584 



SOUND 



harmonica present in any given vowel-sound vary 



with the pitch. 



As with all forms of wave-motion, sound may \x> 

 reflected (see ECHO) and refracted. When a string 

 or air-column is thrown into a steady state of 

 vibration with nodes occurring at regular intervals 

 there is in reality a reflected wave, which, travel- 

 ling hack wards along the vibrating sulmtance, in- 

 terferes with the for ward -travel ling wave in the 

 manner already described. Interference (q.v.) is 

 also shown by the phenomenon of heats. 



The existence of beats is determined by the 

 coexistence of two notes differing very slightly in 

 pitch ; and the number of beats per second is 

 simply the difference of the frequencies. Because 

 of i in- absence of upper harmonics in the note given 

 by a tuning-fork, the phenomenon ia produced in 

 its purest form by means of two tuning-forks origin- 

 ally in unison but thrown slightly out of tune by 

 weighting the one tuning-fork with a small piece 

 of wax attached to it If the tuning-forks, for 

 example, have frequencies 300 and 302, there will 

 be heard two beats per second i.e. the intensity 

 of the resultant sound will vary from zero to a 

 maximum and back to zero again twice every 

 second. The reason of this will lie easily seen if 

 we consider the resultant effect of the two sets of 

 waves at different times. For, since the two 

 sounds have the same velocity, it is clear that 

 across any surface set in the path of the rays of 

 sound the higher note will transmit two more 

 waves per second than the lower note, or one more 

 wave in half a second, or half a wave extra in a 

 quarter of a second. Suppose that at the begin- 

 ning of the second chosen the nearly equal waves 

 combine crest to crest and trough to trough, so as 

 to produce an increased intensity ; then a quarter 

 of a second later the slightly quicker vibration 

 will have gained half a wave-length on the other, 

 crest will fall with trough and trough with crest, 

 and little or no vibratory motion will be the result. 

 At the half second, crest and crest will again coin- 

 cide, and the resultant sound once more reach a 

 maximum ; and so on indefinitely. The transition 

 from maximum to minimum loudness is of course 

 gradual. If notes in which higher harmonics 

 exist are used the beating is not so simple. For 

 example, with open organ -pipes tuned to fre- 

 quencies 300 and 302, not only will the primes beat 

 twice a second, but the second harmonics 600 and 

 604 will beat four times a second, the third har- 

 monics six times a second, and so on. There is 

 generally no difficulty getting beats from a piano- 

 forte note, since the two or three strings that belong 

 to the note are rarely in accurate tune. 



In general if m and are the vibration numbers 

 of two notes sounding together which give beate, 

 (m - ii ) will be the number of 'heats per second. 

 If tlii- beating does not occur oftener than two or 

 three times a second the ear is not distressed. 

 Rapid beating, however, produces very unpleasant 

 sensations even after it has reached a rapidity too 

 high to be counted. \V!n<n the difference of fre- 

 quencies (in - n) is greater than 25 or 30, the 

 note of frequency (m - n) is heard in addition to 

 the two original notes. There is no difficulty in 

 bearing this differential tone, as it is called, when 

 the two notes are sufficiently loud. On instru- 

 ments giving sustained sounds, such as the organ, 

 harmonium, and concertina, very marked differ- 

 ential tones are produced ; and to an ear trained 

 to their i>erception they are recognisable on the 

 piano, when the difference of the frequencies lies 

 between 30 and 100 the rattling of the heato 

 may often be distinguished from the low hum of 

 the differential tone ; so that we are not warranted 

 in regarding these two phenomena as of the same 

 nature. By bringing the two notes by different 



courses to the two ears we can hear the beats, 



luit cannot hear the differential tone. Mu \.-i. 



in addition to the differential tone (m - n), there 

 are other differential tones of frequencies (2m - n ), 

 (i - 2n), &c., and also at times a weak Miiuma- 

 tional tone of frequency (in + n). This last- 

 named tone was discovered by Von Helmholt/., 

 to whom we owe the complete discussion of the 

 origin and significance of these Combinational 

 Tones. MS i hey are collectively termed. Two kinds 

 are distinguished by him. If the vibrations trans- 

 ferred from the vibrating body to the air are vny 

 large the simple law of superposition may not 

 hold. A simple pendular vibration, such as a 

 tuning-fork may give, will when transferred to tin- 

 air lose to some extent its simple harmonic char- 

 acter, and higher harmonics will enter in. If two 

 simple pendular vibrations act powerfully on the air 

 combinational tones will be produced in addition 

 to the higher harmonics of the two original notes. 

 These combinational tones existing in the power- 

 fully disturbed air can lie reinforced by use of 

 resonators. Combinational tones of the second 

 kind cannot be so reinforced, since they are pro- 

 duced in the ear itsdi". They are due to the 

 asymmetric character of the drum of the ear, which 

 cannot respond to two coexisting vibrations with- 

 out producing combinational tones. The fre- 

 quencies of these combinational tones, whether of 

 the first or second kind, are all included under the 

 general formula Mm T Nn, where HI and n are the 

 frequencies of the original notes, and M and N are 

 integers from zero upwards. As experiment shows, 

 only the first few integers are of any importance, 

 and no Minimal ionul tone of higher order than 

 m + n (M and N both unity) has ever been heard. 

 As an example, take two notes having frequencies 

 200 and 315. Their principal combinational tones 

 will have frequencies 115 (315-200) and 85 

 (2 x 200 - 315). This latter may be regarded as 

 the differential tone between the lower prime and 

 the first combinational tone. These two tones can 

 ln h be heard if the intensities of the sounds are- 

 sufficiently strong. Both theory and experiment 

 show that the comparative intensity of combina- 

 tional tones grows rapidly as the 'intensities of 

 the real notes are increased, and also that combina- 

 tional tones of low pitch are most prominent. 



As first brought out clearly by llclmlioltz, 

 combinational tones are of peculiar 'interest when 

 the two notes form a consonant interval. Tims, 

 take any two notes a musical fourth apart. Their 

 vibration numbers may be represented by 3 and 

 4ii. Their principal differential tone will 'have the 

 frequency n ( = 4 - 3n ), and will therefore form 

 the fundamental tone of which the given two are 

 harmonic overtones. Again, take any perfect 

 liiad din. mi, sol) having frequencies 4n, 5n, 611. 

 Each successive pair gives the same differential 

 tone ,- the first and last together give the differ- 

 ential tone 2. Thus we hear the low tone which, 

 is harmonically fundamental to all, and its octave. 

 In some cases the dilVerential tone Uecomes so- 

 loud that the real notes which are being sounded 

 become merged in it as upper harmonics. If the 

 notes of the triad are not in perfect tune the 

 differential tones will not ! harmonically related. 

 On an organ tuned in equal Temperament (q.v.) 

 the chord built upon the treble (' consists of notes 

 having frequencies 522, 6577, 782-1 (see PITCH). 

 The two lo\ve-t differential tones have frequencies 

 13.V7 and 124'4 notes which, sounding together, 

 produce 11 '3 beats per second. The ear that has 

 accustomed itself to the pure harmony of the per- 

 fect triad will easily recognise a certain dissonance 

 in the triads given by pianos and organs. 



Beats always mean the coexistence of two notes, 

 of nearly the same frequency. If on any organ 



