184 



Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



Never was anything* more inaccurate: it would make the whole passage from unison to 

 the minor third a preparation for the grave harmonic of that concord. When the unison or 

 other simple concord is gradually mistuned, the " beating becomes more and more rapid, 

 changes to a violent rattling flutter, and then degenerates into a most disagreeable jar." 

 These phenomena are reversed as continued increase of the interval brings us towards another 

 simple concord. The description is due to Robison, who (iv. 414 — 421) goes through the 

 whole phenomena of the octave. It is clear that Young confounded the two kinds of beat: 

 and even Robison, while animadverting on Young's opinion of Smith, gives strong reason to 

 think that he does not make the distinction. He informs us (iv. 410) that Sauveur had applied 

 beats, and that his method is operose and delicate, " even as simplified and improved by Dr 

 Smith." In common with a great number of other writers, he ventures on no explanation of 

 any beats except those which occur in imperfect unisons, in which Tartini's beat is no other 

 than the vibration of the note itself. When he comes to mention the beats of badly-tuned 

 fifths, he declines explanation, and (iv, 409) states what " Dr Smith demonstrates." He calls 

 the method of beats, and to my mind very justly, the greatest discovery (iv. 411) made in the 

 subject since the time of Galileo: but he goes on to depreciate the value of his own opinion by 

 asserting that the theory of Tartini's harmonic is included in Smith's theory of the beats of 

 imperfect consonances. The great defect of Smith's theory is its exclusion of Tartini's har- 

 monic. Young, in replying, writes as follows (i. 136): "Why then are we obliged to call it 

 Dr Smith's discovery, or indeed any discovery at all .'' Sauveur had already given directions 

 for tuning an organ-pipe by means of the rapidity of the beating with others, Mem. de 

 I' Acad. 1701, 475, ed. Amst. Dr Smith ingeniously enough extended the method; but it appears 

 to me that the extension was perfectly obvious, and wholly undeserving the name either of a 

 discovery or of a theory." This amply proves that neither Robison nor Young had read 

 Smith's theory; and I have very strong doubts that any pei'son who has written on the subject 

 ever did read it, Chladni makes precisely the same mistake as Young. He tells us {Acous- 



* Except, perhaps, Young's reiteration of his own mistake, 

 several years after, in the Course of Lectures ( London, 2 vols. 

 4to, 1807, Vol. I. p. 390). Young here begins by describing the 

 Smith's beat of imperfect unisons, clearly and correctly. He 

 then takes, as his second instance, the Tartini's beat of a well- 

 tuned diatonic semitone, and then repeats the account of the 

 Smith's beats giving the grave harmonic. The terms in which 

 he has spoken of Ur Smith's labours are sucli as can only be 

 met by convicting him of clear and palpable mistake. Those 

 who may be inclined to wonder that Young should have so sig- 

 nally failed in a matter connected with the distinct conception 

 of a complex undulation, may be reminded that many an inves- 

 tigator has fallen into some singular error in the subject which 

 he had, of all others, made completely his own. 



t 1 think this is a mistake. 1 find nothing in Sauveur's 

 memoir of 1701 about beats, except what I have described. 

 Lagrange, in his celebrated Turin memoir on sound, refers 

 (p. 75) to Sauveur's memoir of 1700 (not 1701) in so vague 

 a manner that he might be supposed to have Smith's beats in 

 view. On looking at the volume for 1700, I find, not a memoir 

 by Sauveur, but the description of one, forming part of the 

 abstracts called Histoire. Here we find that Sauveur did 

 actually commence with imperfect unisons, which give Smith's 



beats, that he had a notion of the rationale of such beats, that 

 he had made some experiments, and that a commission of tbe 

 Academy was appointed to inspect their repetition. The ac- 

 count of this experiment is a part of the history of the subject. 

 " M. Sauveur en rendit conte luy-meme et avoUa que pour 

 cette fois elle n'avoit pas bien r^ussi, car d'autres fois, et en 

 presence des plus habiles Musiciens de Paris, elle avoit paru 

 trds juste et tr^s prdcise. La difficult^ de la recommencer, 

 I'appareil qu'il faut pour cela, d'autres occupations plus 

 pressantes de M. Sauveur, et meme d'autres recherches 

 d'Acoustique, ou il a it6 oblige de s'engager par la liaison 

 qu'elles avoient avec le Son fixe, ont e'te cause qu'on en est 

 demeurf? la, mais on sjait qu'en fait d'experiences il ne faut 

 pas se d^courager aistiment, et qu'elles ont pour ainsi dire, leur 

 caprices que I'on surmonte avec le temps " (p. 139). All this 

 means that Sauveur commenced with the beats of imperfect 

 unisons; that he made experiments which satisfied the musi- 

 cians, bat broke down — by caprice — before the academicians; 

 that he had in the mean time commenced his acquaintance with 

 Tartini's beats, and was pursuing the researches which led to 

 thepaper of 1701, in which Smith's beats are wholly abandoned. 

 It is singular that Smith himself did not see this. 



