ACOUSTICS. 



ACOUSTICS. 



70 



note makes three, which is the case with c and its fourth F ; or that of 

 4 to 5, which happens with c and its third E ; the combined effect of 

 the two is agreeable. The same may be said of c and its sixth A, in 

 which the ratio is that of 3 to 5, or of E and its minor sixth [Music] 

 c 1 , in which the ratio is that of 5 to 8 ; or of E and its minor third G, 

 in which the ratio is that of 5 to 6. We write underneath (Fig. 12) 

 the common musical scale in the treble clef, with the denominations of 

 the notes and the fraction of a vibration which is completed while the 

 first c completes one vibration, which fraction is greater than unity, as 

 the notes are rising. Thus while c vibrates once, D vibrates once and 

 one-eighth ; or 8 vibrations of c take place during 9 of those of D. 



This is the musical scale pointed out by nature, since all nations 

 have adopted it, or part of it at least. It fully verifies our assertion 

 that the ear delights in the simplest combinations of vibrations. It 

 would be hardly possible to place between 1 and 2, six increasing 

 fractions whose numerators and denominators should, on the whole, 

 contain smaller numbers. We find, in the six intermediate fractions, 

 only 2. 8, 4, and 5 singly, or multiplied by one another, no product 

 exceeding 15. Neither has the whole of this scale always been 

 adopted. It seems to have been formerly universal to reject F and B, 

 the fourth and nerentk of the scale ; as is proved by the oldest national 



airs of the orientals, the northern nations, and even of the Italians 

 [SCALE]. 



Fig. 12. 



n K F 

 I** 



A B C" D 1 E> F' G 1 A 1 &c. 



t : v a i . I i * v *e. 



The following table will represent the proportions of the lengths of 

 the sonorous waves which yield the preceding notes. These lengths 

 decrease, as we have seen, as the times of vibration decrease, or as the 

 numbers of vibrations in a given time increase. 



Now, let two of these notes be sounded together, for example, c and 

 a, in which two waves of c are equivalent to three of a. The resulting 

 wave is, as we have seen in the preceding part of this article, twice as 

 long as the wave of c, and the curve which represents the condensation 

 and velocity of the particles of air is compounded, as before described, 

 of those of the waves of c and G. The ear ig able to perceive three 



Tig. 18. 



I 



A 



distinct sounds, one of which is almost imperceptible, and indeed 

 inaudible, unless carefully looked for. The two perceptible sounds are 

 those of c and G from which the wave was made ; nor ~are we well able 

 to explain how this can be. Undoubtedly, if the curve, which is the 

 type of the compound wave, were presented to a mathematician, he 

 would be able, with consideration and measurement, to detect its 

 Memento ; and to make that resolution which is done by the most 

 unpractised ear. But we may, perhaps, assert that a savage, or a person 

 totally unused to music, would not separate the sounds, but if c and <; 

 were sounded separately, and afterwards together, would imagine he had 

 heard three distinct notes. The third sound, which is very faint indeed, 

 is that belonging to the whole compound wave, which, being twice as 

 long as the wave of c, belongs to the note called c, an octave below 

 the iirst c of the preceding scale, which may be denoted by c 1 . Wi> 

 may perhaps give an idea of this combination in the following way : 



Fig. 14. 



-iiiplxirie a series of equidistant balls to roll past us at the rate 

 tit two in a second, and another series at the rate of three in a second, 

 and let us moreover suppose that these balls roll in tubes placed one 

 lie other, so that we only see each as it passes an open orifice in 

 it* tube, a in Fig. 14. It is evident that we thus obtain three distinct 

 successions : 1, that by which we might count 3 in a second from the 



tube ; 2, that by which we might count 2 in a second from the 

 upper tul>e ; 3, that by which we might count single seconds, from 



ing when two balls pass together, and waiting till the same hap- 

 pens again. And we must recollect that any sound, however unmusical 

 in itwlf, produces a musical note, if it be repeated regularly and 

 ftfii; M that it is not from the phenomenon itself, but from the 



M.V of it succession at equal intervals, that the pleasant 



11 is derived. Thus in a passage, which has a strong echo, 



that is, where waves are reflected from wall to wall, as in the 



' losed at both ends, already described, if the foot lie struck 

 against the ground, a faint musical note is heard immediately after the 

 echo has ceased. By the action of the foot, shorter waves are excited, 

 :is well as the long wave, by the reflection of which the echo is caused. 

 \on of these would be repeated were it not for the reflection ; but 

 wlii-n the main sound is weakened by reflection, the shorter waves 

 begin to produce the eflect of a musical note, being, as we must 



'. lean weakened than the longer wave. And we may here take 

 occasion to observe, what will be further discussed in the articles PIPE 



IIORD, that it in difficult to excite a perfectly simple .wave, 

 unaccom|>anied by shorter ones, which latter are always contained an 



TiumtiiT Mt timei in the longer. Thus, if the note called C,, or 



ive below c m fy. 12, be struck on a piano-forte, the sounds 

 itid E 1 (see the figure) will be distinctly heard as c becomes weaker, 

 the waved of these notes being respectively one-third and one-fifth of 

 those of c. When two notes are struck together, the effect is not 

 I>lpn-.ins5, except when the numbers of waves per second in the two 

 ber a very simple proportion. 

 We have noticed all the cases which the musicians call twnntt : 



the remainder, though contributing much to the effect of music, being 

 railed discords. Thus, if F and o be sounded together, iu which (jig. 12) 

 v makes | of a vibration while a makes g, or F makes 8 vibrations 

 while G makes 9, the effect is disagreeable, at least if continued for 

 some time. On the piano-forte, in which the notes when struck 

 subside into comparative weakness, this is uot so much perceived ; but 

 on the organ, in which the notes are sustained, the effect is intolerable, 

 and accompanied by an apparent shaking of the note, producing what 

 are called beat*, which we shall presently explain. Nevertheless, it 

 becomes endurable, if not too long continued, provided F, the discordant 

 note, as it is called, is allowed to pass to the nearest sound, which will 

 make one of the more simple combinations of vibrations with o. The 

 nearest such sound is E, which makes 5 vibrations, while G makes 6. 

 For further information, we must here refer to the article HABHOXY. 



We now come to the absolute number of vibrations made by musical 

 notes ; all that we have said hitherto depending only upon the propor- 

 tions which these numbers of vibrations have to one another ; so that 

 any sound might be called c, provided the sound produced by twice as 

 many vibrations in a second were called c', and so on. From the 

 measurements recorded in the 'Memoirs of the Academy of Berlin ' for 

 1823, it appears that the middle A of the treble clef, or the A in///. 12, 

 was produced by the following numbers of waves per second in the 

 following different orchestras, showing a sniall variation between them, 

 but one by no means insensible to the ear : 



Theatre at Berlin ..... 



Paris, French Opera ..... 



Comic Opera ... . . . 



Italian Opera . . . . 



From this we may form an idea how many vibrations are necessary to 

 create the sensation of a musical sound, and also at what point of the 

 scale the vibrations per second would become so numerous that this 

 effect should cease. If we take one of Broadwood's largest pianofortes, 

 and recollect that they are generally tuned (for private purposes) a 

 little below the pitch of the orchestra, we shall not be far wrong in 

 assuming that the A above-mentioned on these instruments is the effect 

 of 420 vibrations per second. The lowest note, which is almost 

 inappreciable (that is, though perfectly audible as a sound, yet hardly 

 distinguishable from the notes nearest to it), is the fourth descending 

 c from this A, and the highest is the third F above it, though the c 

 above that, or the fourth ascending c from the A, can be well heard, 

 and may be had by whistling into a very small key. We must, 

 however, remark, that the point at which a series of undulations ceases 

 to give a sound either from its slowness or rapidity, is different to 

 different ears ; sometimes so much so, that while one person com- 

 plains of a note as too shrill, another cannot hear it at all. We write 

 the above scale below, putting the A, whose vibrations we know, in its 

 proper place, 



C, 2 C, C A C' C 2 



C'. 



On looking at fiij. 12, we see that A makes 5 vibrations, while c makes 

 3 ; that is, A making 420 vibrations per second, c makes 252 ; there- 

 fore, Ci makes the half of this, or 126; Cz makes 68, and cs 81 J. 

 Again, c 1 makes twice as many vibrations per second as o, or 504 ; 

 C* makes 1008, c 3 2016, and c* 4032 vibrations per second. That is to 



