544 



EXPERIMENTS AND INQUIRIES 



XI. Of the Coalescence of musical Sounds. 



It is surprising that so great a mathema- 

 tician as Dr. Smith could have entertained, 

 for a moment, an idea that the vibrations 

 constituting different sounds should be able 

 to cross each other in all directions, without 

 affecting the same individual particles of air 

 by their joint forces : undoubtedly they cross, 

 without disturbing each other's progress ; but 

 lliis can be no otherwise effected . than by 

 each particle's partaking of both motions. If 

 this assertion stood in need of any proof, it 

 might be amply furnished by the: phenomena 

 of beats, and of the grave harmonics ob- 

 served by Romieu and Tartini ; which M. 

 De la Grange has already considered in the 

 same point of view. In the first place, to sim- 

 plify the statement, let us suppose, what pro- 

 bably never precisely happens, that the par- 

 ticles of air, in transmitting the pulses, pro- 

 ceed and return with uniform motions; and, 

 in order to represent tlieir position to the eye, 

 let the uniform progress of time be repre- 

 sented by the increase of the absciss, and the 

 distance of the particle from its original po- 

 sition, by the ordinate, (Fig. 36. .41). Then, 

 by supposing any two or more vibrations in 

 the same direction to be combined, the joint 

 motion will be represented by the sum or dif- 

 ference of the ordinates. When two sounds 

 are of equal strength, and nearly of the same 

 pitch, as in Fig. 39, the joint vibration is 

 alternately very weak and very strong, pro- 

 ducing the effect denominated a beat, (Plate 

 5. Fig. 46, B and C) ; which is slower and 

 more marked, as the sounds approach nearer 

 to each ciher in frequency of vibrations ; and 

 of these beats there may happen to be seve- 

 ral orders, according to the periodical ap- 



proximations of the numbers expressing the 

 proportions of the vibrations. The strength, 

 or rather the momentum, of the joint sound 

 is double that of the simple sound only at the 

 middle of the beat, but not throughout its 

 duration ; and if we estimate the force of 

 sound by the momentum of the particles, it 

 may be inferred, that the strength of sound 

 in a concert will not be in exact proportion 

 to the number of instruments composing it. 

 Could any method be devised for ascertain- 

 ing this by experiment, it would assist in the 

 comparison of sotind with light : but the 

 establishment of the fact would be no proof 

 of a difference in the nature of sound and 

 light ; for there is no reason to suppose the 

 undulations of light continuous : their inter- 

 missions may easily be a million million 

 times greater than the duration of each parcel 

 of undulations. In Plate 4. Fig. 36, letP and 

 Q be the middle points of the progress or re- 

 gress of a particle in two successive com- 

 pound vibrations ; then, CP being = PD, 

 KR = RN, GQ = QH, and MS = SO, 

 twice their distance, 2 RS = 2 RN + 3 

 NM + 2 MS = KN -h NM + NM -I- MO 

 =:KM + NO, is equal to the sum of the 

 distances of the corresponding parts of the 

 simple vibrations. For instance, if the two 

 sounds be as 80 : 81, the joint vibration will 

 be as 80.5; the arithmetical mean between 

 the periods of the single vibrations. The 

 greater the difference in the pitch of two 

 sounds, the more rapid the beats> till at last, 

 hke the distinct puffs of air in the experi- 

 ments already related, they communicate the 

 idea of a continued sound ; and this is the 

 fundamental harmonic described by Tartini. 

 For instance, in Plate 4. Fig. 37. . 40, the 

 vibrations of sounds related as 1 : 2, 4 : 5, 

 9:10, and : 8, are represented ; where 



