MUSIC 



more alike (that is, more nearly resembling a note 

 and its octave) than those forming others of them. 

 By means of suitable instruments, it has been 

 found that certain numerical ratios exist between 

 the different rates of vibration of the notes in the 

 scale, and that a very remarkable relation exists 

 between these ratios and the mental effect of the 

 intervals to which they correspond. When two 

 notes make the interval of an octave with each 

 other, the higher vibrates exactly twice as fast as 

 the lower ; when the interval is a fifth, the higher 

 vibrates 3 times for every 2 vibrations of the 

 lower ; and when the interval is a fourth, the higher 

 vibrates 4 times to every 3 vibrations of the lower. 

 These ratios, 2 : i, 3 : 2, and 4 : 3, are most easily 

 expressed by fractions, and accordingly we call the 

 fractions $, 4, and the -vibration fractions of the 

 intervals of an octave, a fifth, and a fourth re- 

 spectively. The following table shews the vibra- 

 tion fractions for the intervals which occur in the 

 ordinary major and minor scales : 



Major seventh C . B 



Octave... ....C . C 



Y 



Concord. 



The conclusions to be drawn from this table are 

 very plain. The intervals which have the simplest 

 vibration fractions, $, |, , |> &c. are consonant ; 

 'while those whose vibration fractions have numer- 

 ators and denominators of which the least com- 

 mon multiple is large, as f, V, &c. are dissonant. 

 The intervals which have vibration fractions most 

 nearly approaching in simplicity to the vibration 

 fraction of the octave, are just those which most 

 nearly resemble it in smoothness of sound the 

 fifth and fourth. It was for long a very common 

 assertion of musical theorists that these numerical 

 ratios were directly the cause of consonance and 

 dissonance ; that ' the ear delighted in simple 

 numerical ratios,' and therefore took pleasure in 

 the corresponding intervals. This, as we shall 

 see farther on, is by no means a correct statement 

 of the case ; the cause of the smoothness or rough- 

 ness of an interval lies deeper than its vibration 

 fraction ; and, moreover, the intervals which have 

 the simplest ratios are not those in which the ear 

 takes the greatest pleasure. 



The vibration fraction of an interval is constant, 

 no matter what may be the absolute pitch of the 

 notes which form it ; so that the absolute rate of 

 vibration of any note can be at once deduced from 

 the rate of vibration of the key-note of the scale 

 to which it belongs. By extending this reasoning 

 a little further, we are able to establish the fact 

 with which we started, that the pitch of a note 

 depends upon the rate of its vibration, and there- 

 fore upon the length of the air-wave producing it. 



The loudness of a musical sound is a much 

 simpler matter than its pitch. By setting a violin- 

 string or a tuning-fork in motion, we see at once 

 that the largest vibration corresponds to the 

 loudest sound, and that the dying away of the 



sound occurs simultaneously with the visible ces- 

 sation of the vibrations. The pitch of the sound 

 has remained the same throughout ; we know, 

 therefore, that the length of the air-waves has 

 been in no way affected by the decrease of the 

 visible vibrations. The quality of the sound also, 

 of which we shall have more to say presently, has 

 remained unaffected, so that the form of the air- 

 waves has not been altered. It therefore becomes 

 evident, that the extent of the visible vibrations of 

 the fork or string corresponds to the amplitude of 

 the air-waves, and that upon this depends the 

 loudness of the sound. 



We now come to the more difficult question of 

 the quality of musical sounds ; and before we can 

 see how this quality is affected by the wave-form, 

 or mode of vibration, we must say something as 

 to what that quality actually is. We are accus- 

 tomed to speak of the tone of a voice or an instru- 

 ment as if it were a simple thing. In reality, 

 however, there is scarcely such a thing as a simple 

 musical sound to be heard. Almost all musical 

 sounds are compound they are made up of one 

 principal note, from which they take their name, 

 and a number of other and higher notes, not so 

 distinct, but still quite as real. These notes 

 always stand in a certain definite relation to the 

 fundamental tone, the first being an octave above 

 it ; the second, a fifth above the first ; the third, 

 the octave of the first ; the fourth and fifth, 

 respectively a major third, and a fifth above the 

 third. There are a considerable number of these 

 notes ; fig. 2 shews the first nine of them for the 

 note C in the bass clef. The 

 peculiarity about them is, that 

 their rate of vibration in- 

 creases in arithmetical ratio 

 as they ascend. The first of 

 them vibrates twice as fast 

 as the fundamental tone; the 

 second, three times as fast ; 

 the third, four times as fast; 

 and so on. Professor Tyndall 

 has introduced a nomencla- 

 ture for compound sounds 



Fig. 2. 



adapted from Helmholtz ; he calls a compound 

 sound a clang, and the different tones of which it 

 consists, partial tones. Of the partial tones, the 

 lowest is called the fundamental tone, and the 

 others the over-tones. This excellent nomencla- 

 ture has been adopted by all writers upon the 

 subject, and we shall use it here ; it affords us 

 the means of stating in general terms the first 

 proposition in regard to the quality of a musical 

 sound. This is, that 'the quality of a clang depends 

 upon the number, position, and intensity of the 

 partial tones of which it is composed? Fig. 2 

 shews the first nine over-tones which can occur in 

 the clang of which C is the fundamental tone, but 

 it does not follow that the whole of these shall 

 be heard, or shall exist, when that C is sounded. 

 This is a matter which depends entirely upon the 

 instrument used for sounding the C. One of 

 them only may exist, or two, or three, as the case 

 may be. Further, those which do exist need not 

 occur consecutively ; the clang may contain the 

 first, third, and fifth, or the fourth and ninth, or 

 the second, fourth, and sixth over-tones, or any 

 other combination. Further still, the intensity or 

 relative loudness of the different partials may vary 

 to an almost unlimited extent Clangs may differ 



191 



