periodip vibration. The simplest possible pcri-
oilic vibration is like that of a simple pendulnm,
anil it is called "simple harmonic." It is char-
acterlzeii by a ilefinite nnmber of vibrations per
second, i.e.. its "frequency," and by the extent
of the swing, i.e., its "amplitude." If a second
pemlnliiin is suspended from the bob of the first,
and a third from the bob of the second, the vi-
bration of the third and lowest bob is no longer
simple harmonic in general. Its vibration is
called "complex;" and it is evident that it is
the sum of the vibrations of the separate pen-
rtnlnnis. Complex vibr.itions may, therefore,
differ in the number of the component vibrations,
and in their frequencies, amplitudes, and rela-
tive phases.
S()t'Ni> Skn'S.tion'. It would be expected tb.at
there should be some connection between the
nature of the vibrations of the vibrating body,
that of the waves iiroiluced, and that of the
sound heard. Such is the case. .V iu)ise is al-
ways produced l)y an irregular, disconnected dis-
turbance in the air; and this in turn is due to
an ii regular successioti of vibrations, each last-
ing for a brief interval. A simple musical note
is always due to a simple harmonic train of
waves, and this to a simple harmonic vibration.
The loudness of the note varies directly with the
amplitude of tiie waves; whatever increases the
amplitude of the waves increases the loudness of
the sound, and vice versa. It is increaseil, there-
fore, by an increased amplitude of the vibration;
and it decreases as the distance from the ear to
the vibrating body is increa.sed. (It should not
be thought, however, that numerical values can
be given the loudness of a soun<l, or that there
is any lixed numerical relation between the am-
plitude of the waves and the intensity of the
sensation.) The pitch of the note depends upon
the wave-number of the waves entering the ear;
whatever increases the wave-number "raises" the
pitch, and rire versa. Therefore, if the ear and
the vibrating body are at a fixed distance apart,
and at rest with reference to their positions in
space, the iiitoli will vary directly with the fre-
quency of the vibrating body; thus we often use
the expression, "a pitch of 300," meaning the
pitch of a sound produced by a vibrating body
which makes ;:!00 complete vibrations in one sec-
ond. If, however, the vibrating body is ap-
proaching the ear, or if the ear is approaching
the vibrating body, the number of waves enter-
ing the ear is greater than it would be if there
were no such motion; and so the wave-number
is greater than the freqtiency of the vibrating
body, and the pitch of the sound is raised. Sim-
ilarly, if the distance between the ear and the
vibrating body is increasing, the wave-number
is less than the frequency of the vibration, and
the pitch is lowered. This change of pitch, due
to the relative motions of the ear and the vibrat-
ing body in the surrouniling medium, is known
as Dopjilers Principle (q.v.), and is illustrated
by the sudden drop in pitch if one stands on the
platform of a railway station and listens to the
whistle of a locomotive passing at a high speed.
A comjilex musical note is always due to a
complex train of waves, and this, in turn, to a
complex vibration, if there is only one vibrating
body. Further, two notes which dirter in qual-
ity nuvy he shown to be due to complex trains
of waves which differ in complexity. Hut it
should he note<l that all experimental evidence
points to the idea that differences in relative
phases of the component trains of waves do not
cause differences in the quality of the sound
heard. In other words, two complex trains of
waves made up of the same simple waves will
produce the same sound, regardless of the phases
in the two trains. This may be explained by
saying that the ear automatically resolves a
complex train of waves into its simple harmonic
component trains, hears the simple tones due to
each of these, and, therefore, has a complex sen-
sation. This statement is called "Ohm's law for
sound-sensation."
Kl-.D.V.MEN'TAI„ P.VRTI.I,. .WD COMniX.VTroN'-*.!.
Viiii{.Tioxs. Musical instruments may be di-
vided roughly into two classes, wind and string
instruments. In the former class are included
organ-pipes, horns, llutes, etc.; in the latter,
pianos, violins, harps, etc. In all wind instru-
ments a column of air inclosed in a metal or
wooden tube is set in vibration by suitable
means, and this vibrating mass produces the
waves in the surrounding air. In string in-
struments, tiexible .strings are stretched between
pegs fastened to a solid frame — -in general a
wooden board — and they are set in transver.se
vibration by bowing. ])lucking, or striking. As
a result of the vibration of the string, the frame
holding the pegs is itself set in vibrations of the
same frequency, and it, as well as the siring
itself, produces the waves. The importance of
the so-called sounding-board is at once evident.
^ 1
h .
^a
c
b' ■— __i,j d'^_^
•>'!— — i.l
Fio. 3. A stretched flexil)le string, -l B, can vibrate in many waves; as a whole, with its middle point its point of greatest amplitude, as in 1 (fig. 3); in two parts, with its middle point, 6, at rest, and the two halves vibrating like separate strings in opposite phases, as in 2 (lig. 3); in three parts, with two points, c and l>, at rest, di- viding the string into three equal vibrating seg- ments, as in 3 (fig. 3), etc. The frequencies of these different modes of vibration are in the ratios of 1:2:3:4, etc. The vibr.ation of the string as a whole is called the "fundamental;" the others, the " upper partials." The frequency of the transverse vibrations of a stretched flex- ible string is given Ijy the formula where T is the stretching force or tension, m is the mass of each unit l<'ngth of the string. // is the length of the vibrating segment. Thus, in the fundamental, /. is the length of the string; in the lirst upper partial it is one-half the length of the string, etc. When the string is set vi- brating by a random blow or tmwing, it will make complex vibrations, resulting from the combination of the fundamental and some of the upper partials, the number and relative in- tensities of these depending largely on the point where the blow is struck, or the bow applied, anil on the character of the impulse. So, when-
