian tetrachord with this result:
I it I~I
I 9'
but Helmholtz thinks that the principle of tone-relation prevailed in the other tetrachords.
The examples that he gives are the following:
(i) The enharmonic tetrachord of Olympos,
i t... I
:. Ki
The attempt was even made to bisect the small interval between
the lower notes, thus:
(2) The later enharmonic,
* D4
14 t
5
22
Scotch (minor).
Scotchi (major).
I... I