216 KANSAS CITY REVIEW OF SCIENCE. 



tain sounds in rapid succession, superposed upon the principal sound, which 

 sounds are called its harmonics or overtones, the sound produced by the vibra- 

 tion of the whole string being termed the generator. For example the vibration 

 of the whole length of the string is followed by that of half its length which pro- 

 duces the 8th to the fundamental sound, then by one-third of its length, which 

 produces the 5th, then by the one-fourth of its length, which produces the double 

 8th or 15th, then by one-fifth, producing the major 3d to that double 8th, etc., 

 and by proceeding in this manner we learn that the major 3d and the minor 7th 

 belong to the series of harmonics, and are thus naturally prepared whenever 

 their generator is sounded ; our singing or playing this combination of notes is 

 then but a stronger articulation of sounds that are already vibrating in the air. 

 'J'he vibration-numbers of these overtones are connected by a simple law, which 

 is readily deduced from the above relation. 



Suppose the generating note to make 100 vibrations per second, two will 

 make twice as many i. e. 200; (3) being a fifth above (2) will=fX2oo=3oo 

 vibrations; (4) being a fourth above (3) will produce |X3°°=4°° ^^^ (s) fX4°° 

 =500, for (6) fX5°°=6oo. The numbers therefore come out 100, 200, 300, 

 400, 500, 600. A sound may not contain all the overtone series, but never can 

 a tone intermediate in pitch between any two consecutive members of the series 

 make its appearance. Further evidence in support of this most important propo- 

 sition may be enunciated, and in order to do this, the piano forte may be taken 

 as one example. Let tenor C be first silently pressed down, and then C an 

 octave below vigorously struck, and after a few seconds be allowed to rise again. 

 The lower note is at once extinguished, and we now hear its octave sounding 

 Irom the wires of tenor C, but if the damper fall back on these, by releasing the 

 note hitherto held down, the whole sound is at once cut off. Next, by freeing 

 the damper from the wire of ihe 12th or 5th above tenor C and pursuing a sim- 

 ilar course a like result may be obtained, and so on for the 15th and the major 

 and minor 3ds, -but they fall off rapidly in intensity, and by this important fact 

 we learn that the vibrations of any instrument are excited by resonance only 

 when vibrations of the same period are already present in the surrounding air, 

 and as the only sound directly originated in each variation of experiment was C 

 and octave below tenor C this note must have contained the notes successively 

 heard. The enharmonic is the third genus of which I have to treat, which with 

 the Greeks comprised an interval less than the semi-tone, that is to say a note 

 between E and F, higher in pitch than E, but not so high as F, and on keyed 

 instruments it signifies the distinction between two notes of the same sound but 

 having different names, as A flat and G sharp. The Persians for instance, divide 

 their scale of an octave into eighteen sounds and the eastern and southern nations 

 habitually intonate smaller intervals than semi-tones : whereas our modern scale 

 is divisible into only twelve. Again in theory raising a note to its sharp means 

 multiplying the number of vibrations per second of that note by the ratio ff, which 

 is the ratio of the number of vibrations per second of two notes at an interval of 

 a minor semi-tone. To lower it to its flat on the other hand, means to multiply - 



