ever a musical tone is produced by a string
instrument, the ear can recognize in the complex
sound simple tones due to the fundamental and
the upper partials; and differences in the qual-
ity of sounds caused by different string instru-
ments, which have fundamentals of the same
frequency, are due to differences in the number
and character of the upper partials. which de-
pend in turn on the material of the string, the
point where the impulse is applied to set the
string in motion, and the character of this im-
pulse. Similarly, the vibrating column of air in
organ-pipes, horns, etc., can vibrate in different
ways; and in a complex vibration there is a
fundamental and upper partials vrhose frequen-
cies are in the ratios of 1:2:3:4, etc. The
frequency of the vibrations of the fundamental
in an open organ-pipe is given by the formula:
v=
L
where V is the velocity of waves in the gas which fills the pipe, and L is the length of the pipe approximately. The similar formula for a "stopped" pipe is: i L (In stopped organ-pipes the vibrations are in the ratios 1:3:5:7, etc.) In other in- struments than wind and string ones, such as drums, cymbals, etc.. there are upper partials besides the fundamental; but there is no simple matliematical relation between their frequencies. When two organ-])ipes on the same wind-chest are "sounded" loudly, the resulting waves in the air are not diie simply to eacli fundamental and its upper partials, but also to certain extra vi- brations due to the combined action of the two vibrating columns of air on the surrounding air. Tlius, if the fundamentals of the two pipes have frequencies 1000 and 600. there will be present waves showing the existence of vibrations whose frequencies are 1000 -f 600 and 1000— fiOO. The sounds heard owing to these vibrations are called "summational" and "differential" tones, or, in general, "combinational" tones; they are always difficult to hear. The existence of both partial and combinational vibrations may, however, be established by means of resonators (q.v.). Harhony Ayo Discord. If two organ-pipes whose frequencies do not differ much are sounded together, the ear observes a fluctuation in the loudness of the resulting sound. It is first loud, then weak, loud and weak, etc., giving rise to what are called "beats," the number of beats per second being equal to the difference in the frequencies of the pipes. Tlius, two pipes of frequencies 280 and 285 produce 5 beats per second. The explanation of the phenomenon lies in the superposition of the two resulting trains of waves; for, if the wave-number of one train exceeds that of the other by five, it will happen five times in the course of a second that when one train of waves reaches the ear in a certain phase, the other train will reach the ear in an exactly opposite phase; and so the two waves will tend to neutralize each other's action and thus make the sound weak; whereas, in be- tween these instants of weakness there will be others when the two waves reach the ear in the same phase, and so reinforce each other and thus make the sound loud. This is shown diagi-am- matieally in fig. 4, where there are two trains of waves of unequal wave-number which interfere and produce beats. The wave-length of one set is A (I, which is four-fifths of A I, the wave length of the other. The two waves at A are in the same phase, and there is increased sound; but as the motion progresses, one train loses with respect to the other, until they are in opposite phase, as at C and D, where silence ensues. Beats are disagreeable to hear, for the same reason that a flashing light is unpleasant to see, or a tickling feather to feel, namely, the nerves being first stimulated, then allowed to partially re- cover, then again stimulated, etc., are disagree- ably affected. The degi-ee of unpleasantness de- pends in part on the number of beats, but also on the pitch of the note, whose intensity is fluc- tuating. Beats can be formed by the interfer- ence of the upper partials as well as by the fun- damentals, and by the combinational vibrations also. Thus, if two organ-pipes of frequencies 500 and 252 are sounded together, the first upper partial of the pipe whose fundamental is 252, i.e., a note of frequency 504. will beat with the other fundamental whose frequency is 500. it. however, two organ-pipes are sounded who^e fundamentals are such that there are no beats except between the upper partials of high or- ders, the sensation should be a pleasant one; and such is observed. To secure such a condi- tion it is e'ident that the ratios of the frequen- cies of the fundamentals must be simple frac- tions, 1: 1, I: 2, 1: 3, 2: 3, 1: 4, 3: 4, etc. Such combinations of two notes produce what is called "harmony." On the other hand, whenever beats can be expected between two notes or their partials, or their combinational notes, an un- pleasant sensation called "discord" is observc<l, it being possible to predict the degree of the di^* cord from the number of beats which most oc- cur. This explanation of harmony and discord is due to Helmholtz. The explanation of "mel- ody," that is, the pleasant sensation perceived when notes, suitably chosen, are sounded consec- utively, is undoubtedly psychological, not physi- cal. For the discussion of the formation of musi- cal scales liascd on these simple harmonies, see Major: IIixor. Limits of Hearing. Aerial waves of all wave- numbers do not affect the auditory nerves of the normal human ear, it being found by trial that wave-numbers less than 30 do not produce a musical tone, and wave-numbers exceeding about 20.000 do not produce sound at all. For niu~i cal purposes the extremes are about 40 and 40110. To study waves whose wave-numbers e.xceed 10,000 (and in fact for those of much less num-
