THE PHYSICAL BASIS OF MUSICAL HARMONY. 343 



will uot account for the summational tones liave unfortunately some- 

 thing to unlearn — namely, that, when pure tones are used, under no 

 circumstances is a tone ever heard the frequency of which is the sum 

 of the frequencies of the two primary tones. 



The apparatus which Dr. Kcenij^ has brought over enables him to 

 demonstrate in a nmnner audible, I trust, to the whole assembly in this 

 theatre the existence of the beat tones. His lirst illustrations relate 

 to tones of primary beats, some belonging to the inferior, others to the 

 superior set, in the first period. 



He takes here the fork w^^ =2048, five octaves higher thau the great 

 uti. To excite it he may either bow it or strike it with an ivory mallet. 

 With it he will take the fork one note higher, re^ = 2304. When he took 

 the same interval with uti and rci, the number of beats was 8. The ut 

 and re of the next octave higher would have given us 10 beats, that of 

 the next 32, that of the uext 64, of the fourth octave 128, and that of 

 the fifth 256. But 250 per second is a rapidity far too great for the ear 

 to hear as separate sounds. Tf there were 250 separate impulses, they 

 would blend to give us the note m^^ = 256. They are not impulses^ but 

 beats; nevertheless, they blend. Dr. Kcenig strikes the w^g, then the 

 ree, both shrill sounds when you hear them separately; but when he 

 strikes them in quick succession one after the other, at the moment 

 when the mallet strikes the second fork you hear this clear ut^ sound- 

 ing out. I. am not going to waste your time in a disputation as to 

 whether the sound you hear is objective or subjective. It is enough 

 that you hear it, pure and unmistakable in pitch. It is the grave har- 

 monic; and the number 256, which represents its frequenc3', corre- 

 sponds to the positive remainder when you divide 2304 by 2048. 



Now let me give you a beat tone belonging to the superior set ; it also 

 will be a grave harmonic, if you so please to call it; but its frequency 

 will correspond neither to the difference nor to the sum of the frequen- 

 cies of the two primary tones. Dr. Kcpuig takes ?<?g=2048 as pre- 

 viously, and with it s'h = 3840. Let us calculate what the superior beats 

 ought to be : 2048 goes into 3840 twice, less 256. Then, 256 being the 

 negative remainder, we ought to hear from these two forks the beat 

 tone of 256 vibrations, which is ut^i, the same note as in our last experi- 

 ment. He strikes the f(^rks, and you hear the result. The beat tone, 

 which is neither a dilferential tone nor a summational tone, corre- 

 sponds to the calculated number of beats. 



If I take uIq = 2048 and sok = 3072, the two remainders both come out 

 at 1024, which is utr,. Dr. Krenig will first sound nt:, itself, sei)arately, 

 on an ut^ fork, that you may know what sound to listen for. Its sound 

 has died away ; and now he strikes nfti and sok, when at once 3'ou hear 

 ?/f5 ringing out. That sound whicli you all heard corresponds to the 

 calculated number of beats. That is enough for my present purpose. 



The next illustration is a little more complex. 1 select a case in 

 which the beat tones corresponding to the inferior and the superior 



