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# Harmony and Discord

## beats, waves, phase, frequencies and ear

HARMONY AND 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. Thus, two pipes of frequencies 2S0 and 2S5 produce 5 heats 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 tl:e sound loud. This is shown diagram matically in fig. 4, where there are two trains of waves of unequal wave-number which interfere and produce heats. The wave-length of one set is A (1, which is four-fifths of A 1, 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 degree 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, i 1 1 beat with the other fundamental whose frequency is 500. lf, however, two organ-pipes are sounded whose *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 evident that the ratios of the frequen cies of the fundamentals must be simple frac tions, 1 : 1, 1 : 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 observed, it being possible to predict the degree of the dis cord from the number of heats 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 cal scales based on these simple harmonies, see 21AJon;