RESULTANT TONES. 187 



most always done. But the notes of the common major 

 chord, especially the 8th and 5th, are produced without using 

 the valves ; for the column of air in the instrument has a nat- 

 ural tendency to divide itself into its harmonic tones. 



It was ascertained in the middle of the last century, by 

 Sorge and by Tartini, that when two harmonious tones are 

 produced under favorable circumstances, we can hear, in ad- 

 dition to the two sounds, a sound differing from both, and 

 much lower than the lower of the two. This sound is too 

 ]ow for an harmonic, and has been called a resultant tone. 

 The formation of a new sound by combining two sounds of 

 different pitch is analogous to the blending of colors in optics, 

 except that the primary sounds are not lost. The laws of the 

 production of these resultant sounds are very simple. When 

 two tones in harmony are sounded, the resultant tone is equal 

 to the difference between the two primaries. For example, if 

 we sound C, with 48 vibrations, and its 5th, with 72 vibra- 

 tions in a second, the resultant tone is equal to the difference, 

 which is 24 vibrations, and is consequently the octave below 

 C ; or, if we sound C, with 48 vibrations, and its 3d, with 60, 

 we have a resultant tone two octaves below C, the number of 

 vibrations being 12. 1 These resultant tones are very feeble as 

 compared with the primary tones, and can only be heard un- 

 der the most favorable experimental conditions. In addition 

 to these sounds, Helmholtz has discovered sounds, even more 

 feeble, which he calls additional, or summation tones. 2 The 

 value of these is equal to the sum of vibrations of the primary 

 tones. For example, C 24 and its 5th, 36, would give a sum- 

 mation tone of 60 vibrations, or the octave of the 3d ; and C 

 24 with its 3d, 30, would give 54 vibrations, the octave of 

 the 2d. These tones can readily be distinguished by means 

 of the resonators already described. 3 



1 These numbers are used merely in illustration. A sound of 12 vibrations 

 does not come within the musical scale. 



2 HELMHOLTZ, Theorie physiologique de la rmisique, Paris, 1868, p. 192. 



3 See page 184. 



