THEOR Y OF DISSONANCE. 1187 



dissonance will be most disagreeable, even with compound tones having 

 few partials, such as those of the flute, still more with those having 

 numerous partials, like the tones of reeds, as in a harmonium. 1 Helm- 

 holtz has applied this view to an explanation of the satisfying character 

 of certain musical intervals, as contrasted with the discordant character 

 of others. Thus, unison ^, minor third 4, major third -, fourth f , fifth f , 

 minor sixth -f-, major sixth -, and octave -f- , are all concords ; while a 

 second f, minor seventh 1 /, and major seventh y, are discords. The 

 smoothest interval is the octave, next the fifth, then the fourth, major 

 third, and so on. 



There can be no doubt that this explanation satisfies the case, with intervals 

 sounded on instruments giving forth compound tones, but it does not appear 

 to do so when an interval is composed of sounds produced by instruments like 

 large stopped organ pipes, in which there are no over-tones. By careful bow- 

 ing, few partials may be produced by well-constructed tuning-forks, and these 

 soon die away, leaving the pure proper tone of the fork. In such cases the 

 interval of a minor seventh should be as concordant as that of a fifth, but it is 

 not so. On the theory of the cochlea already described, there is no difficulty 

 in explaining these facts, because, if it possesses analytic powers by sympathetic 

 resonance, it must respond to the prime tones and partials of both of the notes 

 forming the interval. .The only thing unexplained is, why the sensation 

 should be disagreeable when two portions of the membrana basilaris sufficiently 

 near are thrown into vibration. The sensation of a beat is not due to the 

 stimulation of one part of the membrana, but the peculiar wave of pressure of 

 two tones, causing a beat, is analysed. While a beat is sounding, listen with 

 an appropriate resonator to one of the constituent tones of the beat, and it will 

 be at once heard. For some unexplained reason, however, if two nerves 

 sufficiently near are simultaneously stimulated, or if they are stimulated in the 

 intermittent manner peculiar to beats, the sensation is disagreeable. Helmholtz 

 compares it with that caused by a flickering light on the eye. Something 

 similar I have found to be produced by simultaneously stimulating the skin, 

 or margin of the lip, by bristles attached to tuning-forks giving both beats. 

 If the frequency of the forks is great, the sensation is that of a most disagree- 

 able tickling. It may be that the instinctive effort at analysis of tones close 

 in pitch causes the disagreeable sensation. 



Combination tones. The law that the oscillations of elastic bodies, 

 and of the air, produced by several sources of sound, are the sum of 

 the individual motions from each source, only holds good when the 

 bodies are of infinitely small dimensions, and when the vibrations 

 themselves are also infinitely small. If vibratile bodies are not 

 infinitely small, then other phenomena may be observed. Helmholtz 2 

 considered these phenomena with reference to the theory of dissonance 

 and the action of the ear. If two tuning-forks, representing a fifth, 

 are properly bowed, sounding the fork of lower pitch first and that 

 of higher pitch afterwards, we may hear a weak lower tone, the pitch of 

 which is an octave below that of the first fork. This is known 

 as a combination tone. Such tones are divided into two classes 

 differential tones, in which the frequency is the difference of the fre- 

 quencies of the generating tones ; and summational tones, having a 

 frequency which is the sum of those of the tones producing them. 



1 For numerous examples, see Sedley Taylor, "Sound and Music," London, 1873, p. 166, 

 el seq.; also, Helmholtz, oj). cit., p. 272. 

 * Op. cit., p. 229. 



