or MESCHED, a city of Persia, the capital of the province of Khorassan, surrounded by a wall, and said by some to be twelve miles in circuit, but by Fraser estimated at not more than six miles. The greatest length to which the town extends is not above two miles, and the enclosed space presents a sad picture of desolation; the approach to the centre of the city, where the inhabitants exclusively reside, being through masses of ruins. There are thirty-two divisions in the city, each being nominally governed by its own magistrate; but of these many are totally devoid of either houses or inhabitants, and the greater part of the rest are very thinly tenanted. The city appears to have been built of sun-dried bricks or mud, so that the whole aspect of the place presents the monotonous gray earthy colour common to all Persian towns; and even the houses which remain entire are miserably poor and mean in their outward appearance, nor are they much better furnished within. The approach to these houses harmonizes with all the other details, being for the most part through dark lanes and narrow alleys, extremely inconvenient and filthy. Fraser informs us, that in his walks he occasionally stumbled upon the strangest holes and corners, where houses peered out that were half hid in filth and rubbish. He adds, that the path among such places sometimes burrows under the earth, or beneath a heap of buildings which have been raised over it, upon a floor of beams, Musbed, and mats; and that, after thus descending, as it seemed, into the bowels of the earth, a door was opened into a small parterre, surrounded with various apartments, and fitted up with reservoirs and fountains of water, trees, and flowers. There is only one street worthy of the name, which extends throughout the whole length of the town, running north-west and south-east. In the centre is a canal, which serves as a receptacle for all the filth of the town, and is in a state of great disrepair. A row of shops extends along the pathway on either side; and there is a bazaar in another quarter, which extends from 500 to 600 yards in one direction. The public buildings of Mushed are very splendid, and highly celebrated for their sanctity. The tomb in which repose the ashes of the Imam Reza and of the Kaliph Haroun Al Raschid, is an extensive and most magnificent structure, which has been embellished and enriched by different princes. This magnificent cluster of domes and minarets is situated in the centre of the city, where there is a noble oblong square, 160 yards in length by 75 in breadth, with gateways at either end, and forms a splendid specimen of the style of eastern architecture. The mausoleum itself comprises a mass of buildings which appear to be of the octagon form; and a silver gate, the gift of Nadir Shah, admits into the passage to the chief apartment, beneath a gilded cupola of magnificent dimensions, rising loftily into a fine dome, from the centre of which depends a huge branched candelstick of solid silver. There are numerous other apartments, a description of which will be found in Fraser's account of his journey to Khorassan. Mushed contains no other religious shrine worthy of notice. There is a large ruined mosque with two minarets; and there are numerous other smaller ones to be found in all quarters of the city. There are also sixteen schools or colleges, ten or twelve public baths, and at least twenty-five or thirty caravanserais; many of them spacious and handsome establishments, whilst others are in ruins. The palace of the prince is a poor fabric, scarcely deserving of notice. Mushed contains the tombs of Nadir Shah and his son, though not their dust; their remains having been dug up by their bitter enemy Mahommmed Khan, and carried to Teheran. This city rose into importance during the contests between the Mahomedan sects of the Soontes and the Sheahs, under the patronage of Tahmareb, a zealous Sheah, and a prince of the Sufianee dynasty, who decorated the tomb of Imam Reza. It was taken and pillaged by the Tartar tribes; and it was again utterly ruined by the Afghans, and scarcely restored to its former magnificence by all the favours which were lavished on it by Nadir Shah. Its commerce is considerable, being an entrepot for that of the surrounding countries, and rich caravans daily arriving from Boekhara, Khuybah, Herat, Kerman, Yezd, Cashan, Isphahan, and other cities. There are in Mushed many merchants, with a considerable number of shopkeepers and tradesmen; and one quarter of the city is appropriated to the Jews, who are here tolerably numerous, and exercise their customary profession of scrap-sellers. Of travellers, whether in the way of religion or of commerce, numbers are to be seen from all the surrounding countries, such as Arabs, Turks, Afghans, Turcomans, Uzbecks, in the different caravanserais and bazzaars of Mushed. It is famed for the manufacture of velvets, which are esteemed the best in Persia; and of sword-blades, which sell, many of them, at from fifteen to a hundred reals a piece, or from Ll. to between L6 and L7. Those of them made by the old workmen sell as high as 2000 reals; and even more is sometimes demanded for a blade of known antiquity and goodness. The vicinity of the turquoise mines gives employment to numerous stone-cutters. The population is estimated by Fraser at 31,000. Long. 57° E. Lat. 37° 35' N. Within the limits necessarily prescribed to this article, it is impossible to do more than touch upon a few points belonging to the subject. A complete treatise upon the theory and practice of music, according to the received doctrines, would contain about six thousand articles, and would fill several volumes. In writing this article, we have frequently availed ourselves of materials offered by the best and latest musical authorities. When so many works have been published by skilful professional musicians upon their art, we have not the presumption to suppose that we can add much that is new; more especially as we have no new theory to propose, and to maintain with Quixotic zeal and recklessness. Whenever we differ from authorities generally followed, we express our dissent, and give our reasons for it. Our main purpose is to direct attention to some useful musical objects, hitherto in general too much overlooked; to point out some errors in the theory and practice of music; and to show the utter uselessness of pursuing the old routine of building up false theories of music, and spending years in the vain study of what is called thorough bass, and is even still considered, by too many persons, as comprehending the whole art and science of music. To attempt to make any one a composer of music by means only of dry treatises upon intervals and chords, is just as absurd as to attempt to make a poet by means of Bysshe's Art of Poetry, or other books of the kind. Genius and observation, and a careful study of the best models, are really the only things that can ever make a good poet, or a good painter, or a good composer of music. The aid of a skilful master will be of great importance, if he is not wrapped up in a theory. In the absence of a master, two or three of the best modern treatises, such as Reicha's and Cherubini's, may help the student to understand the construction of those models of composition which he ought to have constantly before him. We suppose the reader to understand musical notation, and to be able to sing, or to play upon some musical instrument. If this should be the organ, or piano-forte, so much the better for his more easy attainment of a knowledge of harmony; although he must always remember that both these instruments are out of tune, and do not produce perfect intervals or perfect harmony. If the student of musical composition would acquire a real dominion over the materials of his art, he must not trust entirely to his organ or piano-forte. He must learn to read, in silence, any piece of music in score (in partition), and to hear, "in his mind's ear," the effect of the whole; and he must learn to compose in silence, and without the aid of any instrument. All great composers have acquired these powers. This seems, to the vulgar musician, impossible. To mention only one instance of such powers among living artists; Cherubini composes all his music with the aid of no other instruments but pen, ink, and paper. We have seen him at work. An accomplished composer is able to form in his mind, with no aid from any instrument, the whole plan and details of a complicated piece of harmony, before he writes a note of it. In his "mind's eye" he sees the whole score; in his "mind's ear" he hears the effect which the piece would produce if performed. Until the student acquires this power of abstraction, he must consider himself as only on a par with those every-day musicians whom the "fatal facilities" of the organ or piano-forte raise into the ephemeral class of pseudo-composers.

Some persons consider music as a frivolous and useless art. They do not feel nor understand music, and they are not to be blamed for this when nature has denied them a musical ear, any more than a blind man is to be blamed for not admiring painting or sculpture, or a blind and deaf man for not admiring poetry. But really, when musical compositions are frivolous and useless, the fault is in the artist, not in the art. If men choose to write bad poetry, to paint bad pictures, to chisel bad sculptures, this can never prove that poetry, painting, and sculpture, are frivolous and useless arts. Every one of the fine arts may be rendered frivolous and useless by misapplication of its means; nay, some of them may be made highly dangerous and mischievous, as has often happened. No doubt all the fine arts may be considered in one point of view as superfluous things, not at all contributing to the necessaries of human existence. Food, clothing, fire, and shelter, are really all that man's mere animal existence requires to keep him alive. But if poetry and music, and painting and sculpture, cannot till the earth, nor build hovels, nor make clothing, nor kindle coal-fires, they can at least add ornaments to the structure of civilized society, and contribute to the innocent pleasures and happiness of man's transitory life. And it seems to be proved by experience, that the cultivation of these arts, how unimportant soever they may be to mere animal existence, has always tended to divert the attention of mankind from the sole indulgence of their animal appetites, and of their more dangerous passions. If so, it would not be wise to deprive man of such sources of innocent and pleasing occupation, or rather relaxation, and to reduce him to the merely animal state of the savage, who enjoys and admires nothing beyond his animal comforts, and his murderous triumphs over his rivals or his enemies.

Many persons are so constituted, or so trained, as to have no relish for poetry, or painting, or music. So much the worse for them, perhaps, since their want of feeling or imagination deprives them of sources of innocent pleasure open to others. If a mere mathematician should be dissatisfied with the works of a great poet, because these works prove nothing mathematically, a lover of poetry must not take offence at the mathematician. The lover of poetry, perhaps himself a poet, may be totally insensible to the beauties of the most profound mathematical reasonings, or the finest musical compositions. This often happens. But it rarely happens that the real lover of music is not also a lover of poetry and of painting. We have known men high in the literary and scientific world, upon whom the best music produced no other impression than that of an agreeable or a disagreeable noise. But this never made us respect them the less for their own peculiar powers of feeling and thinking. They were not so organized as to feel and think as we did. That was ill. The wiser and more philosophical plan is, not to be angry with any of our fellow-creatures for not feeling and admiring as we do; but to regret that they cannot feel and admire with us, because such communion of feeling and admiration would serve to draw those persons closer to us in human fellowship. To call a man a blockhead, because he does not, or cannot, feel and think in every thing exactly like ourselves, is merely to be at once ill natured, uncharitable, and unphilosophical. It is, "not to know ourselves." Non omnia possimus omnes. We find no fault with men who cannot perceive the beauties of music. We find fault with the perversions of an art which we ourselves feel to be a fine and expressive one, too often deformed and perverted. Most treatises on this subject begin with a definition of music. To persons who already understand music thoroughly, any attempt at such a definition is unnecessary. They have already formed their own ideas of music as an art. To persons ignorant of music, any such definition is quite unintelligible. The extent, the complexity, and the mutability of the art, render all such definitions imperfect and objectionable. The best way in this, as in all others of the fine arts, is to leave the student to form his own definition, after he has thoroughly studied the art. We follow this plan.

In some of the latest and best works on music, we find a definition of it attempted in this manner: "The art of expressing an agreeable play of feelings by means of sounds." But music often expresses the most painful and tragic feelings. Another is, "The art of expressing determinate feelings by means of regulated sounds." And then follows a long description of the nature of all the various branches of music; which is just tantamount to a confession that the definition is unintelligible and useless without the lengthy description.

Leibnitz had a strange metaphysical notion of music, which he thus expressed: "Musica est exercitium arithmetice occultum nescientis se numerare animi; multa enim facit in perceptionibus confusis seu insensibilibus, que distincta, appercepctione notare nequit. Errant enim, qui nihil in anima fieri putant, cujus ipsa non sit conscia. Anima igitur etsi se numerare non sentiat, sentit tamen hujus numerationis insensibilis effectum, seu voluptatem in consonantia, molestiam in dissonantia inde resultante. Ex multis enim congruentiis insensibilibus oritur voluptas," &c. Descartes entertained similar notions; and Euler, in his Tentamen Novae Theoriae Musicae, assures us that the ear is pleasingly or unpleasingly affected by musical intervals, according to its perception of the simplicity or the complexity of their ratios of vibration. His measures of these ratios do not agree with practice. But the absurdity consists in supposing such an auricular arithmetic, by which the ear judges of the ratios of intervals. Does the milk-maid calculate the ratios of the intervals in her untutored song, and take pleasure in it, or the reverse, according to her perception of their simplicity or complexity? In Italy we may hear persons who cannot read music, singing very agreeably in two, or three, or four parts, in harmony. Do such persons know anything of the harmonic ratios of the sounds they combine together in this way? They have no more idea that even an octave is in the ratio of 1:2, than they have of the distance between the earth and the moon. Similar false applications of mathematics have tended greatly to produce that mysterious obscurity which has hitherto been artificially thrown over the beautiful and inviting regions of musical melody and harmony. There, genius and perseverance have called the sweetest flowers; while mathematical investigations have, as yet, only groped among the soil from which these blossoms sprang.

The state of our knowledge of acoustics, one of the most subtle and difficult of sciences, is still too incomplete to permit of the formation of a perfect theory of music, even were music, as a fine art, entirely dependent upon the physico-mathematical science of acoustics, which it is not. Of late years, however, the beautiful experiments of Dr Chladni, M. Oersted, Monsieur Savart, Professor Faraday, and Professor C. Wheatstone, have thrown much light upon some of the obscurer parts of acoustics.

In another work, we have expressed ourselves in the following terms regarding proposed theories of music:

"The mischievous effects of false principles have been experienced in every branch of physical science. The blind rashness of premature generalization has operated with as great absurdity in music as in any other branch of human knowledge. While music was in its infancy, and while the observations and experiments which had been made respecting it were confined within limits by much too narrow to permit the formation of just and comprehensive general principles, musicians, both practical and speculative, misled by a false philosophy, and by erroneous ideas of simplicity, attempted to establish one single principle as the sole basis of musical harmony and composition. Confounding together the essentially distinct methods proper to physical and to mathematical science, they seized upon a particular phenomenon belonging to acoustics, and endeavoured to torture it into a principle which might apply to, and explain, the whole phenomena belonging to musical composition. From a particular fact, which had no necessary connection with musical composition, they attempted, with some ingenuity, and with much sophistry and ineffectual labour, to deduce the whole system of that art; while they were not aware either of the imperfection and incompleteness of the system which then existed, or of the improper method of induction which they had adopted. They employed the synthetical method of induction proper to mathematics, instead of the analytical method of induction, which is the true guide to physical investigation. In mathematics, we make discoveries by reasoning from definitions, axioms, and postulates; in other words, by reasoning from generals to particulars; but in physics, we extend our views and consolidate our knowledge by the opposite method of reasoning from particulars to generals. In physical science, when our observations and experiments have been sufficiently numerous and extensive, we may then, but not till then, establish general laws, or first principles, and reason from these synthetically; but if, on the contrary, the facts from which we generalize have been gathered from a narrow and unenlightened survey of the field of physical science, we shall almost inevitably draw false conclusions, and form principles which involve error and absurdity in relative proportion to the obscurity and contraction belonging to our investigation of particulars.

"It was long ago observed, that a musical string, or wire, capable of rendering a grave and powerful sound when thrown into a state of vibration, produced, in that state, not only a principal sound, corresponding to its length, tension, thickness, &c., but also two audible, concomitant, and accessory sounds, related to the principal sound by the intervals of a twelfth, or double (replicate) fifth, and seventeenth major, or second replicate major third. For example, the fourth string, or largest string of the violoncello, when strongly vibrating, may produce these accessory sounds, or harmonies; which, although feeble in comparison with the principal sound, may, however, be heard by a delicate and attentive musical ear.

"Upon this acoustical phenomenon, Rameau, a French musician, attempted to found his theory of harmony. We shall afterwards quote the opinions of some of the highest authorities in Europe upon this theory, and also upon that of Tartini, to which we now proceed.

"Tartini, in his Trattato di Musica, published at Padua in 1754, informs us, that if two sustained sounds (forming, for example, a third or a fifth) are produced at once from two violins, two trumpets, &c. the result will be the generation of a harmonic third sound, distinctly perceivable by the ear. This phenomenon was observed by Rameau in 1753. Tartini seems to have mistaken the pitch of the third sound, or grave harmonic, produced in this experiment, since M. Serre of Geneva, in his Principles of Harmony, tells us that the grave harmonic sounds produced by major and minor thirds are each an octave lower than those intended by Tartini. This phenomenon gave rise to Tartini's theory of harmony. We now make the quotations which we promised. The first is from the works of..." the late Professor Robison, whose authority, on such a point, is of indubitable weight. He is writing of Rameau's theory. 'Rameau has made this,' the generation of acute harmonics, the foundation of his system of music, asserting that the pleasure of harmony results from the successful imitation of this harmony of nature. But a little logic should convince these theorists that they must be mistaken. A little mathematics, too, or mechanics, would have convinced them. His theory is a very forced accommodation of this principle to the practice of musicians and taste of the public.' Speaking further of Rameau's theory, he says, 'It is a mere whim, proceeding on a false assumption, namely, that a musical sound is essentially accompanied by its octave, twelfth, and seventeenth, in allto. This is not true, though such accompaniment be very frequent, &c. Are these acute harmonics musical sounds or not? He surely will not deny this. Therefore they too are essentially accompanied by their harmonics, and this absolutely and necessarily ad infinitum.' Of Tartini's theory he says, 'Tartini prized this observation, the generation of grave harmonics, as a most important discovery, and considered it as affording a foundation for the whole science of music.' After some farther remarks, he adds, 'The system of harmonious composition which Tartini has, with wonderful labour and address, founded on it, has, therefore, no solidity.'

Dr Chladni, in his celebrated work Traité d'Acoustique, expresses himself as follows regarding the theories which we have just mentioned: 'It is not conformable to nature to desire, like many authors, to derive all harmony from the vibrations of a string, and especially from the co-existence of several sounds with the fundamental sound. A string is only one species of sonorous body.

In many other sonorous bodies the general laws of vibrations, which were not known, are differently modified, consequently the laws of one sonorous body cannot be applied to that which ought to be common to all. A monochord cannot serve to establish the principles of harmony, but only to give an idea of the effect of ratios...... Many authors have regarded the co-existence of sounds comprised in the natural series of numbers (which, according to true principles, is nothing but a particular phenomenon) as an essential difference between a distinct sound and a noise. They have taken this quality for the basis of all harmony, believing that an interval is consonant, because the acute sound may be heard along with the fundamental sound. They do not know that, if more than one sound is heard at the same time, this is nothing more than a consequence of the existence of many species of vibrations; that in many sonorous bodies the series of possible sounds is very different from the natural series of numbers; and that we may produce each manner of vibrations, where there are nodes, without any mixture of other sounds, by touching the nodal points, or lines, which ought to be in motion in other manners of vibrating.

According to their principles, the perfect minor chord—if one does not make use of sophisms—would not be consonant; and, on the bell of the harmonica, the ninth (4:9) would be the first consonance, since it is the first sound which can mingle itself with the fundamental sound, &c. Daniel Bernouilli and Lagrange have sufficiently refuted these false principles.'

With respect to Tartini's theory in particular, he says, 'Tartini pretended that this third sound was more acute by an octave than it really is. He regarded this phenomenon, combined with the pretended co-existence of the series of sounds 1, 2, 3, 4, 5, &c. in each fundamental sound, as the basis of harmony. Mr Mercadier de Belesta has very well refuted some false assertions of Tartini, in his Système de Musique, Paris, 1776.'

Choron, in his work upon composition, says, 'It has been attempted to deduce the laws of succession from the multiples resonance, or from the sub-multiple resonance. Tartini had hardly discovered this last phenomenon, when he hastened, in order to satisfy the taste of his time, to rear up upon it a system, which he gave to the world in a very unintelligible work. J. J. Rousseau, who was almost equally a stranger to geometry and to the science of composition, produced, without having even comprehended it, a very imperfect analysis of it (Tartini's system) in his shapeless dictionary, and exalted it to the utmost of his power, for the pleasure of mortifying Rameau, with whom he had some quarrel.

With the phenomenon of the multiple resonance, of which he had considered no more than the three first terms, Rameau had propped up his system of the fundamental bass. Without entering more into detail, I shall remark, that this phenomenon has no connection with the laws of harmony. That if one absolutely would apply it to them, it would be necessary, first, in order to be consistent, to suppose, at least implicitly, that the sounds of the system are those of the series of aliquots: first absurdity. Second, That all the notes of the bass ought to be accompanied by all their aliquots, moving in a parallel manner with each other: second absurdity. Every other consequence is illegitimate, and tends, not to give a foundation in nature to the rules of harmony, but to reconcile, as one best can, the phenomena with the rules of harmony, which is a very indifferent matter.'

In 1753, M. Serre, a miniature painter at Geneva, published his Essais sur les Principes de l'Harmonie. He assumed three essential fundamentals in the scale; the tonic, the fifth, and the fourth. He described the nature and use of what he termed diacomatic intervals, or slides necessary to perfect intonation in various modulations; and he laid down as a principle, that it depended upon the nature of the intervals of a chord whether that chord should have one or two, or even three fundamentals. These opinions of M. Serre's have been of late years, and with some modifications, reproduced as new. In some works recently published, we have observed an analogy pointed out, as new, between the harmonics above mentioned and the curious phenomena of complementary colours. In Blackwood's Edinburgh Magazine for February 1823 (pp. 159-162), will be found a letter of ours in Italian, in which this analogy is particularly noticed, and a short description of some of the phenomena given. In the same letter there are some remarks upon the analogy between the harmonic series 1, 3, 5, 7, &c. and the progression of numbers 1, 3, 5, 7, ascribed by Newton to the squares of the diameters of the coloured circles produced by him on applying to the plane side of a plano-convex lens one of the convex sides of a double convex lens.

A strange error has long prevailed regarding the co-existent vibrations of a musical string. The total vibration which gives the gravest sound of the string, can by no means co-exist with the vibrations of the aliquot parts of the same string. The thing is physically impossible, as could be easily demonstrated. In fact, to assert that a vibrating string can move in a number of different and opposite directions at the same instant of time, is as absurd as to maintain that a man can run backwards and forwards, to the right and to the left, &c. all at the same moment.

The following diagrams represent the three primary curves of the harmonic series 1, 2, 3. The co-existence of all these curves is a physical impossibility. For how can ACB coincide with AdefB, or with AfgdB? It is needless to go further. There may be many co-existent vibrations of traction and torsion in the string, but not any co-existent vibrations in directions quite opposite to each other.

The musical treatises of Choron, Catel, and Momigny, &c., among the French, and of Reicha and others among the Germans, are still too much tinctured with peculiar and arbitrary theories and systems, for which there are no sufficient grounds in either acoustics or aesthetics. By this last term the Germans have long chosen to designate, not very appropriately, the theory of taste in the fine arts. It is indeed impossible, by any purely mechanical and mathematical theories, or even by any metaphysical ones, to explain all the varieties of human sensations, affections, passions, that enter into our perceptions of beauty, sublimity, &c., in poetry, painting, or music. It cannot be too often repeated, that all the rules laid down by theorists for the construction of works belonging to the fine arts, are drawn from models of art previously in existence, and relate merely to the mechanical portions of these arts.

Had the rigid rules formed for (and from) the ancient Greek drama been always adhered to, we should never have possessed Shakespeare's plays. The magnificent musical works of Haydn, Mozart, and Beethoven, not to speak of many other great German and Italian composers, were not produced by blind adherence to old rules of art, but by an enlightened view of things, far beyond what the authors of these rules contemplated. Büllé has remarked, that the mechanical rules laid down in treatises on the fine arts may be compared to telescopes, which assist the vision of those persons who already see. A remarkable instance of this is found in the case of Beethoven, who happened to be placed under a master destitute of genius for melody; but a profound harmonist, and a learned writer of fugues and canons, &c. Under this man, Beethoven laboured most industriously, and went through the whole drudgery of thorough bass, and all the rigid ancient rules of composition; but evidently with frequent misgivings as to the general truth and application of what was taught to him. But the result was, that these lessons and rules served him as a "telescope," to enable him to perceive a wide field of composition far beyond them all. In short, he was a man of first-rate musical genius, and therefore by nature a great melodist; and, fortunately for the world, his injudicious training could not extinguish his passionate feeling for melody, and his charming expression of it in his best works. In some of his works, especially among his last, we find unpleasing traces of the predominance of his early training over his native genius. But his latest works were composed when he had been for many years perfectly deaf.

Notwithstanding the laborious investigations of many eminent anatomists and physiologists, from Comparetti downwards to Magendie, the uses and functions of all the various parts that compose the human ear are by no means well understood.

The perceptive powers of the ear differ considerably among mankind, especially as regards the perception of the various qualities and relations of musical sounds. In like manner, we find that the perceptions of form, proportion, colour, &c., are by no means always the same in every human eye. Perfection of the eye is requisite to the painter; perfection of the ear to the musician. Sometimes persons are found who cannot distinguish colours, or shades of colour, from each other. Perhaps more frequently instances are met with of persons whose perceptions of the differences between musical sounds are very imperfect. We have been informed that Mr. Pond, the late astronomer royal, though a real lover of music, and capable of hearing distinctly sounds of a grave pitch, or of a middle pitch, was incapable of hearing very acute sounds, whether musical or not, which were perfectly audible to other persons; for example, the loud chirping of a number of crickets in a room, and the very piercing sound produced by turning round the ground-glass stopper of a bottle containing calomel. The stopper was turned round close to Mr. Pond's ear without producing any sensation. All this clearly proved that Mr. Pond's ear, however perfect in other respects, was incapable of conveying any perception of very acute sounds.

All vibrations sufficiently rapid and powerful to act upon the auditory organs produce the sensation of sound. To enable us to hear slow vibrations as well as more rapid ones, it would be necessary, according to Riccati, that the intensity of each simple vibration should be in proportion to its duration. For this reason, says Chladni, and on account of the different organization of each individual, and each kind of animal, there exist no absolute limits to the perceptibility of sounds.

The ear does not distinguish the very small differences of the exact ratios between sounds. Were it not for this illusion, music would have no existence. But this is not to make us seek the less for true intonation, wherever it can be obtained. Few persons are aware how great is the difference between the true intonations of a fine voice, or a violin, &c., and the false intonations of such instruments of fixed sounds as the organ, piano-forte, &c. Many singers, trained to the intonation of a piano-forte, have their ear and voice so misled that they can never afterwards learn to sing in tune. The famous Madame Mara condemned the use of the piano-forte in learning to sing. She said every singer ought to learn to play on the violin, in order to know what true intonation is.

The different quality (timbre) of a musical sound and its articulations, says Chladni, are among the most remarkable objects of audition. They do not appear to depend on the manner of vibration, nor (or very little) upon the form of the sonorous body; but rather on the matter of the sonorous body, and that of the body by which it is rubbed or struck, as well as on the matter which propagates the sound. We have not the least idea of the nature of these different characters of sound, nor of their propagation.

The limits usually assigned to musical sounds, reckoning from grave to acute, or the contrary, are as follows:

Two octaves higher than written.

Two octaves lower than written.

The lowest of these sounds will be such as is produced by an open organ-pipe of 32 feet in length, and the num- ber of vibrations of the reed will be 32 in a second of time. The next octave above will be produced by an open organ-pipe of 16 feet, and the number of vibrations in a second will be 64; the next octave above that, pipe 8 feet, and 128 vibrations; and so on. The highest sound above noted will have 16,384 vibrations in a second. This last sound is not to be taken absolutely as indicating the extreme limit of acute sounds that may be used in music, and may be appreciated by the human ear; for it has been calculated that a sound produced by 24,000 vibrations in a second is appreciable. We shall give, in a wood cut, p. 618, a copy of a very useful table of the compass of voices and instruments, published by Monsieur Choron, in his large and expensive work upon composition. As all the plates of that work were destroyed some years ago, copies are now extremely rare and valuable.

If we take a vibrating musical string or wire, perfectly uniform in thickness, and homogeneous throughout, and divide it into its aliquot parts, its half, its third, its fourth, its fifth, and so on, we shall obtain, by this division of the monochord, as it is called, a great many of the sounds belonging to our musical system. A number of these sounds can be obtained from it by lightly touching it at these divisions, as happens when we produce harmonics on the open string of a violin, &c.; and all these sounds are true, or nearly true, if the string is perfect; otherwise they are not. This frequent imperfection in the uniformity and homogeneity of strings is one great obstacle to perfect intonation. Again, if we take an organ-pipe, or a French horn, &c., and blow into it in such a manner as to produce its natural series of sounds, we shall have, beginning with the lowest, a series corresponding (in ratio of vibrations of the column of air contained in it) to the arithmetical series 1, 2, 3, 4, 5, 6, 7, 8, &c.; thus:

These sounds also are true harmonics, supposing the sonorous tube properly constructed, and the force of the blast suitably regulated. If we push this harmonic generation of sounds still farther, we may obtain a number of other sounds, some of which, though apparently false as regards our artificial temperament on instruments of fixed sounds (such as the organ, piano-forte, &c.), are yet true, or nearly so, as regards the intervals which occur in true intonation.

There is no room here to enter into a discussion of this curious and intricate subject. We shall content ourselves with giving a table of the harmonics obtainable from an open cylindrical glass tube furnished with a suitable mouthpiece, and fitted to an organ-bellows. This table shows that the gravest sounds obtainable from the tube are removed from each other by wide intervals. Thus the two first sounds, C₁ and C₂, are separated by the interval of an octave. G₃ is a fifth above C₂, and C₄ is a fourth above G₂, and so on. This is the series followed by the natural sounds of such instruments as the horn, trumpet, serpent, &c.; but it is extremely difficult, or nearly impossible, to produce on these instruments the sound corresponding to C₁. Even C₂ is very difficult to produce; and the first sound that usually occurs corresponds with G₂. As the sounds become more acute, that is, as the column of air divides itself into a greater number of parts, they approximate each other more and more. By and by, chromatic intervals occur, represented by flats and sharps; and, at last, intervals so small that they cannot be represented by any of the common signs of musical notation. But these smaller intervals are necessary to perfect musical intonation, and are employed by the best singers and performers on instruments of the violin kind. In France, a number of experiments were tried with Vioti's performance, and it was ascertained that he employed a vast number of very minute intervals, in order to play perfectly in tune in all keys.

In treating of the musical sounds produced by sonorous bodies, such as vibrating strings, or wires, or springs, or columns of air in tubes, &c., it is rarely kept in view that in these, as in all mechanical phenomena, allowance must be made for the mechanical conditions which may render the actual phenomena not exactly correspondent to the mathematically calculated results. From want of attention to this, many false theories of musical intonation have been adopted. It is quite true, mathematically speaking, that if all the hypothetical conditions of a sonorous body were, as they are assumed to be, in the course of a mathematical reasoning regarding them, the physico-mathematical result deduced by such reasoning, supposing this reasoning accurate, must be perfectly correct. But, in general, such reasoning and deduction are carried on with abstraction made of some of the inevitable physical circumstances which attend the real phenomena. In this way, pure mathematical reasoning is often, in some degree, at variance with mechanical phenomena. A badly formed string, or wire, &c. will not conform to the mathematical calculations as to the sounds that it must produce when divided in such and such ways. Neither will its real vibrations agree with those mathematically calculated. In like manner, the friction, and inequalities and imperfections, of any piece of machinery, will, in the real operations of the latter, produce results in some respects contradictory to the abstract mathematical theory of what the operations of the mechanism ought to be.

A musical interval consists in the difference between two given sounds, in respect to their relative acuteness and intensity. Thus it is evident that the unison is not an interval, although it is often improperly so called. Aristotle, in the tenth section of his thirty-ninth problem, very correctly designates the unison as being "only the same sound multiplied." But the slightest departure from unison, by one of the sounds becoming a little more acute, or the other a little more grave, forms an interval, though it may be so very small as not to belong to those intervals generally recognised in melody and harmony. The measure of the relative lengths, or vibrations, of two musical strings producing an interval, will be the difference of their respective logarithms, as has been remarked by Dr Smith in his "Harmonics," and by various other subsequent writers. Among these, the late Professor Robison, of Edinburgh, University, pointed out some useful applications of the logarithmic subdivision of the circumference of a pasteboard circle, fitted with a moveable concentric circle, &c., as described in his article Temperament, in the present work. He adds: "Or a straight line may be so divided, and repeated thrice; then a sliding ruler, divided in the same manner, and applied to it, will answer the same purpose."

We may remark, that these suggestions of Professor Robison have been employed in the construction of similar instruments, without any acknowledgment; and also, that Professor Robison's experiment, by applying a stop-cock to an organ-pipe, and producing various sounds from the regular and rapid opening and shutting of the stop-cock, bears great analogy to the syren instrument recently constructed by M. Cagniard de la Tour. Professor Robison, speaking of his stop-cock apparatus, says, "The intelligent reader will see here an opening made to great additions to practical music, and the means of producing musical sounds, of which we have at present scarcely any conception," &c. We have already mentioned, very briefly, how intervals are produced by the subdivisions of a sonorous string or wire, &c., or of the column of air in a wind-instrument, or in the glass tube before described.

If we suppose such a tube to produce, as its gravest sound, its primary harmonics will be

viz. octave, replicate of fifth, and replicate of third. Carrying the series farther, we shall have one similar to that already given in the table of harmonics. Supposing two other such tubes, the one having for its gravest sound , and the other , the harmonics resulting from these respectively will be , and , and so on, as in the case of the first.

By bringing closer together these dispersed primary harmonics, by means of their octaves above or below, we shall obtain the following series of sounds:

and so on. If to the last of these series we add at the top, we shall have the major diatonic scale of C. By carrying farther the series of harmonic products of these three tubes, we shall obtain a number of other intervals (see the table), and among these the ones suitable to the scale of C minor:

The subject of intervals has been involved in frightful confusion by the number and complexity of names introduced, and the contradictory statements of various writers upon music. We have no space to devote to the clearing up of this chaos; but we may remark, in general, that in this matter, as well as in many others connected with music, there is great need of a reformation in terminology. To glance at only three or four of the misnomers in daily use, without meddling with the more abstruse terms: the fifth diatonic sound of an ascending scale is called the dominant of the scale, which signifies the ruling or governing sound; while, in fact, the tonic, or key-note, and no other, is the chief and ruling sound of the scale, the sound from which, as a common centre, all the others of the scale may diverge, or to which they may converge, like the radii of a circle. This use of the term dominant seems to have arisen from the predominance of the fifth of the key in ancient church-chants. The term subdominant is improper, too, as applied to the diatonic fourth of a scale. The terms double octave, double third, double fourth, &c., are wrong as applied to intervals, because they imply that these intervals are doubled in the unison, while it is meant to express only the acuter or graver replicates of these intervals in the octave. The same with regard to triple octave, &c. When we read of diminished or augmented unisons, false fifths, superfluous fifths, seconds, &c., &c., we must regret the obscurity of such terms; but meantime we shall use the common terms as we find them, because to introduce new ones abruptly would only add to the confusion already existing.

Table of Intervals in general use.

Inversions of the above Intervals.

Inversion of an interval takes place when the graver sound is carried an octave higher, or the acuter an octave lower. The effect of inversion of intervals may be represented by the following two rows of figures, where it will be seen that 1 or unison, becomes 8 or octave; 2, or second, becomes 7 or seventh, and so on.

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---| | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |

By looking at the above table of intervals, and their inversions, we perceive that minor intervals inverted become major, and major, minor; diminished intervals become augmented, and augmented, diminished. We have placed Db before Cz, and Eb before Dz, and so on; because, contrary to the common opinion, the Db in the above series is a graver sound than the Cz, and the Eb than the Dz, and so of the others. This is not easily understood by a mere player on the organ or piano-forte, but can be exemplified by any accomplished singer or violinist.

In writing for voices, especially in the strict or serious