544 HISTORY OF SCIENCE. 



Now, the vibration ratios of an octave of the natural scale are 

 24:27:30:32:36:40:45:48, 



and by doubling these numbers we obtain the figures representing the 

 vibrations for the next higher octave, and so on. It will be seen that the 

 partial tones of the string up to the 6th, form harmonious combinations 

 with the i st ; for they are octaves, or octaves of the third and of the 

 fifth of the fundamental tone. The 7th partial does not belong to the 

 scale, and forms a discord. Now, the makers of pianofortes, guided 

 only by practice, in order to obtain a fine tone, have long made the 

 hammers of their instruments to strike the strings at a particular part 

 of their length, where the vibrations that would give the yth partial 

 are annulled. This circumstance may serve to illustrate the scientific 

 importance of such discoveries as those of Helmholtz. We may say 

 of it what Tyndall (" Sound," Lect. vii.) says, speaking more generally 

 of the progress of music : " The musicians engaged in this work knew 

 nothing of the physical facts and principles involved in their efforts ; 

 they knew no more about them than the inventors of gunpowder knew 

 about the law of atomic proportions. They tried and tried until they 

 obtained a satisfactory result ; and now, when the scientific mind is 

 brought to bear upon the subject, order is seen rising through the con- 

 fusion, and the results of pure empiricism are found to be in harmony 

 with natural law." 



When it is understood that every sounding body has its own pecu- 

 liar system of partial tones, which may vary according to certain laws 

 with the mode in which the vibrations are excited, the cause of the 

 variety in the qualities of the composite resulting tones no longer 

 remains a mystery. The case of strings is mentioned here as a simple 

 instance. With other sounding bodies more complicated laws appear. 

 Thus, for example, the ist upper partial tone of a tuning-fork has a 

 vibration rate 6 that of the fundamental tone. Helmholtz has shown 

 also that the various vowel sounds depend upon combinations of dif- 

 ferent partial tones, and that for each vowel there is a particular part 

 of the scale where the characteristics of the sound are best brought 

 out. The space at our command will not allow us to do more than 

 mention Helmholtz's examination of differential tones, which were first 

 discovered by a German organist named SORGE in 1740, and after- 

 wards brought into notice by Tartini. These are the sounds some- 

 times called grave harmonics, which are heard when two musical tones 

 of different pitch are sounded loudly and continuously, and they cor- 

 respond with the difference of the vibration rates of the two notes. 

 A third kind of tones were discovered by Helmholtz himself, namely, 

 those with a vibration rate equal to the sum of the rates of the gene- 

 rating tones, and were called by him summational tones. 



There are two quite recent inventions which in the most remarkable 



