scale-pan attached to the end of the string (fig. 154). On sounding the tuning-fork with a bow, the cord is thrown into one or more segments as the case may be : and by reducing the tension to a half or a third, we increase the number of segments four or nine times. All the laws of vibrating strings may be thus experimentally established. 535. The simple scale of sound which nature thus gives by the spontaneous dividing of a single string, corresponds with the series of harmonics, as explained above, and is called a chord. The vacancies in it, compared with the common musical scale, may be filled up by joining to it the notes of two additional strings, one a third shorter, and therefore giving sounds sharper or more acute than it, and the other a third longer, and therefore giving tones more grave. Of these additional notes, while part agree, or are in unison with certain notes of the principal chord, the remainder just serve to fill up its larger intervals, and to complete a scale of nearly uniform interval-as three ladders, having unequal intervals between their steps, might still, if placed together, complete a stair of easy ascent. The relation between these strings or chords is such, that the principal beats thrice for twice of the low chord, and the high chord beats thrice for twice of the principal; and in the usual scale of notes, the principal is the fifth note above the lower and fifth note below the higher. The numbers which express the relations of beats among the notes of an octave are easily found, from our knowing the relative number of beats in the notes of any one simple chord, and the relation as above described of the three chords forming the compound scale. The following table exhibits on the first line these relations, or the arithmetical expression for the beats of an octave ; in the second line the corresponding lengths of a given string required to produce them; in the third line the English designation of the notes by letters, and in the fourth line the continental designation by names, these names being the first syllables of certain verses formerly sung by learners : 536. If the intervals in the musical scale were all equal, a per 356 Melody, Harmony, and Time in Music. former might choose indifferently any note as a fundamental or key note, and would only have to attend to the number of intervals above and below it; but, in fact, the relation of the three constituent chords is such, that the third and seventh intervals in ascending from a key note are only about half as large as the others. It is owing to this circumstance that in changing the key on an instrument, certain notes belonging to other keys are about half a note too low or too high, that is, too flat or too sharp, and must be changed accordingly. And hence, when an instrument is to be used to play in all keys, its larger intervals must be divided into two parts at least. The fact of these unequal intervals, ill understood, is what gives an appearance of great complexity and difficulty to musical science. 537. Melody, in music, is when notes, having the simple numerical relations of beat which we have been describing, are played in succession: harmony is when two or more such notes are sounded together. The effect of both is delightfully increased by what is called measure, viz., making the duration of the notes or strain cor. respond with certain regular divisions of time. This gives to the listener an anticipation, to a certain degree, of what is coming, with the pleasure of having expectation realized, as happens similarly in regard to the metre and rhyme of poetry: it moreover enables the memory to retain musical combinations of sound. The airs of the Æolian harp, which observe no time, cannot be learned by rote or repeated. The music of a single drum is chiefly that of time. The accompaniment of an air afforded to a singer by one or more instruments, and which is so pleasing, is chiefly the sounding simultaneously, in a subdued manner, some other notes of the chords to which the several vocal notes belong. Duets and more complicated concert-pieces have their origin from the same source: and highly cultivated musical sense can even follow and enjoy several melodies played together. Musical notes, by whatever instrument produced, have to each other the same numerical relations in the beats or vibrations which constitute them. The different qualities of tone, therefore, from different instruments, can depend only on peculiarities of the single beats, as to whether they are sharp or soft, strong or weak, and accompanied or not by their natural harmonics. Such is the extraordinary nicety of perception which the human ear possesses in this respect, that it can not only distinguish different kinds of instruments, as a flute and clarionet, playing the same note, but Classification of Musical Instruments. 357 with respect to the human voice, goes to the extent of recognizing almost each one of many voices singing the same air. One of the greatest charms of concert-music is that a particular voice and the different instruments may take up, separately, parts of the strain suited to their individual expression: the flute and hautboy, for instance, breathe softness; the trumpet and drum arouse; the harp rolls forth its brilliant chords; the violin leads the clear sound through rapid and endless variety; and so of the rest. That there might be correspondence in instruments when played together, and a known pitch when played apart, it became necessary to fix on some tone or certain number of vibrations as a point of comparison. Hence tuning-forks are made of steel, with length of prongs calculated to produce a certain note (see Art. 536). This note is usually the fourth A or la, from the bass of the pianoforte, and vibrates about 440 times in the second ;-and when the note of the same name on any instrument is tuned in unison with this, the other notes can be easily adjusted according to the harmonic relations above explained. 538. Almost every substance or contrivance that can produce a uniform continued sound may enter into the composition of a musical instrument: hence the almost endless variety which the world has seen. The chief classes of instruments are stringed instruments, wind instruments, and bells or rods. The guitar, as affording an accompaniment to vocal music, has many advantages. It is not too loud, yet the strains are very distinct : it admits of most touching expression; command of it is easily learned to the extent desirable as an accompaniment, by any one who should attempt to perform music at all; it is portable and cheap. The great facility of accompaniment on it depends on this, that the player is able by one position of the hand so to touch the strings, that the sounds of all the six shall belong to the same chord:-three positions of the hand, therefore, for one key, produce all the notes and chords which a simple accompaniment requires ; and the hand soon falls into these so readily, that the player is hardly sensible of exerting volition in regard to them. Sounding Rods and Plates. 539. Rods of wood, metal, glass, &c., fixed at both ends, will vibrate and divide into ventral segments after the same manner as the strings above described: but the same simple relations between the pitch of the note produced and the number of segments do not 358 Vibrations of Sounding Rods and Plates. obtain. When a string of catgut tightly stretched divides into two segments, each half sounds the octave to the note produced by the whole vibrating string; but when a rod thus divides into two, each half vibrates more than twice as fast as the whole, and hence the note sounded by each half, is higher than the octave to the prime or fundamental note of the rod; the reason being that the force tending to restore the rod to its position of rest or equilibrium, is greater than that in the case of the string. If a steel or brass rod of two or three feet be fixed at one end only, and made to vibrate, either by striking or by means of a fiddlebow, it will divide into wave-sections, like a sounding string. If the rod be three feet in length, it will vibrate about once in a second, and the motion will be plainly visible, but too slow to be audible. As we shorten the rod, the rapidity of quiver increases very fast; being four, nine, &c., times as fast when the length is reduced to a half, a third, &c. At about four inches, the vibrations begin to fuse together into a sound, and a steel rod one inch long will give nearly 1300 vibrations per second. An instrument to sound the notes of the gamut or scale might obviously be constructed out of eight rods whose lengths were connected by these relations. The metal tongues of the common musical-box belong to this class of sounding bodies. 540. The motions of the free end of a vibrating rod fixed at the other end are much more complicated than might at first be supposed they may be easily followed by fixing on the end of the rod a common glass bead, silvered inside, and watching the fiery path which the bead will trace out in presence of a candle-flame or strong sunlight. By striking the rod at different places or in different ways, the luminous track may be made almost infinitely various: sometimes curves of exceeding beauty are obtained. This method of studying the vibrations of a rod by the rapid motion of a luminous point is the invention of the late Sir Charles Wheatstone. The common harmonica, and the claque-bois of the French, consist of a series of glass, wood, or brass rods, graduated in lengths so as to give a musical sequence of notes, and laid loosely on two strings or rods. 541. The tuning-fork is simply a bent steel rod whose ends are free to vibrate, and whose constancy of vibration-period is insured by the stability of character which good-tempered steel possesses: so that it may at any moment be appealed to as a referee for the pitch of any required tone. When a tuning-fork vibrates, each prong may divide into two or Chladni's Sonorous Figures. 359 more ventral segments, just as a string or a straight rod may do. The deepest or fundamental tone is got when each prong sways to and fro as a whole: the first overtone or harmonic is got when each prong divides into two, the second when each divides into three, and so on. But, in accordance with the law already stated for vibrating rods, the first overtone is not the octave of the lowest or prime tone, but one which makes twenty-five vibrations while the ürst makes four. Thus the first overtone of a C fork, which makes 256 vibrations per second, will be a note making 1600 per second : and the others rise in the following proportions :-While the first overtone makes 9 vibrations, the second, third, fourth, &c., will make 25, 49, 81, &c. That is, the vibration-rates of the whole series of overtones are as the squares of the successive odd numbers 3, 5, 7, 9, &c. The vibrations of a tuning-fork may be readily shown by bringing it near to a light ball or a small bead suspended by a thread; the bead will be projected to a considerable distance. "Chladni's Sonorous Figures." 542. Plates of wood, glass, or metal may be made to give forth musical sounds like rods, by fixing them at one part with a clamp, or in a vice, and drawing a violin bow across the edge. The famous musician, Chladni, in 1785 discovered a simple method of rendering the vibrations of plates visible. He fixed the plates horizontally, and strewed some fine sand over them. On sounding the plates with a fiddle-bow, the sand was set in vibration and collected in regular heaps along the lines of no vibration, or the nodal lines, that is, the lines where parts of the plate in opposite states of vibration meet, or where the parts are at rest. In fig. 155 these nodal lines are seen in a variety of forms. The sand assumes the position indicated by the lines across the squares. A square metal or glass plate, firmly fixed to a metal or wooden rod at the centre, a violin bow, and some "silver sand," are all the apparatus required to produce these remarkable figures. When the bow is drawn down the edge of the plate near one corner, we get |