THE PHYSICAL BASIS OF MUSICAL HARMONY. 347 



take their origin from some one organ or portion of vibratile matter 

 vibrating in a single but more complex mode. To this objection an ex- 

 perimental answer has been returned by Dr. Koenig in the following 

 way. He takes a prismatic bar of steel, about 9 inches in length, and 

 files it to a rectangular section, so as to give, when it is struck at the 

 middle of a face to evoke transversal vibrations, a sound of some well- 

 defined pitch. By carefully adjusting the sides of the rectangular sec- 

 tion in proper proportions, the same steel bar can be made to give two 

 difl'erent notes when struck in 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 «f6=2,048, with 

 re6=2,394, Dr. Koenig will 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, he strikes it 

 on the corner, so as to evoke both notes at once, you immediately hear 

 the strong boom of 1^^3=256, the inferior beat-tone. If Dr. Koenig takes 

 a second bar tuned to wfgand siG=3,840, you hear also ut^, this time the 

 superior beat-tone. If he takes a bar tuned to ufe and the 11th har- 

 monic of ut3 (in the ra tio 8 : 11) you hear the two beat-tones soU and mis 

 (in ratios of 3 and 5 respectively), precisely as you did when two sepa- 

 rate 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 al- 

 ways produce a tone the i)itch of which corresponds simply to the fre- 

 quency of such maxima and minima. A series of beats may be regarded 

 us such maxima and minima of sound ; but there are other ways of pro- 

 ducing the effect than by beats. Dr. Koenig will now illustrate some 

 of these to you. 



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

 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 closely resembling beats, but which, if the rotation is suffi- 

 ciently rapid, blend to a tone of definite pitch. Dr. Koenig uses a large 

 zinc disk with IG holes, each about 1 inch in diameter. In one set of 

 experiments this disk was driven at 8 revolutions per second, giving 

 rise to 128 intermittences. The forks used were of all different pitches 

 from ut^ = 25G to ^lt^ = 4096. In all cases there was heard the low note 

 ^iti corresponding to 128 vibrations per second. In another series of 

 experiments, using forks ufz and ut^, the number of intermittences was 

 varied from 128 to 25G by increasing the speed, when the low note rose 

 also from ut^ to iit^. 



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

 tensity of a tone is caused to vary in a periodic manner. For this pur- 



