232 



NATURE 



\yan. 20, 1876 



The Minister of the Admiralty has promised to send 

 every instrument which should appear suitable for the 

 purpose. The Committee of the German Chemical 

 Society have elected superintendents to procure a worthy 

 representation of chemical apparatus and specimens of 

 chemical compounds of scientific interest. It has been 

 decided to address an invitation to the members of the 

 society to send such specimens to Berlin, in order to form 

 a systematical and uniform collection of rare and inte- 

 resting chemicals. 



We may state generally that the Governments of 

 Belgium, France, Germany, Italy, the Netherlands, and 

 Switzerland, have now appointed committees to act in 

 concert with the general committee in London. The 

 Government of the United States has, through Mr. 

 Fish, intimated that it is in communication with the 

 various Departments and Scientific Institutions, with the 

 object of forwarding the Exhibition. 



BEATS IN MUSIC* 



II. 



THE second kind of beat differs from the first in that it 

 arises from the imperfection, not of unisons, but of 

 wide-apart consonances, such as the third, fourth, fifth, 

 sixth, and octave. 



This beat is well known practically to organ tuner?, 

 and may be soon rendered appreciable to any musical 

 ear. Taking the fifth as an example, let the two notes 



be sounded'on an organ, or any instrument 



of sustained tones. If they are perfectly in tune, the 

 united sound will be smooth and even ; but if one of them 

 be sharpened or flattened a little, a beat will be heard just 

 as in the case of the imperfect unison ; and which, like 

 it, will increase in rapidity as the note is made more and 

 more out of tune. That this is not the same beat as 

 Tartini's is obvious from the failure cf the rule for the 

 latter when applied to the former ; for example, when the 

 concord is in tune the upper note vibrates 768, and the 

 lower one 512 vibrations per second, therefore there 

 ought to be 256 beats per second ; but in reality there are 

 no beats at all, they only begin when the notes are put 

 out of tune ; hence the Tartini-beat rule is useless and 

 inapplicable in this case. 



This beat may be called the consonance beat, and it 

 has also been termed " Smith's beat," from its having been 

 first investigated by him. 



The theory of Smith's beat, as given by Smith himself, 

 is complicated and difficult to describe ; but we will en- 

 deavour to give some idea of its nature and cause. 



We must return to the illustration of the coffin-makers. 

 Suppose one of them to have sold his business to another 

 man in the trade, who was so much more active and 

 energetic that he could drive his nails half as fast again 

 as ordinary workmen. Call him A, and suppose that 

 when he began to work it was found that he struck exactly 

 three blows to two of his neighbour B. As B struck 100 

 per minute, A will now strike 150. And assume that on 

 a certain day they both begin exactly together. The 

 passer-by will hear that every third blow of A exactly 

 coincides with every second of B ; so that he will no- 

 tice fifty coincidences in a minute ; or to describe them 

 more correctly, he will notice per minute fifty phases of 

 compound effects, in each of which there is a coincidence. 

 This phase constitutes Tartini's beat, but now very much 

 augmented in rapidity from what it was before : then 

 there were only one or two coincidences per minute, novt 

 there are fifty. 



Now suppose that A, from some slight exhilarating 



* By W. Pole, F.R.S., Mus. Doc. Oxon. Continued from p. 214. 



cause, begins to strike a little faster ; i.e. that he 

 makes 151 blows in a minute instead of 150. Let us 

 endeavour to find out what will be the result on the 

 listener. Still supposing the two strokes to begin with a 

 coincidence, the third blow of A will still coincide 7iery 

 nearly indeed with the second of B ; it will only differ 

 from it by f^^xi of 3, minute, a quantity inappreciable to 

 the ear. Hence the Tartini phase will at this time be 

 practically unharmed. But after a few repetitions the 

 divergence of the blows will be so great as to become 

 appreciable, and the listener will begin to notice a series 

 of changes of form of the Tartini phase, in which there 

 is now no coincidence of the blows, but only a variation of 

 their arrangement, which, moreover, is itself constantly 

 varying. After a time, however, these changes will 

 exhaust their possible varieties, the listener will notice 

 that two of the blows begin to approach again, and at 

 last will coincide, as they did before. He thus notices a 

 long cycle of the Tartini beats, and this long cycle is the 

 Sfnith's beat. It is, in fact, a beat of what mathematicians 

 would call the second order ; the first, or Tartini's beat, 

 is a cycle of differing periods ; the second, or Smith's 

 beat, is a cycle of differing cycles. 



Let us next attempt a numerical estimation of the 

 length of this second cycle in the case of the coffin- 

 makers. To effect this we must inquire when the coin- 

 cidences of two blows will recur. It is plain that they 

 will recur at the end of the minute, i.e. if the first blow of 

 A coincided with the first of B, then the 151st of A will 

 coincide with the looth of B. This will give one long 

 cycle, or one Smith's beat, per minute. A careful com- 

 parison of the times of the respective blows will show, 

 moreover, that (since 100 and 151 are prime to each other) 

 there will be no other exz-Ct coincidence during the minute; 

 and a hasty reasoner may conclude that one beat per 

 minute will be the proper number. But if the listener be 

 asked to describe what he hears, he will dissent from this 

 and say confidently that there are ttuo places in the 

 minute where he hears a coincidence. To test his asser- 

 tion, let us apply Young's principle mentioned before, and 

 inquire whether in the course of the minute there is any 

 other place where the blows so nearly coincide that the 

 ear may mistake them for real coincidences. The 74th 

 blow of A will occur at y'/t of a minute after starting, 

 whereas the 49th blow of B will occur at ^^^^ of a minute. 

 Ttie difference between these is only TXiOff of ^ minute, 

 which is quite inappreciable. Hence, practically, there 

 will be two parts of the minute where the blows coincide, 

 and there will be consequently two Smith's beats in the 

 same time.* If we were to suppose A to make 152 blows 

 per minute (or 148, for a deficiency would produce the 

 same result as an excess) to B's 100, we should, calcu- 

 lating on the above plan, find fottr cycles or beats per 

 minute. Or we may alter the proportion : suppose for 

 example. A, intending to make five blows to B's four, 

 makes really 126 per minute instead of 125 as he ought 

 to do ; it will be found by calculation that there will be 

 four places of coincidence in the minute, or four Smith's 

 beats ; if he strikes 127 blows, there will be eight Smith's 

 beats — and so on. 



We hope the foregoing homely illustration will help to 

 render clear the nature of the Smith's beat as applied to 

 sounds. Although the Tartini beat may not hz really 

 converted, as Young supposed, into the Tartini harmonic, 

 but, according to Helmholtz, remains as a beat, inappre- 

 ciable by reason of its great rapidity, it certainly has a 

 physical and mathematical existence ; and it as certainly 

 changes its phases by reason of the small divergence of 

 the times of vibration from those due to the true concord, 

 and it is the recurrence of similar phases in a long cycle 



* Mr. De Morgan, in his admirable paper elucidating Smith's profound 

 investigation, unfortunately omits to notice this important element of the 

 af>ptoximate coincidences. The consequence is that Ins e.xplanation is not 

 easy to follow, and indeed would appear wrong, although his results are 

 perfectly correct. 



