THR PHYSICAL BASIS OF MUSICAL HARMONY. 339 



ear to discriminate between the degrees of harmouiousness possessed 

 hy such intervals (fifths, sixths, etc.) as consist of two tones too widely 

 apart on the scale to give beats of a discontinuous character. He also 

 considers that such combinational tones are chiefly effective in pro- 

 ducing beats, the summational tones of the jirimaries beating with 

 their upper partial tones ; and that this is the way in which they make 

 an interval more or less harmonious. 



The whole fabric of the theory of harmony as laid down by von 

 Helmholtz is thus seen to repose upon the presence or absence of 

 beats; and the beats themselves are in turn made to depeiul, not upon 

 the mere interval between two notes, but upon tlie timbres also of those 

 notes, as to what upper partials they contain, and whether those par- 

 tials can beat with the summational tone of the primaries. It becomes, 

 then, of the utmost importance to ascertain the precise facts about the 

 beats and about the supposed combinational tones. What the numbers 

 of beats are in any given case, whetlier they do or do not correspond 

 to the alleged differential and sumnnitional tones, these are vital to 

 the theory of harmony. Equally vital is it to know what the timbres 

 of sounds are, and whether they can be accurately or adequately repre- 

 sented by the sum of a set of pure harmonics corresponding to the 

 terms of a Fourier series. 



In investigating beats and combinational tones, Dr. Koenig deemed 

 it of the highest importance to work with instruments producing the 

 purest tones; not with harmonium reeds or with polyphonic sirens, 

 the tones of which are avowedly complex in timbre, but with massive 

 steel tuning forks, the pendular movements of which are of the sim- 

 plest possible character. Massive tuning-forks properly excited by 

 bowing with a violoncello bow, or, in the case of those of high pitch, 

 by striking them with an ivory mallet, emit tones remarkably free from 

 all sounds of subdivision, and of so truly pendular a character (uidess 

 over-excited) that none of the harmonics corresponding to the members 

 of a Fourier series can be detected. No living soul has had a tithe of 

 the experience of Dr. Kcenig in the handling of tuning forks. Tens of 

 thousands of them have passed through 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 phenomenon of beats through 

 every i)ossible degree of pitch, even beyond the ordinary limits of audi- 

 bility, 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 a 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 inter- 

 ference of two simple tones simultaneously sounded. 



When two simple tones interfere, the ])rimary beats always belong 

 to one or other of two sets, called an inferior and a superior set, cor- 

 responding respectively in number to the two remainders, positive and 



