234 



NATURE 



\yan, 20, 1876 



instead of 160, i.e. five vibrations too sharp, the two upper 

 E's in the treble stave will clash, and a beat will result. 



In all these three cases Smith's beats also will natu- 

 rally be present, and it is curious that in each case when 

 we come to determine the rapidity of the beats, we find it 

 come out the same, whether we calculate it by Smith's 

 formula or by the unison beats of Helmholtz's overtones. 

 We have added the vibration-numbers to the notes, to 

 facilitate the calculation, and we find the number of beats 

 per second to be — 



For the imperfect octave =* 5 



For the imperfect fifth =10 



For the imperfect major third =20 



each arising from a sharpness of five vibrations in the 

 upper note of the concord. 



Hence we may lay it down as a principle that in conso- 

 nances slightly out of tune, the beat given by the two 

 fundamentals on Smith's plan, and those given by the 

 first corresponding overtones on Helmholtz's principle, 

 are synchronous, and may be considered identical. 



The two kinds of beats, however, must not be con- 

 founded, as their cause is so distinct. The Helmholtz 

 beats arise from the overtones only, whereas Smith's ex- 

 planation applies to the fundamental notes, independently 

 of the overtones altogether. 



Helmholtz notices (Ellis's translation, pages 302-3) 

 that beats of consonances will occur when sounded by 

 simple tones, but accounts for them in another and very 

 ingenious way, namely, by calling in the aid of the grave 

 harmonics, or, as he calls them, the combination tones. 



Taking our first example of the octave consonance 

 given above, when the two notes of 128 and 261 vibra- 

 tions are sounded together, they will give rise to a com- 

 bination tone of 123 vibrations, and this, clashing with 

 the 128 note, will give beats at the rate of five per second. 



For the next example, the consonance of the fifth, this 

 explanation will not suffice, and Helmholtz has to resort 

 to a cause of the second order, namely, the beat of a 

 grave harmonic, not with an imperfect unison, but with 

 an imperfect octave. Taking our former example, an 

 out-of-tune fifth C and G, of 128 and 197 vibrations re- 

 spectively ; these two notes will give a combination or 

 difference tone of 69 vibrations, or an octave below the 

 C, but out of tune. Then Helmholtz says this lower C 

 will beat with its imperfect octave, on account of a new 

 or second order of difference-tones formed from them, as 

 in the former case. 



In a similar but still more remote way, Helmholtz 

 accounts for the beats of other consonances, the fourth, 

 third, &c. 



Without questioning the sufficiency of these explana- 

 tions, I must say they seem to me somewhat far-fetched, 

 and less satisfactory than Smith's, which account for the 

 beats by a more positive and direct method, without calling 

 in the aid of any sounds but the simple fundamental ones. 

 There is at any rate the satisfaction that whichever ex- 

 planation be adopted, the numerical value of the number 

 of beats per second comes out the same and agrees with 

 the fact ; so that in a practical point of view it is imma- 

 terial which explanation be adopted. 



I have alluded above to one important practical use of 

 beats, namely, in tuning ; but there is another use of 

 them, also very interesting, i.e., that they furnish a means 

 of ascertaining the positive number of vibrations per 

 second corresponding to any musical note. This may be 

 done either by the unison or by Smith's beat, and I will 

 give both methods. 



For the unison beat : — Take two notes in unison on an 

 organ, a harmonium, or other instrument of sustained 

 sounds, and put one of them a little out of tune, so as to 

 produce beats when they are sounded together. Let V 

 and V represent the vibration numbers of the upper and 

 lower notes respectively. Then by means of a mono- 



chord it will be easy to determine the ratio ->, which call 



V 



m. Count the number of beats per second, which call ^. 

 Then, since /3 = V — v,-w& obtain the simple equation, 



m — I 

 which gives the actual number of vibrations per second 

 of the lower note of the two. 



The method of deducing the vibration-number from the 

 Smith's beat was pointed out by Smith himself ; but as this 

 method, so far as I know, is not to be found anywhere, ex- 

 cept buried under the mass of ponderous learning contained 

 in his work ; I give it here in a simple algebraical form. If 



— represents the true ratio of the interval, N the actual 

 n ' 



number of vibrations per second of the lower note, and 



M the same number for the upper one, the formula for 



Smith's beats becomes 



N 



or N = 



,M 

 N 



Now, as m and n are both known for any given concord, 

 if we can tell by any independent means the actual ratio 



of the notes -, we may, by simply counting the beats, 



calculate the actual number of vibrations A' of the lower 

 note. If the interval is too flat, ^ must be -|- ; if too sharp, 

 it must be — . 



The following example will show how this maybe done. 

 Let it be required to determine how many vibrations per 



second are given by the note 



on an organ. 



Tune three perfect fifths upwards, and then a perfect major 

 sixth ^downwards, thus — aTzinlzifi: 



:2^z±: 



=zznri, w 



I 



which 



will give the C an octave above the original note. But, 

 by the laws of harmony, we know that this octave will 

 not be in tune ; the upper C will be too sharp, the ratio 



being f J,^instead off, as it ought to be. Hence --. — fj, 



and — = 2_ Count the beats made by this imperfect 



octave, and suppose them = 192 per minute, or 3"2 per 

 second ; then, as the interval is sharp, 



i.e. the note Q ^ ^ - H is making 128 double vibrations 



per second. 



This method has the advantage of dispensing with the 

 use of the monochord, which was necessary in the former 

 case. 



NOTES 



A Meteorological Commission, appointed by the Ministers 

 of Public Instruction, Agriculture and Commerce, Marine, and 

 Public Works, to inquire into the possibility and practicability 

 of a more intimate co-operation being effected among the various 

 meteorological systems of Italy, have just issued their report. 

 The Commission consisted of fourteen members, including most 

 of the well-known meteorologists of Italy, with Prof. Cantoni as 

 president, and Prof. Pittel as secretary, and met daily at Palerma 

 from Aug. 30 to Sept. 6, 1875. The more important of the con- 

 clusions arrived at are these : — That all methods of observing at 

 the stations of the various systems connected with the State be 



