LAWS OF SONOROUS VIBRATIONS. 745 



are richer and more perfect when the strings are attacked at this point, is 

 that the harmonious overtones are full and perfect, and certain of the dis- 

 cordant overtones are suppressed. 



When two harmonious notes are produced under favorable conditions, 

 one can hear, in addition to the two sounds, a sound differing from both and 

 much lower than the lower of the two. This sound is too low for a har- 

 monic, and it has been called a resultant tone. The formation of a new 

 sound by combining two sounds of different pitch is analogous to the blend- 

 ing 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 

 notes in harmony are sounded, the resultant tone is equal to the difference 

 between the two primaries. For example, C, with 48 vibrations, and its 5th, 

 with 72 vibrations in a second, give a resultant tone equal to the difference, 

 which is 24 vibrations, and it is consequently the octave below C. These result- 

 ant tones are very feeble as compared with the primary tones, and they can 

 be heard under only the most favorable experimental conditions. In addition 

 to these sounds, Helmholtz has discovered sounds, even more feeble, which he 

 calls additional, or summation tones. The value of these is equal to the sum 

 of vibrations of the primary tones. For example, C (48) and its 5th (72) 

 would give a summation tone of 120 vibrations, or the octave of the 3d ; and 

 C (48) with its 3d (60) would give 108 vibrations, the octave of the 2d. 

 These tones can be distinguished by means of resonators. 



It is thus seen that musical sounds are complex. With single sounds 

 there is an infinite variety and number of harmonics, or overtones, and in 

 chords there are series of resultants, which are lower than the primary 

 notes, and series of additional, or summation tones, which are higher ; but 

 both the resultant and the summation tones bear exact mathematical relations 

 to the primary notes of the chord. 



Harmony. Overtones, resultant tones and summation tones of strings 

 have been discussed rather fully, for the reason that in studying the physiol- 

 ogy of audition, it will be seen that the ear is capable of recognizing single 

 sounds or successions of single sounds ; but at the same time certain com- 

 binations of sounds are appreciated and are even more agreeable than those 

 which are apparently produced by simple vibrations. Combinations of tones 

 which thus produce an agreeable impression are called harmonious. Th,ey 

 seem to become blended with each other into a complete sound of peculiar 

 quality, all of the different vibrations entering into their composition being 

 simultaneously appreciated by the ear. The blending of tones which bear 

 to each other certain mathematical relations is called harmony ; but two or 

 more tones, though each one be musical, are not necessarily harmonious. 

 The most prominent overtone, except the octave, is the 5th, with its octaves, 

 and this is called the dominant. The next is the 3d, with its octaves. The 

 other overtones are comparatively feeble. Reasoning, now, from a knowledge 

 of the relations of overtones, it might be inferred that the re-enforcement 

 of the 5th and 3d by other notes bearing similar relations to the tonic would 

 be agreeable. This is the fact, and it was ascertained empirically long before 



