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

THE ORIGIN AND DEVELOPMEVT OF MUSICAL SCALES. 335
(c) On these two facts hangs the law of selection according to tone-relationship, which is the formal statement of the
familiar thought that we seek to associate together tones that
seem akin to each other. The law of relation is: Two clangs
are to be considered related in the first degree when they have
.an identical partial, and the nearness of the relation is propor
tionate to the loudness of that partial in each clang compared
with the loudness of the other, unidentical partials; and two
clangs are to be considered related in the second degree when
both are in the first degree to a third clang.
Thus C (264 vibrations per second) and G (396) are related in the first
degree because the third partial of C and the second of G are identical
(792); C and A (44o) also are in the first degree (but further removed from
each other than C and G) because the fifth partial of C is identical with the
third of A (I 320). B (495), however, is in the second degree to C because
both stand in the first degree to G, the fifth partial of G being identical
with the fourth of B (I980).
The law of selection is: Within certain limits, in selecting
tones to compose a scale we choose those most closely related
to the tonic or starting-point.
It is now easy to see why the octave is the most universally
recognized of intervals. It joins the nearest of kin to the tonic,
and indeed bears a unique relation to the latter. This relation
is described by Helmholtz thus: "Let any melody be executed
on any instrument that has a good musical quality of tone, such
as the human voice; the hearer must have heard not only the
primes of the compound tones, but also their upper octaves, and
less strongly the remaining upper partials. When, then, a higher
voice afterwards executes the same melody an octave higher,
we hear again a part of what we heard before-namely, the even
partials of the former compound tones-and at the same time we
hear nothing that we have not previously heard.... Hence the
first and chief division of our scale is that into a series of
octaves. In reference to both melody and harmony we assume
tones of different octaves which bear the same name to have the
same value, and in the sense intended and up to a certain point
this assumption is correct." Imitation at the twelfth or at the
double octave would possess the same peculiarity as imitation at