1890.] on the Phijsical Foundation of Music. 211 



througli his hands. He is accustomed to tune them himself, making 

 use of the phenomenon of beats to test their accuracy. He has traced 

 out the phenomena of beats through every possible degree of pitch, even 

 beyond the ordinary limits of audibility, with a thoroughness utterly 

 impossible to surpass or to equal. Hence, when he states the results 

 of his experience, it is idle to contest the facts gathered on such 

 an unique basis. 



The results of Dr. Koenig's observations on beats are easily 

 stated. He has observed primary beats, as well as beats of secondary 

 and higher orders, from the interference of two simple tones simul- 

 taneously sounded. When two simple tones interfere, the j^rimary 

 beats always belong to one or other of two sets, called an inferior and 

 a superior set, corresponding respectively in number to the two 

 remainders, positive and negative, to be found by dividing the 

 frequency of the higher tone by that of the lower. 



This mode of stating the facts is a little strange to those trained 

 in English modes of expressing arithmetical calculations : but an 

 example or two will make it plain. Let tbere be as the two primary 

 sounds two low tones having the respective frequencies of 40 

 vibrations and 74 vibrations. What are the two remainders, positive 

 and negative, which result from dividing the higher number, 74 by the 

 lower number 40 ? Our English way of stating it is to say that 40 

 goes into 74 once, and leaves over a (positive) remainder of 34. But 

 it is equally correct to say that 40 goes into 74 twice, all but 6 : or 

 that there is a negative remainder of 6. Well, Dr. Koenig finds that 

 when these two tuning-forks are tried, the ear can distinguish two 

 sets of beats, one rapid, at 34 per second, and one slow, at 6 per 

 second. 



Again, if the forks chosen are of frequencies 100 and 512, we 

 may calculate thus : 100 goes into 512 five times, plus 12 ; or 100 

 goes into 512 six times, minus 88. In this actual case the 12 beats 

 belonging to the inferior set would be well heard: the 88 beats 

 belonging to the superior set would probably be almost indistin- 

 guishable. As a rule the inferior beat is heard best when its 

 number is less than half the frequency of the lower primary, whilst, 

 when its number is greater^ the superior beat is then better heard. 

 Dr. Koenig has never been able toJaear any primary beat which did 

 not fall within the arithmetical rule which I have previously stated. 



I will now illustrate to you the beats, inferior and superior, as 

 produced by these two massive tuning-forks, each weighing about 50 

 pounds, and each provided with a large resonating cavity consisting 

 of a metal cylinder with an adjustable piston. One of them is tuned 

 to the note ut^ = 64. The other also sounds ut^ ; but by sliding 

 down its prongs the adjustable weights of gun-metal, and screwing in 

 the piston, I can raise its pitch a w^hole tone to re^ = 72. I excite 

 them with the 'cello bow, first separately, that you may hear their 

 individual tones, then together. At once you hear an intolerable 

 beating — the beats coming 8 per second. This is the inferior beat, 



p 2 



