132 



Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



about 1714. And even when they give a sound, it will still be convenient to call them Tar- 

 tini's beats. These beats are in their perfect theoretical existence when a consonance is quite 

 true, and they owe their usual existence to its approximate truth. Tartini* used to tell his 

 pupils that their thirds could not be in tune when they played or sang together, unless they 

 heard the low note: assuming, doubtless, that their perceptions were as acute as his own. 



The second kind of beats I shall call SmitJCs beats, because Dr Smith first made use of 

 them, and gave their theory. They are entirely the consequence of the imperfection of a con- 

 sonance, and become more rapid and more disagreeable as the imperfection increases, vanishing 

 entirely when the consonance is perfectly true. 



I cannot find the means of affirming that Smith was acquainted with Tartini's grave har- 

 monic. In the place in which one would have expected him to mention it, namely, when he 

 mentions the Jlutterings, as he calls them, which I name Tartinis beats, he does not make 

 the slightest reference to those flutterings becoming rapid enough to yield a note, though he 

 complains that he could hardly count them. 



Smith accuses Sauveur of confounding the beats of an imperfect consonance with the 

 flutteringsf of a perfect one. It is true that Sauveur makes the same use of Tartini's beat 



• Tartini published his treatise on harmony at Padua in 

 1754. D'Alembert's account of this work ia so precisely what 

 he might have written of Smith, that I quote it. " Son livre 

 est e'crit d'une maniere si obscure, qu'il nous est impossible 

 d'en porter aucun jugement : et nous apprenons que des Savans 

 illustres en ont pense' de menie. II seroit k souhaiter que 

 I'Auteur engageat quelque homme de lettres verse' dans la 

 Musique et dans I'art d'&rire, ^ developper des idees qu'il 

 n'a pas rendues assez nettement, et dont Part tireroit peut-etre 

 un grand fruit, si elles e'toient mises dans le jour convenable." 

 M. Romieu, of Montpellier, published a memoir in 1751, in 

 which he described Tartini's grave harmonic : and hence some 

 have made him the first discoverer. But Tartini had been 

 teaching the violin, on which instrument he was the head of a 

 celebrated school, a great many years : that he should not have 

 published t'he grave harmonic to every pupil whom he taught 

 to tune by fifths, is incredible. He himself affirms in his 

 work that he always did so from 1728, when he established 

 his school : and further, that he made the discovery on his 

 violin, at Ancona, in 1714; this was the year after he dreamed 

 the Devil's Sonata. As it is stated that he told how the devil 

 played to him in his sleep, many years after, to Lalande, who 

 could make astronomical gossip of any thing, I should not be 

 at all surprised if a certain four-volume work contained evidence 

 of the date of the grave harmonic. 



Kameau, not Romieu, is the natural counterpart of Tartini. 

 In 1750 he published his celebrated treatise on harmony, the 

 completion of a system which he had sketched in previous 

 works : and he and Tartini are thus related. Tartini makes 

 his grave note the natural and necessary bass to the consonance 

 which produces it : Rameau makes the harmonics of any given 

 note the natural and necessary treble of the given note as a bass. 

 These contemporary counter-systems are now exploded : they 

 have an uncertain coiuiexion with the truth, no doubt ; but the 

 are demands and obtains a great number of combinations which 

 neither system will allow. 



It is due, however, to Rameau to observe that his discovery, 



which appears independent of Tartini's, is that of a physical 

 philosopher, and is developed in a masterly manner. He gave 

 the theory, and detected the beats which occur when the grave 

 harmonic becomes inaudible by lowness. His memoir was pub- 

 lished by the Royal Society of Montpellier in 1751, in a coUec 

 tion headed Assemblie Publique &c. I have never seen this 

 memoir. There ia a long extract from it in a curious and ex- 

 cellent work, which I never see quoted, the Essai sur la musique 

 ancienne et moderne^ Paris, 1780, 4 vols. 4to, attributed by 

 Brunet to Jean Benjamin de la Borde. 



Chladni {Acotistique, p. 253) says that the first mention of 

 the grave harmonic which he knew of is by G. A. Sorge (An- 

 weisung zur Stimmung der Orgelwerke, Hamburg, 1744), who 

 asks why fifths always give a third sound, the lower octave of 

 the lower note, and concludes that nature will put 1 before 2, 3, 

 that the order may be perfect. If Tartini's evidence in his own 

 favour be disallowed, then Sorge becomes the first observer. 

 But to me the uncontradicted assertion of a teacher whose 

 pupils were scattered through Europe, and included men so 

 well known on the violin as Nardini, Pugnani, Lahoussaye, 

 &c. &c., that he had pointed out the third sound to all his 

 school from 1728 to 1750, is real evidence. Chladni's mention 

 of Tartini is as uncandid as possible: — ' Tartini, auquel on 

 a voulu attribuer cette de'couverte, en fait mention dans son 

 Trattato . . .' Mentions it .' No one knew better than Chladni 

 himself (as he proceeds to show, the moment the paragraph 

 about priority is finished) that Tartini's whole book is a system 

 founded upon it. D'Alembert, La Borde, Rousseau, &c. do not 

 dispute Tartini's claim ; and the common voice of Europe 

 gives no other name to the discovery. 



On this subject in general see the Article Fondamental in 

 the Encyclopedia, by D'Alembert ; Rousseau's Musical Dic- 

 tionary, Ilarmonie and Systime ; Matthew Young's Enquiry, 

 &c. 



f Smith does not, so far as 1 can find, attempt to explain 

 these Jlutterings; though I think it may be collected that he 

 knew their cause. 



