214 Professor Sllvanus P. Thompson [June 13, 



sounds being dependent on the manner in wliicli the card is pressed 

 against the wheel — that is to say, on the nature of the individual 

 impulses themselves. The opponents of the view that beats blend 

 into a tone, state plainly enough that, in their opinion, a mere 

 succession of alternate sounds and silences cannot blend into a tone 

 different from that of the beating tone. Having said that the beats 

 cannot blend, they then add that they do not blend ; for, say they, 

 the combinational tones are a purely subjective phenomenon. Lastly, 

 they say that if even the beats blend they will not so exi)lain the ex- 

 istence of combinational tones, because the combinational tones have 

 frequencies Avhich do not correspond to the number of the beats. 



In the teeth of all these views and ojunions. Dr. Koenig — 

 without dogmatising as to how or why it is — emphatically affirms 

 that beats do jDroduce heat-tones; and he has pursued the matter 

 down to a point that leaves no room for doubting the general truth 

 of the fact. The alleged discrepancy between the frequency of the 

 observed combinational tones and that of the beats disappears when 

 closely scrutinised. Those who count the beats by merely taking 

 the difference between the frequencies of the two primary tones, 

 instead of calculating the two remainders, will assuredly find that 

 their numbers do not agree in pitch wdth the actual sounds heard. 

 But that is the fault of their miscalculation. Those who use harmo- 

 nium reeds or polyphonic sirens instead of tuning-forks to produce 

 their primary tones must not expect from such impure sources to 

 reproduce the effects to be obtained from pure tones. And those 

 who say that the beats calculated truly from the two remainders will 

 not account for the summational tones, have unfortunately something 

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

 stances is a tone ever heard, the frequency of which is the sum of the 

 frequencies of the two primary tones. 



The apparatus before you enables me to demonstrate, in a manner 

 audible, I trust, to the whole assembly in this theatre, the existence 

 of the beat-tones. My first illustrations relate to tones of primary 

 beats, some belonging to the inferior, others to the superior set, in 

 the first period. 



I take here the fork w^g = 2048, five octaves higher than the 

 great ut^. To excite it, I may either bow it or strike it with this 

 ivory mallet. With it I will take the fork one note higher, rcg = 2304. 

 When we took the same interval with ut^ and re^, the number of beats 

 was 8. The ut and re of the next octave higher would have given us 

 16 beats, that of the next 32, that of the next 64, of the fourth octave 

 128, and that of the fifth octave higher 256. But 256 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 ut^ = 256. They are not impulses, but heats: nevertheless, 

 they blend. I strike the ut^, then the rCg, both shrill sounds when 

 you hear them separately ; but when I strike them in quick suc- 

 cession one after the other, at the moment when the mallet strikes 



