RESPECTING SOUND AND LIGHT. 



5^5 



the beats, if the sounds are not taken too 

 grave, constitute a distinct sound, which 

 corresponds with the time elapsing between 

 two successive coincidences, or near ap- 

 proaches to coincidence ; for, that such a 

 tempered interval still produces a harmonic, 

 appears from Plate 4. Fig. 41. But, besides 

 tliis primary harmonic, a secondary note is 

 sometimes heard, where the intermediate 

 compound vibrations occur at a certain in- 

 terval, though interruptedly ; for instance, 

 in the coalescence of two sounds related to 

 each other, as 4 : 5, there is a recurrence of 

 a similar state of the joint motion, nearly at 

 the interval of y of the whole period, three of 

 the joint vibrations occupying ^l and leaving 

 ■^g : hence, in the concord of a major third, 

 the fourth below the key note is heard as dis- 

 tinctly as the double octave, as is seen in some 

 degree in Plate 4. Fig. 38; AB being nearly 

 two thirds of CD. If the angles of all the 

 figures resulting from the motion thus assumed 

 be rounded off, they will approach more near- 

 ly to a representation of the actual circum- 

 stances ; but, as the laws, by which the mo- 

 tion of the particles of air is rvigulated, differ 

 according to the different origin and nature 

 of the sound, it is impossible to adapt a de- 

 monstration to them all : if, however, the 

 particles be supposed to follow the law of the 

 harmonic curve, derived from uniform cir- 

 cular motion, the compound vibration will 

 be the harmonic instead of the arithmetical 

 mean ; and the secondary sound of the inter- 

 rupted vibrations will be more accurately 

 formed, and more strongly marked: thus, in 

 the concord 4 : 5, instead of -^ of the whole 

 period, the compound vibration will become 

 J, and three such vibrations, occupying |, 

 will leave exactly y. (Plate 5. Fig. 44, 45.) 

 The demonstration is deduciblefrom the pro- 



VOL. II. 



perties of the circle ; and in the same man- 

 ner if the sounds are related as 7 C 8, or as 

 5 : 7, each compound vibration will occupy 

 JLj or -i\^ ; and deducting 5 or 4 vibrations 

 from the whole period, we shall have a re- 

 mainder of y. This explanation is satisfac- 

 tory enough with regard to the concord of a 

 major third ; but the same harmonic is some- 

 times produced by taking the minor sixth 

 below the key note: in this case it might be 

 supposed that the superior octave, which 

 usually accompanies every sound as a se- 

 condary note, supplies the place of the ma- 

 jor third; but I have found that the experi- 

 ment succeeds even with stopped pipes, 

 which produce no octaves as harmonics. We 

 must therefore necessarily suppose that in 

 this case, if not in the former, the sound in 

 question is simply produced as a grave har- 

 monic, by the combination of some of the 

 acute harmonics, which always accompa- 

 ny the primitive notes. It is remarkable> 

 that the law, by which the motion of the 

 particles is governed, is capable of some 

 singular alterations by a combination of 

 vibrations. If we add to a given sound 

 other similar sounds, related to it in fre- 

 quency as the series of odd numbers, and 

 in strength inversely in the same ratios, we 

 may convert the right lines indicating a uni- 

 form motion very nearly into figures of sines, 

 and the figures of sines into right lines, as in 

 Plate 4. Fig. 42, 43. 



XII. Of the Frequency of Vibrations consti- 

 tuting a given Note. 



The number of vibrations, performed by a 

 given sound in a second, has been variously 

 ascertained; firet, by Sauveur, by a very in- 

 genious inference from the beats of two 

 sounds; and since, by the same observer and 

 4 A 



