THE PHYSICAL BASIS OF MUSICAL HARMONY. 343 



will uot account for tbe siiramational tones have unfortunately some- 

 tljing 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. K(Bni«j^ has brought over enables him to 

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

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

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

 superior set, in the first period. 



He takes here the fork w^u =1304:8, five octaves higher than 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 uty and rei, the number of beats was 8. The wt 

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

 the next 32, that of the next 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. If there were 256 separate impulses, they 

 would blend to give us the note M^t = 25(5. They are not impulses, but 

 beats; nevertheless, they blend. Dr. Kcenig strikes the utg, then the 

 reg, 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 utj sound- 

 ing out. I am uot 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 aud unmistakable in pitch. It is the grave har- 

 monic; and tiie number 25(5, which represents its frequency, 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. Kcenig takes iif^ = 20iS as pre- 

 viously, and with it sio = 3840. Let us calculate what the superior beats 

 ought to be : 2048 goes into 3840 twice, less 25(5. Then, 256 being the 

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

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

 ment. He strikes the fnks, and you hear the result. The beat tone, 

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

 sponds to the calculated number of beats. 



If I take utc = 2048 and sol,-, = 3072, the two renuunders buth come out 

 at 1024, which is ut:,. Dr. K<enig will first sound nf-, itself, separately, 

 on an w/5 fork, that you may know what sound to listen for. Its sound 

 has died away ; and now ho strikes iit^ and .w/,j, when at once you hear 

 lit:, ringing out. That souiul wliicli you all heard correspoiuls to the 

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



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

 which the beat tones corresi)onding to the inferior and the superior 



