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-173] Compound Musical Tones and Harmonics.

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rapidly separating the two legs by means of a steel rod, as shown in the figure. The vibration produces a note which is always the same for the same tuning fork. The note is strengthened by fixing the tuning fork on a box open at one end, called a resonance box.

It has been remarked for some years that not only has the pitch of the tuning fork, that is, concert pitch, been getting higher in the large theatres of Europe, but also that it is not the same in London, Paris, Vienna, Milan, etc. This is a source of great inconvenience both to composers and singers, and a commission was appointed to establish in France a tuning fork of uniform pitch, and to prepare a standard which would serve as an invariable type. In accordance with the recommendations of that body, a normal tuning fork has been established, which is compulsory on all musical establishments in France, and a standard has been deposited in the Conservatory of Music in Paris.

It performs 870 single vibrations per second, and gives the standard note a, or the a in the treble stave. Consequently, with reference to this standard, the middle C would result from 261 double vibrations per second.

173. On compound musical tones and harmonics.--When any given note (say C) is sounded on most musical instruments, not that tone alone is produced, but a series of tones, each being of less intensity than the one preceding it. If C, which may be called the primary tone, is denoted by unity, the whole series is given by the numbers 1, 2, 3, 4, 5, 6, 7, etc. ; in other words, first the primary C is sounded, then its octave becomes audible, then the fifth to that octave, then the second octave, then the third, fifth, and a note between the sixth and seventh to the second octave, and so on. These secondary tones are called the harmonics of the primary tone. Though feeble in comparison with the primary tone, they may, with a little practice, be heard, when the primary tone is produced on most musical instruments; when, for instance, one of the lower notes is sounded on the pianoforte. Helmholtz's researches show that the different timbre or quality of the sounds yielded by different musical instruments is due to the different intensities of the harmonics which accompany the primary tones of those sounds. The leading results of these researches may be thus stated :

i. Simple tones, as those produced by a tuning fork with a resonance box, and by wide covered pipes, are soft and agreeable without any roughness, but weak, and in the deeper notes dull.

ii. Musical sounds accompanied by a series of harmonics, say up to the sixth, in moderate strength are full and musical. In comparison with simple tones they are grander, richer, and more sonorous. Such are the sounds of open organ pipes, of the pianoforte, etc.

iii. If only the uneven harmonics are present, as in the case of narrow covered pipes, of pianoforte strings struck in the middle, clarionets, etc., the sound becomes indistinct; and when a greater number of harmonics are audible the sound acquires a nasal character.

iv. If the harmonics beyond the sixth and seventh are very distinct, the sound becomes sharp and rough. If less strong the harmonics are not prejudicial to the musical usefulness of the notes. On the contrary, they are useful as imparting character and expression to the music. Of this kind are most stringed instruments, and most pipes furnished with tongues, etc. Sounds in which the harmonics are particularly strong acquire thereby a peculiarly penetrating character ; such are those yielded by brass instruments.

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CHAPTER III.

TRANSVERSE VIBRATIONS OF STRINGS. STRINGED INSTRUMENTS.

174. Transverse vibrations of strings.—We have already seen (153), that when an elastic string, stretched at the ends, is removed from its position of equilibrium, it reverts to it as soon as it is let go, making a series of vibrations which produce a sound. The strings used in music are commonly of catgut or metallic wire. The vibrations which strings experience may be either transversal or longitudinal, but practically the former are alone important. Transversal vibrations may be produced by drawing a bow across the string, as in the case of the violin; or by striking the string, as in the case of the pianoforte ; or by pulling them transversely and then letting them go suddenly, as in the case of the guitar and the harp.

175. Laws of the transverse vibrations of strings.--The number of transverse vibrations which a string can give in a

-176] Laws of the Vibrations of Strings. 169 certain time, that is, the sound it yields, vary with its length, its diameter, its tension, and with its specific gravity, in the following manner:

The tension being constant, the number of vibrations in a second is inversely as the length ; that is, that if a string makes 18 vibrations in a second for instance, it will make 36 if its length is halved, 54 if its length is one-third, and so on. On this property depends the violin, the contre basso, etc., for in these instruments, by pressing the string with a finger, the length is reduced or increased at pleasure, and the number of vibrations, and therewith the note, is regulated.

With strings of the same length and tension the number of vibrations in a second is inversely as the radius of the string ; that is, the thinner a string, the greater its number of vibrations, and the higher its pitch. In the violin, the treble string, which is the thinnest, makes double the number of vibrations of that which would be made by a string twice its size, that is to say, the diameter of which is twice as great.

The number of vibrations in a second is directly as the square root of the stretching weight or tension; that is, that when the tension of a string is four times as great, the number of vibrations is doubled; when the tension is nine times as great, the number is trebled, and so on. This, then, furnishes a means of altering the character of a note by stretching, as is done in stringed instruments.

Other strings being equal, the number of vibrations in a second of a string is inversely as the square root of its density. Hence, the greater the density of the materials of which strings are made, the less easily they vibrate, and the deeper are the sounds they yield.

From the preceding laws it will be seen how easy it is to vary the number of the vibrations of strings and make them yield an extreme variety of sounds, from the deepest to the highest, used in music.

176. Verification of the laws of the vibrations of strings. Sonometer.-This may be effected by means of an instrument called the sonometer, or monochord. It consists of a thin wooden box to strengthen the sound. On this there are two fixed bridges A and B (fig. 144), over which pass the strings AB, CD, which are commonly metallic wires. These are fastened at one end, and stretched at the other by a weight P, which can be increased at will. By means of a third movable bridge D, the length of that

portion of the wire which is to be put in vibration can be altered at pleasure.

If two strings are taken, which are identical in all respects and are

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Fig. 144. stretched by equal weights, they will be found, on being struck, to yield the same sound. If now one of them be divided by the movable bridge D into two equal parts, the sound yielded by CD will be the higher octave of that yielded by the entire string AB, which shows that the number of vibrations is doubled, and thus verifies the law.

To verify the second law, the bridge D is removed. If the string AB is taken so that it has double the size of the other, but both stretched by the same weight, it will be found that the sound which the thinnest string yields is the next higher octave of that yielded by AB; proving thus that the number of vibrations is doubled.

The two strings being of the same diameter, and the same length, if the weight which stretches the one be four times that which stretches the other, the sound yielded by the first is the higher octave of that of the second, which shows that the number of vibrations is doubled; when the weight is nine times as great, the sound is the higher octave of the fifth of the former.

The fourth law is established by using strings of different densities, but of the same dimensions, and stretched to the same extent.

177. Stringed instruments.—Stringed musical instruments depend on the production of transverse vibrations. In some, such as the piano, the sounds are constant, and each note requires a separate string : in others, such as the violin and guitar, the sounds are varied by the fingering, and can be produced by fewer strings.

In the piano the vibrations of the strings are produced by the stroke of the hammer, which is moved by a series of bent levers

-178]
Wind Instruments.

171 communicating with the keys. The sound is strengthened by the vibrations of the air in the sounding board on which the strings are stretched. Whenever a key is struck, a damper is raised, which falls when the finger is removed from the key and stops the vibrations of the corresponding string. By means of a pedal all the dampers can be simultaneously raised, and the vibrations then last for some time.

The harp is a sort of transition from the instruments with constant to those with variable sounds. Its strings correspond to the natural notes of the scale: by means of the pedals the lengths of the vibrating parts can be changed, so as to produce sharps and flats. The sound is strengthened by the sounding box, and by the vibrations of all the strings harmonic with those played.

In the violin and guitar each string can give a great number of sounds, according to the length of the vibrating part, which is determined by the pressure of the fingers of the left hand while the right hand plays the bow, or the strings themselves. In both these instruments the vibrations are communicated to the upper face of the sounding box, by means of the bridge over which the strings pass. These vibrations are communicated from the upper to the lower face of the box, either by the sides, or by an intermediate piece called the sound post. The air in the interior is set in vibration by both faces, and the strengthening of the sound is produced by all these simultaneous vibrations. The value of the instrument consists in the perfection with which all possible sounds are intensified, which depends essentially on the quality of the wood, and the relative arrangement of the parts.

Instruments of the class of the violin are very difficult to play, and require a very delicate ear; but in the hands of skilful artists, they produce marvellous effects. They are the very soul of an orchestra, and the most beautiful pieces of music have been composed for them.

CHAPTER IV. SOUNDING TUBES AND WIND INSTRUMENTS. 178. Production of sound in pipes.—Sounding pipes are hollow pipes or tubes in which sounds are produced by making the enclosed column of air vibrate. In the cases hitherto considered the sound

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