424 Dr. R. Konig on the Simultaneous 



Of all these intermediate notes I have been able to discover 

 no trace ; and the note c" — 2 v. d. (1020 v. s.) has besides com- 

 paratively so little intensity, even when its beats are most 

 distinctly audible with C, that it seems absolutely impossible 

 that it should produce any (even the very smallest) practical 

 combination-note in connexion with other notes ; and it would 

 be still more incredible that it should be the origin of a whole 

 series of combination-notes. It is therefore far more natural 

 to derive the beats of the harmonic interval, as well as those 

 of the unison, directly from the formation of the sound-waves, 

 and to consider that they arise from the periodically inter- 

 changing coincidences of the similar maxima of the notes n 

 and n f j and of the maxima which have opposite signs. 



In the beats of these harmonic intervals, as well as in those 

 of the unison, the similar maxima will either come exactly 

 together, or else with two successive vibrations of the funda- 

 mental note ; maxima of compression of the higher notes will 

 slightly precede the maximum of compression of the first vibra- 

 tion and follow the second, so that the centre of beating will 

 lie between these two ; in both cases, however, the effect upon 

 the ear will be exactly the same, as a beating is no momen- 

 tary phenomenon, but arises from the constant ebb and flow 

 of the intensity of the note. To give a clearer idea of the 

 order of vibrations in the beats of these harmonic intervals, 

 I have reduced to writing the vibrations of the interval 

 mhn and nihn+y (h=l, 2, . . . 8) by means of my well- 

 known apparatus, with which, according to the method first 

 applied by Lissajous and Desains, one of the tuning-forks 

 whose vibrations are to be calculated has attached to it a piece 

 of smoked glass which vibrates with it, and the other carries 

 the pencil which marks the figures upon this plate. If 

 we look at the common characters of these figures, we find 

 that the beats of the imperfect intervals 1 : 3, 1 : 5, 1 : 7, as 

 well as the beats of the unison, are shown by periodical 

 maxima and minima of the amplitude of the vibrations which 

 very clearly declare their direct audibleness. In the perfect 

 intervals, 1:2, 1:4, 1:6, and 1 : 8, a maximum of compres- 

 sion is constantly changing with a maximum of dilatation, as is 

 the case in ordinary sound-waves, and every entire period may 

 therefore be equally considered as a single united wave of air ; 

 and there can be nothing remarkable in such air-waves being 

 considered singly as beats, as the notes of the great organ- 

 pipe of the 32-foot octave may very easily be heard as single 

 air-beats, and we receive the impression of a series of beats 

 also if we apply the ear to the prongs of a large tuning-fork 

 which gives less than 32 v. d. 



