1890.] on the Physical Foundation of Music. 223 



of motion and will be heard as series of beats ; tbe rapidity with 

 which, they succeed one another being proportional to the velocity 

 of the movement of the fork, the fork I am using is ut^, which gives 

 well-marked beats, slow when I move my arm slowly, quick when 

 I move it quickly. There are limits to the speed at which the 

 human arm can be moved, and the quickest speed that I can give to 

 mine fails to make the beats blend to a tone. But if I take soZ^, 

 vibrating 1^ times as fast, and strike it, and move it away from the 

 wall with the fastest sj^eed that my arm will permit, the beats blend 

 into a short low growl, a non-uniform tone of low pitch, but still 

 having true continuity. 



This first portion of my discourse may then be summarised by 

 saying that in all circumstances where beats, either natural or 

 artificial, can be produced with sufficient rapidity, they blend to 

 form a beat-tone of a pitch corresj)onding to their frequency. 



I now pass to the further part of the researches of Dr. Koenig 

 which relates to the timbre of sounds. Prior to the researches of 

 Dr. Kcenig, it had been supposed that in the reception by the ear 

 of sounds of complex timbre the ear took no account of, and indeed 

 was incapable of perceiving, any differences in phase in the 

 constituent partial tones. For example, in the case of a note and 

 its octave sounded together, it was supposed and believed that the 

 sensation in the ear, when the difference in phase of the two com- 

 ponents was equivalent to one-half of the more rapid wave, was the 

 same as when that difference of phase was one-quarter, or three- 

 quarters, or zero. I had myself, in the year 1876, shown reason for 

 holding that the ear does nevertheless take cognizance of such 

 differences of phase. Moreover, the peculiar rolling or revolving 

 effect to be noticed in slow beats is a proof that the ear perceives 

 some difference due to difference of phase. Dr. Koenig is, however, 

 the first to put this matter on a distinct basis of observation. That 

 such differences of phase occur in the tones of musical instruments 

 is certain : they arise inevitably in every case where the sounds of 

 subdivision are such that they do not agree rigidly with the 

 theoretical harmonics. Fig. 5 depicts a graphic record taken by 

 Dr. Koenig from a vibrating steel wire, in which a note and its 

 octave had been simultaneously excited. The two sounds were 

 scarcely preceptibly different from their true interval, but the higher 

 note was just sufficiently sharper than the true harmonic octave to 

 gain about one wave in 180. The graphic trace has in Fig. 5 been 

 split up into 5 pieces to facilitate insertion in the text. It will be 

 seen that as the phase gradually changes, the form of the waves 

 undergoes a slow change from wave to wave. 



Now, it is usually assumed that in the vibrations of symmetrical 

 systems, such as stretched cords and open columns of air, the sounds 

 of subdivision agree with the theoretical harmonics. For examj)le, 

 it is assumed that when a stretched string breaks up into a nodal 

 vibration of four parts, each of a quarter its length, the vibration is 



