number of vibrations per second which is made by the sounding body. It is easy to notice the fact that pitch depends upon the rapidity with which the impulses succeed each other. As I pass this card round the milled edge of this coin, you hear, when the movement is slow, the distinct impulses; but when the motion is quicker, these give place to a noise in which you may recognize a grave note; and when I make the movement yet more rapidly, you will distinguish a rise in the pitch of the sound. Now, the reason the shorter teeth of the steel plate in your musical box yield the higher notes is because the vibrations of a rod (like the oscillations of a pendulum) are quicker the shorter the rod is." "The vibrations," said Louisa, "are, I suppose, much too rapid to be seen?

"You will remember, from our experience with the thaumatrope, that the impressions made on the eye remain for a certain time, and for that reason, when a body oscillates so as to occupy the same position oftener than eight times in a second, we receive a permament impression from it. Now, when I stretch this piece of cord very tightly by tying it to one of the legs of the table, and passing it round the other, you will notice that when I pluck the cord in the middle-so-it gives out a musical note, just as the string of the harp does, but fainter, from the absence of a sounding-board. You will notice that the string produces on the eye, while vibrating, the effect of a spindleshaped body, with a semi-transparent gauzy appearance in the central parts. You hear, when I tighten the string at the same time that I pluck, how the note of the twang rises."

"Yes," said Louisa, "I know that the tighter the cord or wire, the higher is the note; for I have seen the pianoforte tuner tighten or release the wires, according as the note required to be made sharper or flatter. And I have also noticed in the harp and piano that the cords and wires which produce the highest notes are the shortest."

"It has been found,” replied Mr. Seymour, "that with the same tension in the same wire or cord, a piece of half the length will give a note exactly an octave higher-that is, it will perform just double the number of vibrations as a cord of twice its own length."

"This fact was discovered," observed the vicar, "by Pythagoras more than five centuries before the Christian era. He stretched a string, and having divided it into three equal lengths.

THE MUSIC OF THE SPHERES.

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he fixed it firmly at one of the points of division, so that it thus virtually became two strings similar in every respect, except that one was twice the length of the other. He then found, on sounding the two sections at the same time, that a perfect concord was produced. He also found, when he divided his string in other proportions, that concordant notes were yielded only when the lengths of the sounding parts had some very simple ratio to each other, as, for example, 2 to 3, or 3 to 4."

"I am thankful to the vicar for reminding us of this," said Mr. Seymour, "for this discovery of Pythagoras had no little influence in ancient philosophy, as it probably led him to the axiom that the universe is governed by numbers and harmony; and Pythagoras, who was the first in Europe to teach that the planets revolve about the sun, also supposed that their distances therefrom were in the same ratio that express the relations of the seven notes of the musical scale. Hence, perhaps, the notion of the music of the spheres, which was imagined to be produced by the rotation of the crystalline spheres that carried the heavenly bodies, and to be so loud and sweet that by mortal ears it was unappreciable, and therefore unheard by men. The real physical cause of the concordance of certain musical notes has only quite lately been made out."

"Do tell us, papa, what it is!" cried Louisa.

"The subject is not one we can enter into, but I may just indicate to you the direction in which its explanation is to be found. Tell me what is the higher note which gives the most perfect harmony with the c near the middle of the keyboard of the piano?"

"The c of the next octave."

"And what other concords are there in the octave-I mean, what notes struck with the c would produce the most pleasing effect on the ear?"

"There is G, which is the fifth, and E, which is the third note higher."

"Now, it is known that the note c of which I have been speaking corresponds with a rate of vibration of 528 per second."

"I cannot understand how movements so extremely rapid can possibly be counted," said Louisa.

"The principle is simple," replied Mr. Seymour. "Suppose that I have a wheel with 132 teeth, and I press a spring against

the wheel in such a manner that a sound is made as each tooth passes; then, as I have already said, when the separate impulses occur with a certain rapidity, which is about twenty in a second, a low musical note is heard; and if the velocity of the wheel be increased, the note will rise. It is quite easy to attach to such a wheel a train which shall register the number of revolutions per minute; and when the wheel is turned so that it makes four revolutions per second, the note will be in unison with the note of the pianoforte; if the rate be quickened to eight revolutions, when 1,056 teeth would pass under the spring in a second, the note would be the octave higher. But here I will show a table of the number of vibrations corresponding with each note of the octave.* Look, now, at the number of vibrations in c and G, and observe that they have the simple ratio to each other of 2:3; while those of c and E are to each other as 4: 5; and in every case in which a fundamental note is compared with its higher third, fifth, and octave, the numbers of vibrations are found to have these simple ratios to each other, 4: 5: 6: 8.”

"And I suppose," said Louisa, "that if the numbers of vibrations do not quite answer to these proportions, the harmony would not be quite perfect, but would be better the nearer the numbers agreed."

"On the contrary," replied Mr. Seymour, "the discord is most grating and harsh when the numbers differ but little, and the combinations which give the most inharmonious effects are precisely those which nearly but not quite agree with the proportions I have stated. Now, from these and other facts, by a long course of beautiful experiments and clear reasonings, a living man of science has succeeded in giving a satisfactory theory of the physical causes of harmony, but you are not yet in a condition to understand such investigations."

Louisa here wished to know how it was that the same note was always yielded by the wires of the pianoforte, whether the key of the instrument were struck quickly or slowly.

"To understand that fact," replied her father, "you must know that the wire does not alter the period of its vibration with the force of the blow. The only effect is to impart a greater swing to the wire, it is made to traverse a greater space, and as by the laws of all vibrating bodies it performs the vibration

*NOTE A.-Musical Pitch.

QUALITY OF SOUNDS.

III

always in the same time, its velocity is greater; hence the loudness of the note is increased, but the pitch remains the same."

"There is," said Louisa, "another circumstance which I do not think your theory explains. When the same note is sounded by a violin, a flute, a concertina, and other instruments, one recognizes the fact of their pitch being the same, while the tones themselves are quite distinct and different, each having a quality of its own."

"The reason is, that no sounding body sends out a single set of regular undulations. A vibrating wire, for example, vibrates not only as a whole, but divides itself into shorter vibrating segments, each of which contributes to the actual sound. It is the mixture of these (overtones, as they are called) with the fundamental tone, which gives the character to the sound of different musical instruments. It is true that the overtones are rarely noticed, but the reason is that an attentive ear is required to pick them out of the general mass of sound, and it seldom or ever happens that attention is directed to them, as even a musician is not required to analyse the sound of his instrument,-the general effect upon the ear suffices for him. The overtones may be very numerous, and may vary greatly in relative intensities, and thus the variety of qualities in sounds may be accounted for. For instance, the same voice singing in the same note the different vowels, and different voices singing the same vowel in the same note, all produce different effects, which are easily explained by the various mixtures of overtones with the fundamental note. You will, perhaps, more easily understand the operation of such mixtures and superposition of vibrations by the analogical case of colours. If every note emitted by the voice or by musical instruments was simple, and gave rise only to one set of soundundulations, the only possible difference in sounds would be in pitch and loudness, and a piece of music would in such a case be similar to a painting which would be produced by an artist. who was compelled to spread on his canvas no other tints than the pure unmixed colours of blue, red, yellow, green, &c. How much we should miss the varied and broken hues which now charm the eye!"

"Upon my word, Mr. Seymour," cried Mr. Goodenough, "you are getting out of your depth; pray let us take leave of this part of the subject, for I am quite sure that my young friends have already received more than they can carry away."

"I submit, my good sir," said Mr. Seymour; "but I should like to say a few words upon the different kinds of musical instruments, and you will, perhaps, kindly give us your opinion as to the antiquity of the various forms of these. It appears to me that musical instruments may be classed under three heads -stringed instruments, such as the harp, violin, &c.; wind instruments, as the flute and trumpet; and instruments of percussion, as the tabor and drum."

"And what kind is considered the most ancient?" asked Louisa.

"Wind instruments, most unquestionably," cried Mr. Goodenough. "Diodorus ascribed their invention to the accidental notice of the whistling of the wind in the reeds on the banks of the Nile, and the poet Lucretius maintained a similar opinion:

666

"Et Zephyri cava per calamorum sibila primum,
Agrestes docuere cavas inflare cicutas.""

"I really, my dear sir, cannot see any good reason for giving this preference, in point of antiquity, to wind instruments,” said Mr. Seymour. "The lyre or harp is, surely, as ancient as any instrument on record. The mythologist ascribes the idea of producing sound by the vibration of a string to Apollo, which is said to have suggested itself to him on his hearing the twang of the bow of his sister Diana. With respect to instruments of percussion, it may be reasonably supposed that the sonorous ringing of hollow bodies, when struck, must have very soon suggested their invention to mankind; but I really consider any research into a question of such obscurity as uninteresting as it must be hopeless: let us rather devote our attention to the philosophy of these instruments. I have stated that they may be referred to three principal classes; but I must, at the same time, .observe that, in some cases, the vibrations of solid bodies are made to co-operate with those of a given portion of air; for example, trumpets and various horns may be said to be mixed wind instruments, since their sound is produced by the joint vibrations of the air and a solid body; and in certain stringed instruments, as in the violin, the immediate effect of the strings is increased by means of a sounding-board, which acts more powerfully on the air than the strings could do alone."

"I apprehend that this mixture must obtain more or less in all instruments," said the vicar.