jaiu 13, 



1876] 



NATURE 



213 



Investigation. Hence a correct discrimination between 

 them is ver>' desirable. 



The first and simplest kind of beat we will distinguish 

 ty the name of the Union Beat, it being produced by the 

 concurrence of two sounds nearly, but not quite, in unison 

 with each other. Let two organ-pipes, or any other sus- 

 tained soundi, be tuned first exactly in unison ; the com- 

 bined effect will be equable and smooth, undistinguishable 

 from a single sound. But now let one of the notes be 

 put out of tune, at first very slightly ; the result will be a 

 peculiar effect of wavy pulsation or beating. The exact 

 description varies according to the fancy of different 

 hearers, but it is usually said to resemble an alternation 

 of different vowel sounds, like luaii', waw, ivatu, orj'a,ya, 

 ya. The beating, when the notes are but slightly out of 

 tune, will be slow ; if the error is made worse, the pulsa- 

 tions will increase in rapidity, till they become too quick 

 to be counted. 



This fact is very commonly known, and its experi- 

 mental exhibition is exceedingly simple and easy. If 

 organ-pipes and the means of tuning them cannot be had, 

 the two sounds may be produced on any wind instrument, 

 which can be easily put into the adjustment necessar}'. Or 

 one of two unison reeds of a harmonium may be thrown 

 out of tune by weighting it with a little bit of wax ; indeed, 

 in the drawing-room instruments one unison stop is 

 usunlly made purposely out of tune with another, the com- 

 position giving a tremulous effect resembling the shaking 

 of the voice, the stop being named '•' voix celeste " (in 

 Italian organs a stop called the " vox humana " is foimed 

 of two pipes tuned in a similar way). These are real 

 unison beats with so short a period as to produce 

 the tremulous effect in question. Two unison tuning- 

 forks may also be thrown out of tune by attaching wax 

 to the arm of one of them, which will make it a little 

 flatter. 



The beats may aiso be well produced on a violin. 

 Stop A with the fourth finger on the third string, and 

 play it along with the second string open, when the 

 adjustment of the former may be made with the greatest 

 nicety, and if it be put cut of tune the resulting beat will 

 sometimes be so prominent as almost to shake the instru- 

 ment under the chin. On a pianoforte the beats may also 

 be observed when one wire of a note is a little sharp or 

 flat of another, although tkis case is not so favourable 

 for observation, from the sovjids not being sustained. 



Nov.-, in seeking for the txplanation of this pheno- 

 menon, a homely preliminary illustration wUl be useful 

 Suppose two coffin-makers liv« next door to each other, 

 and suppose that on seme particjlar day they both strike 

 the blows on their nails at exaaly the same rate, and 

 begin exactly together ; the effect on a passer-by will be 

 that the sounds of the two will reach his ear simul- 

 taneously, smoothly, and regularly, md he will have dif5- 

 calty in disticguishing the combintd sound from what 

 mrald be produced by one workaan only. But now 

 suppose that by some change in the %ncy of one of the 

 men. A, he begins to strike a little faster than his neigh- 

 bour, making, we will say, eleven strikes to ten of the 

 other, B, The effect on the passer-by will be changed, 

 the sounds will reach his ear irregular.y, and, which is 

 the important thing, there will be regular periodical 

 phases appreciable ; for it is obvious that at every 

 tenth blow of B, or every eleventh of A, \he blows will 

 coincide, after which they wUl diverge and become irre- 

 gular till they coalesce again. 



To apply this illustration to the case of ^e sounds, it 

 must be borne in mind that a sound is traismttted to 

 the ear by waves of the air, each of which comists of an 

 alternate condensation and rarefaction. Tie comci- 

 dences of sound-waves give rise to peculiar tffects of 

 interference of various kinds, but it will suffice her; to say 

 that when two condensations coincide, the effect vill be 

 :" ^"erent to that when the condensation of one way* coin- 



cides with the rarefaction of another. It will be easily 

 seen that when the vibrations producing two sounds are 

 a little unequal in time, as if, for example, one vibrates 

 eleven times while the other is vibrating ten ; there will be 

 periodical coincidences corresponding to those of the 

 blows just mentioned, and it is these periodical coin- 

 cidences that produce the effect of what is called the beat 

 on the ear. 



Having thus established the nature of the beat, we may 

 now go a little further, and see what we can find out 

 about its time, or the length of the period which it 

 involves, and this is a matter which requires careful 

 attention. 



We will go back to the illustration of the coffin-makers, 

 and will now assume that the slower workman, B, makes 

 100 blows in a minute, whereas the quicker workman, A, 

 makes loi. It will be evident that a coincidence will 

 take place exactly at the end of every minute of time, so 

 that, for these numbers, the periods of coincidence (cor- 

 responding to our beats) will be one per minute. Let now 

 A increase his speed to 102 blows per minute, the other 

 remaining the same ; here there will be one coincidence 

 every fiftieth blow of B, or every fiftj--first of A, i.e. there 

 will be two coincidences per minute. 



It is easy to apply this to the sound-v-ibrations. Let 

 one note make 100 double vibrations per second, and let 

 the other note be sharpened to make loi. Here there 

 will be one coincidence, or, what is the same thing, one 

 beat per second. If the second note is sharpened a little 

 more, so as to make 102 vibrations, there will be two 

 coincidences, or two beats per second. 



Hence the rule has been derived, that th^ number of 

 beats per second is equal to the difference of the number of 

 vibrations per second of the two sounds. 



This is a simple rule, and it happens to ht practically 

 an accurate one ; but hasty writers, who have deduced it 

 firom one or two simple examples, have omitted to see a 

 curious theoretical difficulty that attends it. Let us go a 

 step further, and suppose the higher note to make 103 

 vibrations per second, while the other makes 100 : how 

 many beats per second would this give ? The rule says 

 three, but if we examine very carefully the succession of 

 sounds, we shall find there will not be three coincidences 

 per second, there would be only one, and hence the rule 

 will appear to fail. But if we try the experiment we 

 shall hear that there u-ill be three beats, and hence the 

 theory and the fact do not correspond. 



To explain the discrepancy, let us revert again to the 

 illustration of the coffm-makers : supposing A to make 

 103, and B 100 strokes per minute, the interval between 

 A's stroke is = tti of a minute, and between B's = ^^77. 

 The 33rd stroke of B wiU take place after ^4^ of a minute, 

 and the 34th of A after tsVi and as these fractions are 

 not the same, it is clear that the blows will not coincide, 

 neither will the 66th and 68th ; in fact, there wiU be only 

 one point in each minute when the blows will be heard 

 exactly together. 



Yet if the passer-by be asked to count with his watch 

 how many coincidences per minute he hears, he will 

 assuredly say three, and this discrepancy between theory 

 and fact demands to be reconciled. 



The explanation was cleverly hit upon by Dr. Young, 

 who (" Experiments and Inquiries respecting Sound and 

 Light," sec. xi.), treating of the subject, mentions " coinci- 

 dences, or near approaches to coincidences^ He saw that, 

 so far as the ear was practically concerned, a near ap- 

 proach would answer the purpose of an exact coincidence 

 equally well ; and this clears up all the difiaculty. For 

 although the 32rd and 34th blows do not come together 

 with theoretical exacttuss, they come practically so nearly 

 together that the difference between ^'5 and x4z is only 

 T oitt , that is, the difference in time between the two 

 blows is only ToaTrrr of a minute, which no passer-by could 

 appreciate, and he may therefore say they coincide. 



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