VIII. § 80.] DISSONANCE OF TWO CLANGS.
165
two co-existing sounds are no longer simple tones,
but composite clangs, eacb consisting of a series of
well developed partial-tones, the case becomes alto¬
gether different. Let us examine the state of things
which then presents itself.
The sounds of most musical instruments do not
contain more than the first six partial-tones ; we
will, therefore, assume this to be the case with the
clangs before us. No beats can then arise between
partial-tones of the same clang for the reason as¬
signed on p. 164. Dissonance due to beats will,
however, be produced if a partial-tone belonging
to one clang is within the specified distance of a
partial-tone belonging to the other clang. Several
pairs of tones may be thus situated, and, if so, each
will contribute its share of roughness to the general
effect. The intensity of the roughness due to any
such pair will depend chiefly on the respective
orders to which the beating partial-tones belong,
and on the interval between them. The lowest
partial-tones being the loudest, produce the most
powerful beats, and half-tone beats are, in general,
harsher than those of a whole tone. In determining
the general effect of a combination of two clangs, we
have to ascertain what pairs of partial-tones come
within beating-distance, and to estimate the amount
of roughness due to each pair. The sum of all these