VOL. XC.] PHILOSOPHICAL TRANSACTIONS. 6\? 



in fig. 36, the joint vibration is alternately very weak and very strong, producing 

 the effect denominated a beat, pi. 10, fig. 43, b and c; which is slower and more 

 marked, as the sounds approach nearer to each other in frequency of vibrations; 

 and, of these beats there may happen to be several orders, according to the peri- 

 odical approximations of the numbers expressing the proportions of the vibrations. 

 The strength of the joint sound is double that of the simple sound only at the 

 middle of the beat, but not throughout its duration; and 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 ascertaining this by 

 experiment, it would assist in the comparison of sound with light. In pi. 9, fig. 

 33, let p and a be the middle points of the progress or regress of a particle in 2 

 successive compound vibrations: then, cp being = pd, kr = rn, gq= qh, and 

 ms = so, twice their distance, 2rs = 2rn + 2nm + 2ms = kn -f- NM + nm + 

 mo = km -f- no, is equal to the sum of the distances of the corresponding parts of 

 the simple vibrations. For instance, if the 2 sounds be as 80 to 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 2 sounds, the more rapid the 

 beats, till at last, like the distinct puffs of air in the experiments already related, 

 they communicate the idea of a continued sound ; and this is the fundamental har- 

 monic described by Tartini. For instance, in pi. 9, fig. 34 — 37, the vibrations of 

 sounds related as 1 to 2, 4 to 5, 9 to 10, and 5 to 8, are represented; where the 

 beats, if the sounds be not taken too grave, constitute a distinct sound, which cor- 

 responds with the time elapsing between 2 successive coincidences, or near ap- 

 proaches to coincidence; for, that such a tempered interval still produces a har- 

 monic, appears from pi. 9, fig. 38. But, besides this primary harmonic, a secondary 

 note is sometimes heard, where the intermediate compound vibrations occur at a 

 certain interval, though interruptedly; for instance, in the coalescence of 2 sounds 

 related to each other as 7 to 8, 5 to 7, or 4 to 5, there is a recurrence of a similar 

 state of the joint motion, nearly at the interval of T 5 T , -V* or -?- of the whole 

 period; hence, in the concord of a major 3d, the 4th below the key note is heard 

 as distinctly as the double octave, as is seen in some degree in pi. 9, fig. 35 ; ab 

 being nearly ■§- of cd. The same sound is sometimes produced by taking the 

 minor 6th below the key note; probably because this 6th, like every other note, is 

 almost always attended by an octave, as a harmonic. If the angles of all the figures 

 resulting from the motion thus assumed be rounded off, they will approach more 

 nearly to a representation of the actual circumstances; but, as the laws by which 

 the motion of the particles of air is regulated, differ according to the different 

 origin and nature of the sound, it is impossible to adapt a demonstration to them 

 all ; however, if the particles be supposed to follow the law of the harmonic curve, 

 derived from uniform circular motion, the compound vibration will be the harmonic 

 instead of the arithmetical mean; and the secondary sound of the interrupted 

 vibrations will be more accurately formed, and more strongly marked, pi. 10, figs. 

 vol. win, 4 K 



