832 SPECIAL SENSES. 



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

 these sounds, Helmholtz has discovered sounds, even more feeble, which he calls addi- 

 tional, or summation tones. The value of these is equal to the sum of vibrations of the 

 primary tones. For example, (24) and its 5th (36) would give a summation tone of 60 

 vibrations, or the octave of the 3d ; and (24) with its 3d (30) would give 54 vibrations, 

 the octave of the 2d. These tones can readily be distinguished by means of the reso- 

 nators already described. 



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

 have an infinite variety and number of harmonics, or overtones, and in chords, which 

 will be treated of more fully under the head of harmony, we have a series of resultants, 

 which are lower than the primary tones, and a series of additional, or summation tones, 

 which are higher ; but both the resultant and the summation tones bear an exact mathe- 

 matical relation to the primary tones of the chord. 



Harmony. We have discussed the overtones, resultant tones, and summation tones 

 of strings rather fully, for the reason that, in the physiology of audition, we shall see that 

 the ear is capable of recognizing single sounds or successions of single sounds ; but, at 

 the same time, certain combinations of sounds are appreciated and are even more agree- 

 able than those which are apparently produced by simple vibrations. Combinations of 

 tones which thus produce an agreeable impression are called harmonious. They seem to 

 become blended with each other into a complete sound of peculiar quality, all of the dif- 

 ferent vibrations entering into their composition being simultaneously appreciated by the 

 ear. From what we have learned of overtones, it is evident that few musical sounds are 

 really simple, and that those which are simple are wanting in richness, while they are per- 

 fectly pure. The blending of tones which bear to each other a certain mathematical rela- 

 tion is called harmony ; but two or more tones, though each one be musical, are not neces- 

 sarily 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 our knowledge of the 

 relations of overtones, we might infer that the reenforcement 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 the pleasing impression produced by such com- 

 binations was explained mathematically. We do not propose to enter into a full discus- 

 sion of the laws of harmony, but a knowledge of certain of these laws is essential to the 

 comprehension of the physiology of audition. These are very simple, now that we have 

 analyzed the sound of a single vibrating body. 



It is a law in music, that the more simple the ratio between the number of vibrations 

 in two sounds, the more perfect is the harmony. The simplest relation, of course, is 

 1 : 1, when the two sounds are said to be in unison. The next in order is 1 : 2. If we 

 sound C and its 8th, we have, for example, 24 vibrations of one to 48 of the other. 

 These sounds can produce no discord, because the waves never interfere with each other, 

 and the two sounds can be prolonged indefinitely, always maintaining the same relations. 

 The combined impression is therefore continuous. The next in order is the 1st and 5th, 

 their relations being 2:3. In other words, with the 1st and 5th, for two waves of the 

 1st we have three waves of the 5th. The two sounds may thus progress indefinitely, 

 for the waves coincide for every second wave of the 1st and every third wave of the 5th. 

 The next in order, if we sound at the same time the 1st, 5th, and 8th, is the 3d. The 3d 

 of C has the 8th of C for its 5th, and the 5th of C for its minor 3d. The 1st, 3d, 5th, 

 and 8th form the common major chord ; and the waves of each tone blend with each 

 other at such short intervals of time that the ear experiences a continuous impression, 

 and no discord is appreciated. This explanation of the common major chord illustrates 

 the law that, the smaller the ratio of vibration between different tones, the more perfect 

 is their harmony. Sounded with the 1st, the 4th is more harmonious than the 3d ; but 



