Sounding of tivo Notes. 513 



dissimilar order, namely the n + 2nd note of the deeper clang, 

 and the n + 1st of the higher one, whose beat-note is equal to 

 the sum of the fundamental notes. Thus, for example, in the 

 minor sixth, 5 : 8, the seventh note of 5 (35) and the sixth of 

 8 (48) gave the beat-note m, which is equal to the sum 5 + 8. 



It might perhaps appear strange that we should have re- 

 marked especially the beat-notes of overtones of two sounds 

 whose vibrations were equal to the sums of both fundamental 

 notes, while the beat-notes of many other overtones must also be 

 audible; but it must be remarked that the number of these 

 notes which are audible is by no means so great as we 

 might be disposed to assume without a closer inspection. Thus 

 the overtones of a fifth interval, up to the fifth, allow no beat- 

 note, excepting the note 5, to be heard higher than the funda- 

 mental note, which does not coincide with one of the overtones 

 of the two clangs. In the fourth, except the note 7, only the 

 beat-note 5 of the first seven overtones arising from 15 and 20 

 does not coincide with the notes which are already contained 

 in the clangs; and the ratio is the same in the other intervals. 



The beat-notes in all the above-mentioned cases are equal to 

 the difference of the notes by which they were formed, and there- 

 fore coincide with the difference-notes of these same notes ; if, 

 however, we take into consideration the great intensity that two 

 primary notes must have in order to produce only a very weak 

 difference-note, we may assume with tolerable certainty that 

 the intensity of the difference-notes produced by the over- 

 tones must be very much less than that of the beat-notes with 

 which they coincide. 



It is further to be observed that on the siren and the har- 

 monium not only are the singly produced notes accompanied 

 by overtones, but also, if two sounds are produced at the same 

 time, neither of them can be perceived except as produced by 

 a series of similar impulses succeeding one another; for if the 

 openings of two concentric circles of the siren are opened at the 

 same time, the intensity of the impulse is not double as great 

 as it would be if only one circle of holes were opened ; and 

 this diminution of intensity of the impulses at the moment of 

 their coexistence, which is only produced by the disposition of 

 the instruments employed, is sufficient alone to produce phe- 

 nomena which have nothing to do with those of the simulta- 

 neous sounding of simple notes produced by isolated sound- 

 waves (Tonempfind. Yol.iii. p. 627; Terquem, Annates de VEcole 

 JSTormale, vii. 1870). If, therefore, we wish to be sure that 

 we really have to do with combination-notes of simple pri- 

 mary notes, we must set aside both the many-voiced siren 

 and the reed-pipes, and only make use of simple tuning-forks. 



Phil. Mag. S. 5. No. 7. Suppl. Vol. 1. 2 M 



