370 THE SPECIAL SENSES. 



ing to Helmholtz, to the beats produced when two dissonant notes 

 are sounded together. On the physical side the beat, that is, a 

 rhythmical variation in the intensity of the sound, is due to the 

 phenomenon of interference. If the rates of vibration of two notes 

 are such that at certain intervals the crests of the waves fall to- 

 gether and again the crest of one coincides with the hollow of the 

 other, the sound sensations will be periodically increased and 

 decreased. While there is no fundamental explanation for the 

 fact that a regularly varying intensity of sound is disagreeable, it 

 is a well-known phenomenon and it finds analogies in the other 

 sensations, for instance, in the very disagreeable effect of a flick- 

 ering light. When two notes are sounded together the number of 

 beats varies with the difference between the rates of vibration; 

 thus, two notes, one of 128 vibrations and the other of 136 vibra- 

 tions, give 8 beats per second. When the number of beats 

 rises to 33 per second the discord is most disagreeable; if, however, 

 the rate of interference is more rapid, the unpleasant sensation 

 becomes less perceptible, and beyond 132 per second is not notice- 

 able. When the rates of vibrations of two tones are such that 

 neither the fundamentals nor any of the overtones give beats, the 

 effect is that of harmony, the vibrations of one note strengthening 

 that of the other. The most perfect harmony is that of a note 

 sounded simultaneously with another of the same rate, ratio 1:1, 

 or with its octave, ratio 1: 2. The various intervals which in 

 music have been found to be perfectly consonant or which vary so 

 little from it as to be usable in harmonies are those whose vibra- 

 tions bear a simple ratio to each other. Thus, the octave of any 

 note has the ratio of 1:2, the double octave 1 : 4, the twelfth 1 : 3. 

 These three intervals give absolutely consonant sounds. Other 

 intervals such as the fifth, 2:3, or the major third, 4: 5 give 

 a less perfect consonance. Three or more notes bearing such rela- 

 tions to each other constitute a chord, the vibrations in the 

 major chord being, for instance, in the ratios 4:5:6, c' (128), 

 e' (160), g' (192). 



The Limits of Hearing. The rates of vibration that can be 

 perceived by the ear as musical tones lie between fairly well- 

 defined limits, although in this organ, as in the case of the eye, 

 there are individual variations, variations, indeed, which are 

 more marked in the case of the ear, since its range of appreciation 

 is larger. The lowest rate of vibration that can cause a musical 

 sensation is usually placed at 28 to 30 per second, although some 

 ears can still respond to an octave lower, about 16 per second. To 

 most ears vibrations below 30 per second are felt, if perceived at all, 

 as single pulses that stimulate the sensory nerves of the tympanic 

 membrane itself. The cochlea does not respond. It may happen, 



