1890.] on the Physical Foundation of Music. 209 



lower or fimdamental tone, are denominated upper partial /owes, thereby 

 signifying that they are higher in the scale, and that they corre- 

 spond to vibrations in parts. It is to be regretted that Professor 

 Tyndall in his Lectures on Sound, rendered von Helmholtz's 

 cAerpartialtone by the term overtones, omitting the most significant 

 half of the word. To avoid all confusion in the use of such a terra 

 I shall rather follow Koenig in speaking of these as sounds of sub- 

 division. And I must protest emphatically against calling these 

 sounds harmonics, for the simple reason that in many cases they 

 are very inharmonious. It is a matter to which I shall recur 

 hereafter. 



Eeturning to the subject of beats, the question arises : What 

 becomes of the beats when they occur so rapidly that they cease 

 to produce a discontinuous sensation upon the ear ? The view 

 which I have to put before you to-night in the name of Dr. Koenig 

 is that they blend to make a tone of their own. Earlier acousticians 

 have propounded, in accordance with this view, that the grave 

 harmonic of Tartini (a sound which corresponds to a frequency of 

 vibration which is the difference between those of the two tones pro- 

 ducing it) is due to this cause. Von Helmholtz has taken a different 

 view, denying that the beats can blend to form a sound, giving reasons 

 presently to be examined. Von Helmholtz considered that he had 

 discovered a new species of combinational tone, namely one corre- 

 sponding in frequency to the sum of the frequencies of the two tones, 

 whereas that discovered by Tartini (and before him by Sorge) 

 corresponded to their difference. Accordingly he includes under the 

 term of combinational tones the differential tone, of Tartini and the 

 summational tone which he considered himself to have discovered. 

 To the existence of such combinational tones he ascribed a very impor- 

 tant part in determining the character, harmonious or otherwise, of 

 chords; and to them also he attributes the ability of the ear to 

 discriminate between the degrees of harmoniousness possessed by 

 such intervals (fifths, sixths, &c.) as consist of two t(mes 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 primaries 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 tlie<»ry 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 depend not 

 upon the mere interval between two notes but upon the timbres also 

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

 those partials can beat with the summational tone of the primaries. 

 It becomes then of the utmost importance to ascertain the precise 

 facis about the beats and about the supposed combinational tones. 

 What the numbers of beats are in any given case : whether they do 

 or do not correspond to the alleged differential and summational 

 Vol. XIII. (No. 84.) p 



