512 Dr. R. Konig on the Simultaneous 



from the ear, the four beats with the note 7 = 448 v. s. were 

 audible. In the same way I was able, by the clang of the 

 notes d and d" (4 : 9), to prove through the beats the ex- 

 istence of the very soft note e' (5) with the help of a tuning- 

 fork of 648 v. b., and by the clang of the notes c' and /"'• 

 (3 : 8) that of a soft a! (5), with a fork of 860 v. s. 



As regards the observation of the summation-notes, Helm- 

 holtz has remarked that " these are only to be heard under 

 peculiarly favourable circumstances — for instance, on the har- 

 monium and on the many-voiced siren (Tonempjind. vol. iii. 

 p. 244). But even if it is really sometimes possible, on sound- 

 ing simultaneously two clangs on a siren or on a reed instru- 

 ment, to distinguish notes of which the pitch is equal to the 

 sum of the primary fundamental notes of both sounds, still this 

 is not sufficient to prove the existence of the summation-notes, 

 as neither sirens nor reed instruments produce simple notes, 

 but sounds which are rich in overtones ; and a slight examina- 

 tion shows that in consequence of this the mere beat-notes 

 which must be produced by the overtones are sufficient to 

 prove the existence of notes whose vibrations are equal to the 

 sum of the vibrations of the fundamental notes of these sounds. 



Two sounds in the interval of a fifth contain these two 

 series of notes, 



2, 4, 6, 8, 10, 



3, 6, 9, 12, 15 ; 



and the fifth notes of both sounds (10 and 15) produce a beat- 

 note m= m' = 5, which is equal to the sum 2 + 3 of the roots. 

 In the fourth, 3 : 4, Ave have the two series of notes, 



3, 6, 9, 12, 15, 18, 21, 



4, 8, 12, 16, 20, 24, 28; 



and it is here the seventh notes of these sounds which produce 

 a beat-note that is equal to the sum 3 + 4. In the third, 4 : 5, 

 we have the overtones 36 and 45, from which a beat-note 

 must ensue which will equal the sum 4 + 5 ; and thus in every 

 ratio of the form n : n + 1 the beat-note of the 2n + 1st notes 

 of both clangs is equal to the sum of the fundamental notes. 



In intervals of the form n : n + 2 there are also two notes of 

 the same order, namely the n + 1st of both clangs, whose beat- 

 note is equal to the sum of the fundamental notes. Thus the 

 sixth, 3 : 5, gives the notes 



3, 6, 9, 12, 



5, 10, 15, 20, 



where the beat-note m is produced by 12 and 20 = 8 = 5 + 3. 

 Lastly, in intervals of the form n : n + 3 there are notes of 



