Jan. 20, 18 76 J 



NATURE 



23: 



which gives rise to the phenomenon in question. Smith 

 contrived, with profound ability, to account for and calcu- 

 late the beat independently of the Tartini beat, or what- 

 ever it may be called, but the introduction of this by De 

 Morgan has wonderfully simplified the comprehension of 

 the thing. 



The accurate rule for finding how many beats per 

 second will result from the concord being any given 

 quantity out of tune ; or for finding how much out of 

 tune any concord is when it makes a certain number of 

 beats per second, is remarkably simple. 



Let n represent the denominator of the fraction, ex- 

 pressing, in the lowest terms, the true ratio of the concord 

 {e.g. for the fifth f , « = 2 ; for the minor sixth 4, // = 5, 

 and so on) ; then let q = the number of vibrations per 

 second either in excess or deficiency of the number which 

 would make the interval perfectly in tune ; also let jS = 

 the number of Smith's beats per second : 



then fi = nq, 



/3 

 or q = - . 

 n 



A few examples will show the easy application of these 



formulse. Take the concord of the fifth, 



When this is true the upper note should make 768 vibra- 

 tions per second to 512 of the lower one; but if it is 

 tuned by equal temperament the upper note will be 

 slightly flat, making I^T^S- Hence ^ = o"85 ; and as 

 « = 2, we shall get ^ = 17, i.e. there will be 102 Smith's 

 beats per minute. 

 Again, suppose we find the concord of the major third 



give 120 beats per minute (=2 per second), 



how much is it out of tune 

 have 



As in this case ;/ = 4, we 



t.e. the upper note vibrates half a vibration per second 

 either more or less than it ought to do. 



The number of beats per second due to imperfections 

 in the various consonances will be as follows, g being 

 always the number of vibrations by which the upper note 

 is untrue : — 



TartinVs Beat. 



In the case of the unison, the Tartini beat and the Smith 

 beat are synonymous, and this identity is the reason 

 why so many writers on beats have gone wrong ; they 

 have so often taken unison sounds as the easiest and 

 ■simplest for popular illustration, and have either assumed, 

 without further investigation, that the same principles 

 would apply for other consonances also, or have omitted 

 notice of the other consonances altogether. 



It will now be easy to understand why beats are capable 

 of such great utility in a practical point of view — namely, 

 as giving a means of measuring, with great ease and 

 positive certainty, the most delicate shades of adjustment 

 m the tuning of concordant intervals. To get, for ex- 

 ample, an octave, a fifth, or a third perfectly in tune, the 

 tuner has only to watch when the beats vanish, which he 

 can observe with the greatest ease, and which will give 

 him far more accuracy than he could possibly get by the 

 «ar alone. Whereas if he desires to adopt any fixed 



temperament, he has only to calculate the velocity of 

 beats corresponding to the minute error which should be 

 given to each concord, and the required note may be 

 tuned to its proper pitch with a precision and facility 

 which would be impossible by the unaided ear. 



The delicacy of this method of tuning would hardly be 

 believed, if it did not rest on proof beyond question. To 

 recur to our example, the difference between 95 and icx) 

 beats per minute would be appreciable by anyone with a 

 seconds watch in his hand ; and yet this would corre- 

 spond to a difference of only ^^ of a vibration per second, 

 or in pitch less than toVo of ^ semitone I 



This use of beats has been long practised by organ- 

 tuners to some extent, but its capabilities, as amplified by 

 the aid of calculation, are certainly not appreciated or 

 used as they ought to be. 



The third kind of beat is what we may call the overtone 

 beat, and was brought into prominent notice in 1862 by 

 Helmholtz, who uses it for important purposes in regard 

 to his musical theories. 



It is known that nearly all musical sounds are com- 

 pound ; they consist of a fundamental note, which is 

 usually the strongest (and by which the pitch of the note 

 is identified), but which is accompanied with several 

 fainter and higher /larmonic notes, or, as Helmholtz calls 

 them, overtones. The first of these is an octave above 

 the fundamental, the second a twelfth above, the third a 

 fifteenth, the fourth and fifth seventeenth and nineteenth 

 respectively, and there are others still higher which we 

 need not mention here. The number and strength of the 

 overtones vary for different kinds of sounds, but the five 

 lowest ones are very fcommonly present and distinguish- 

 able. Now, suppose we sound two notes, having su:h a 

 relation to each other that any of the overtones of one 

 will come within beating distance either of the other fun- 

 damental, or of any of its overtones, then a beat will be 

 set up, which is the kind of beat now in question. 



A lew examples will make this clear. The bass funda- 

 mental C shown by a minim in the following example, 



856W--tS>-gSl 



has its overtone (W:^ '\ an octave above, as 



shown by the crotchet. Now if another fundamental 

 C be sounded an octave above the former one, as the 

 second minim, and if it be a little out of tune, there will 

 be a unison beat between it and the overtone of the first 

 note. This is one of Helmholtz's beats, and the simplest 

 of them. 



Again, take an interval of a fifth ; the fundamental 

 notes being shown in minims m the following illustration, 

 and their respective overtones in crotchets : — 



(si) 3S* -<I B-394 ! 



r^ -i- ' 



Here, if the G is not a perfect concord with the C, the 

 two G's in the treble stave will be also out of tune with 

 each other, and a unison beat will ensue. This is another 

 Helmholtz beat, and a little more complex than the last, 

 as both the beating notes are overtones. 



Again, take an interval of a major third, expressing the 

 notes and such of their respective overtones as we require 

 in the same way as before, thus : — 



i 



^ 





3 



Here, if we suppose the fundamental E to vibrate 165 



