1890.] on the Physical Foundation of Music. 219 



steel bar can be made to give two different notes when struck in the 

 two directions respectively parallel to the long and short sides of the 

 rectangle. A set of such tuned steel bars are here before you. 

 Taking one tuned to the note of utg = 2048, with re^ = 2304, 1 give you 

 the notes separately by striking the bar with a small steel hammer 

 when it is lying on two little bridges of wood, first on one face, then 

 on the other face. If, now, I strike it on the corner, so as to evoke 

 both notes at once, you immediately hear the strong boom of w^3 = 256, 

 the inferior beat- tone. If I take a second bar tuned to utg and s^g = 

 3840, you hear also ut^, this time the superior beat-tone. If I take a 

 bar tuned to ut^ and the 11th harmonic of ut^ (in the ratio 8 : 11), 

 you hear the two beat-tones sol^ and mi^ (in ratios of 3 and 5 respec- 

 tively) precisely as you did when two sej^arate forks were used instead 

 of one tuned bar. 



Dr. Koenig goes beyond the mere statement that beats blend to a 

 tone, and lays down the wider proposition that any series of maxima 

 and minima of sounds of any pitch, if isochronous and similar, will 

 always produce a, tone the pitch of which corresponds simply to the 

 frequency of such maxima and minima. A series of beats may be 

 regarded as such maxima and minima of sound ; but there are other 

 ways of producing the effect than by beats. Let me illustrate some 

 of these to you. 



If a shrill note, produced by a small organ-pipe or reed, be con- 

 veyed along a tube, the end of which terminates behind a rotating 

 disk pierced with large, equidistant apertures, the sound will be 

 periodically stopped and transmitted, giving rise, if the intermittences 

 are slow enough, to effects which closely resemble beats, but which, 

 if the rotation is sufficiently rapid, blend to a tone of definite pitch. 

 Dr. Koenig uses a large zinc disk with 16 holes, each about 1 inch in 

 diameter. In one set of experiments this disk was driven at 8 revolu- 

 tions per second, giving rise to 128 intermittences. The forks used 

 were all of different pitches from w^3 = 256 to w^7 = 4096. In all cases 

 there was heard the low note ut2 corresponding to 128 vibrations per 

 second. In another series of experiments, using forks ut2 and w^g, the 

 number of intermittences was varied from 128 to 256 by increasing 

 the speed, when the low note rose also from ut^ to ut^. 



From these experiments it is but a, step to the next, in which the 

 intensity of a tone is caused to vary in a periodic manner. For this 

 purpose Dr. Koenig has constructed a siren-disk (Fig. 1), pierced with 

 holes arranged at equal distances around seven concentric circles ; 

 but the sizes of the holes are made to vary periodically from small to 

 large. In each circle are 192 equidistant holes, and the number of 

 maxima in the respective circles was 12, 16, 24, 32, 48, 64, and 96. 

 On rotating this disk, and blowing from behind through a small 

 tube opposite the outermost circle, there are heard, if the rotation is 

 slow, a note corresponding to the number of holes passing per second, 

 and a beat corresponding to the number of maxima per second. With 

 more rapid rotation two notes are heard — a shrill one, and another 



