CHAMBERS'S INFORMATION FOR THE PEOPLE. 



in one or in all of these respects that is, in the 

 number, position, and relative loudness of their 

 constituent partials ; so that the number of differ- 

 ent clangs which can exist having the same funda- 

 mental tone is almost indefinitely great. Each of 

 these clangs gives rise to a distinct quality of 

 tone. It may be added here, that not only can 

 clangs be resolved by suitable apparatus into their 

 constituent elements, but they can also be built 

 up. A tuning-fork emits a sound which is very 

 nearly simple; and by sounding together a 

 number of tuning-forks corresponding to a series 

 of partial tones, the effect produced is that of 

 one tone of great richness, and not of a number 

 of different tones forming a chord. The same 

 effect, but in a less remarkable degree, can be 

 observed by striking all the notes in fig. 2 simul- 

 taneously upon a piano. 



It is not difficult to see the connection between 

 the clang and the air-wave which conveys that 

 clang to our ears. We have shewn how the pitch 

 of the sound is determined by the length of its air- 

 wave, and the loudness of the sound by its ampli- 

 tude. There is now left the third element in each, 

 the quality of the sound and the wave-form, so 

 that the effect of each variety of clang must be to 

 produce air-waves of different form. The form of 

 the wave produced by any particular clang is 

 determined by the combined action of the waves 

 of all the different tones of which it consists ; and 

 this action, for any given clang, will produce a 

 number of different wave-forms, according to the 

 relative position of the crests and troughs of the 

 simple waves when first they come in contact. 

 This might at first sight seem to disprove what 

 we have been saying, were it not for the two re- 

 markable facts ist, that although one clang 

 gives rise to many wave-forms, yet each wave- 

 form can only arise from one particular clang ; 

 and 2d, that the ear possesses the faculty of 

 analysing the wave-form, or breaking it up into 

 its different elements, and thus is never misled as 

 to what clang it belongs to. The conclusion of 

 the whole matter is, therefore, that the form of 

 an air-wave determines its quality, but that the 

 quality does not, conversely, indicate the form. 



Consonance and Dissonance Beats. 



We have now seen what are the physical con- 

 ditions which accompany a musical sound, and 

 make it appear to us high or low, loud or soft, full 

 and round, or hard and wiry. It will be necessary 

 to follow the subject a little farther, in order to 

 understand the physical conditions accompanying 

 a combination of sounds which translate them- 

 selves to us as consonance and dissonance, quali- 

 ties which, as we have said, have been often, by 

 theorists, erroneously attributed to the vibration 

 fraction of the intervals. When two simple tones 

 are sounded together, they give rise to a com- 

 pound air-wave, just in the same way as two of the 

 partial tones of a clang. An investigation of the 

 resultant air-waves (which can very easily be 

 made, but for which we have not space here) 

 shews that they go through a regular series 

 of changes in amplitude becoming alternately 

 larger and smaller. The sound produced, there- 

 fore, must necessarily become louder and softer 

 correspondingly, with the same regularity 'its 

 greatest intensity exceeding, and its least intensity 

 falling short of, that of the louder of the two 



692 



tones.' Each recurrence of maximum loudness 

 is called a beat; and the number of beats per 

 second due to the sounding together of two simple 

 tones, is equal to the difference between the num- 

 ber of vibrations they respectively make in the 

 same period. Thus, if one of the tones vibrates 

 512 times in a second, and the other 576, the num- 

 ber of beats produced when they are sounded 

 together will be 576 - 512, or 64. Suppose, now, 

 that we have two tubes capable of producing the 

 same simple musical tone when air is blown 

 through them, and one of which can be shortened 

 at pleasure, so as to produce higher tones. If air 

 be blown through the tubes when they are of the 

 same length, we hear, of course, one musical tone 

 only, of perfectly uniform loudness. If, now, one of 

 them be gradually shortened, we notice at once 

 the variation of loudness which we have called a 

 beat. At first, the beats are very slow, and scarcely 

 unpleasant, but they become rapidly faster, till 

 we can no longer count them. We still recognise 

 their existence most distinctly, however, and they 

 produce upon the ear an exceedingly harsh and 

 disagreeable effect. They become faster and 

 faster as the tube is shortened, until a point is 

 reached at which the disagreeable effect entirely 

 ceases, and we should no longer recognise their 

 existence, were it not that we feel their effects in 

 the altered (but now pleasant) effect of the sounds 

 produced, as compared with the tone with which 

 we started. If we translate the effect of the 

 whole process into the universally understood 

 musical language, we say that we began with 

 notes, in unison, and that the disagreeable sounds 

 were dissonant, and the agreeable ones consonant. 

 The experiment, however, shews that these terms 

 denote differences of degree only, and not of kind. 

 We gather from it that the recurring beats are alike 

 the cause of dissonance and consonance, and that 

 when they are slow they give us the former feel- 

 ing, and when extremely fast, the latter. On ex- 

 amining separately the notes produced by the two 

 tubes, we find that the most unpleasant effect on 

 our ears occurred when the interval between them 

 was about a semitone, and that the effect ceased 

 to be unpleasant when that interval had increased 

 to a minor third. In order, therefore, that two- 

 notes may be dissonant, the interval between them 

 must be less than a minor third. This interval 

 Mr Sedley Taylor has called the 'beating dis- 

 tance,' and in his language we may therefore say 

 that ' dissonance can arise directly between two 

 simple tones only when they are within beating 

 distance of one another!* But although direct dis- 

 sonance can only arise between simple tones 

 within this limit, the case is different in regard to 

 clangs, to which category it will be remembered 

 belong almost all the musical sounds with which 

 we are acquainted. Here dissonance occurs 

 whenever any of the partial tones of the one clang 

 come within beating distance of those of the 

 other. This will be found to occur, to a certain 

 extent, with every interval except the octave. The 



* As the rapidity of the beats depends upon the absolute diffe 

 ence between the vibration numbers of the two tones, and as 

 sense of dissonance depends upon the absolute rapidity of 

 beats, it follows that the ' beating distance ' must vary very much 

 with the pitch of the notes, being greater for low notes, and less 

 for high ones. That this is really the case, any one may convince 

 himself by noticing the exceedingly coarse and disagreeable 

 effect of even a fourth or fifth in the lowest octave of a pianoforte, 

 and the comparatively smooth effect of a second in its highest 

 octave. 



