When the considerate reader reflects on the large and almost numberless dissertations on this subject, by the most eminent philosophers, mathematicians, and artists, both of ancient and modern times, and the important points which divided, and still divide, their opinions, he will not surely expect, in a Work like our's, the decision of a question which has hitherto eluded their researches. He will rather be disposed, perhaps, to wonder how a subject of this nature ever acquired such importance in the minds of persons of acknowledged talents (for surely no person will refuse this claim to Pythagoras, to Aristotle, Euclid, Ptolemy, Galileo, Wallis, Euler, and many others, who have written elaborate treatises on the subject); and his surprise will increase, when he knows that the treatises on the scale of music are as numerous and voluminous in China, without any appearance of their being borrowed from the ingenious and speculative Greeks.

The ingenious, in all cultivated nations, have remarked the great influence of music; and they found no difficulty in persuading the nations that it was a gift of the gods. Apollo and his sacred choir are perhaps the most respectable inhabitants of the mythological heavens of the Greeks. Therefore all nations have considered music as a proper part of their religious worship. We doubt not but that they found it fit for exciting or supporting those emotions and sentiments which were suited to adoration, thanks, or petition. Nor would the Greeks have admitted music into their serious dramas, if they had not perceived that it heightened the effect. The same experience made them employ it as an aid to military enthusiasm; and it is recorded as one of the respectable accomplishments of Epaminondas, that he had the musical instructions of the first masters, and was eminent as a performer.

Thus was the study of music ennobled, and recommended to the attention of the greatest philosophers. Its cultivation was held an object of national concern, and its professors were not allowed to corrupt it in order to gratify the fastidious taste of the luxurious or the sensualist, who sought from it nothing but amusement. But its influence was not confined to these public purposes; and, while the men of speculation found in music an inexhaustible fund of employment for their genius and penetration, and their poets felt its aid in their compositions, it was hailed by persons of all ranks as the foother of the cares and anxieties, and sweetener of the labours of life. O Phoebi decus laborum dulce lenimen. Poor Ovid, the victim of what remained of good in the cold heart of Octavius, found its balm.

Exul eram (says he): requiesque mihi, non fama petita est. Mentis intenta fuit ne foret ulque malis. Hoc est cur cantet victoriae quoque compede fassor. Indolentis numero cum grave molit opus. Cantat et innatens limosa proua arena. Adverso tardum qui trahit amne catem, Quoque ferens pariter lentos ad pictura remos, In numerum pulsa brachia werfat aquas. Fefuis ut incubat baculo, faxove rufedit. Pafior; arundines carmine mulct ovem. Cantantis pariter, pariter data pensa trabentis. Fallitur ancilla, decipiturque labor.

It is chiefly in this humble department of musical influence that we propose at present to lend our aid; music. It requires sufficient for informing the reader of what is received as the scale of music, and the inequality of its different flaps, the tones major and minor, fennone, comma, &c. We shall only observe, that what is there delivered on temperament by M. d'Alembert, after Rameau, bears the evident mark of uncertainty or want of confidence in the principle adopted as the rule of temperament; and we have learned, since the printing of that article, that the instructions there delivered have not that perspicuity and precision that are necessary for enabling a person to execute the temperament recommended by Rameau; that is, to tune a keyed instrument with certainty, according to that system or construction of the scale.

If such be the case, we are in some measure disappointed; because we selected that treatise of D'Alembert as the performance of a man of great eminence as a mathematician and philosopher, aiming at public instruction more than his own fame, by this elementary abstract of the great work of the most eminent musician in France.

To be able to tune a harpsichord with certainty and few inaccuracies, seems an indispensable qualification of any person who is worthy of the name of a musician. It would certainly be thought an unpardonable deficiency in a violin performer if he could not tune his instrument; yet we are well informed, that many professional performers on the harpsichord cannot do it, or cannot do it any other way than by uncertain and painful trial, and, as it were, groping. groping in the dark; and that the tuning of harpsichords and organs is committed entirely to tuners by profession. This is a great inconvenience to persons residing in the country; and therefore many take lessons from the professed harpsichord tuners, who also profess to teach this art. We have been present during some of these lessons; but it did not appear to us that the instructions were such as could enable the scholar to tune an instrument when alone, unless the lessons had been so frequent as to form the ear to an instantaneous judgment of tune by the same habit that had instructed the teacher. There seemed to be little principle that could be treasured up and recollected when wanted.

Yet we cannot help thinking that there are phenomena or facts in music, sufficiently precise to furnish principles of absolute certainty for enabling us to produce temperaments of the scale which shall have determined characters, and among which we may choose such a one as shall be preferable to the others, according to the purposes we have in view; and we think that these principles are of such easy application, that any person, of moderate sensibility to just intonation, may, without much knowledge or practice in music, tune his harpsichord with all desirable accuracy. We propose to lay these before the reader. We might content ourselves with simply giving the practical rules deduced from the principles; but it is surely more desirable to perceive the validity of the principles. This will give us confidence in the deduced rules of practice.

In the employment of sacred music, an inspired writer counsels us to sing, not only "with the heart, but with the understanding also." We may, without irreverence, recommend the same thing here. Let us therefore attend a little to the dictates of untutored Nature, and see how she teaches all mankind to form the scale of melody.

It is a most remarkable fact, that, in all nations, however they may differ in the structure of that chant which we call the accent, or tone, or twang, in the colloquial language of a particular nation, or in the favourite phrases or passages which are most frequent in their songs, all men make use of the same rises and falls, or inflexions of voice, in their musical language or airs. We have heard the songs of the Iroquois, the Cherokee, and the Esquimaux, of the Carib, and the inhabitant of Paraguay; of the African of Negroland and of the Cape, and of the Hindoo, the Malay, and the native of Otaheite—and we found none that made use of a different scale from our own, although several seemed to be very sorry performers by any scale. There must be some natural foundation for this uniformity. We may never discover this; but we may be fortunate enough to discover facts in the phenomena of sound which invariably accompany certain modifications of musical sentiment. If we succeed, we are entitled to suppose that such inseparable companions are naturally connected; and to conclude, that if we can infer the appearance of those facts in sound, we shall also give occasion to those musical sentiments or impressions.

There is a quality in lengthened or continued sound which we call its pitch or note, by which it may be accounted shrill or hoarse. It may be very hoarse in the beginning, and during its continuance it may grow more and more shrill by imperceptible gradations. In this case we are sensible of a kind of progress from the one state of sound to the other. Thus, while we gently draw the bow across the string of a bass viol, if we at the same time slide the finger slowly along the string, from the nut toward the bridge, the sound, from being hoarse, becomes gradually acute or shrill. Hoarse and shrill therefore are not different qualities, although they have different names, but are different states or degrees of the same quality, like cold and heat, near and far, early and late, or, what is common to all these, little and great. A certain state of the air is accounted neither hot nor cold. All states on one side of this are called warm, or hot; and all on the other are cold. In like manner, a certain sound is the boundary between those that are called hoarse and those called shrill. The chemist is accustomed to say, that the temperature of a body is higher when it is warmer, and lower when colder. In like manner, we are accustomed to say, that a person raises or depresses the pitch of his voice when it becomes more shrill or more hoarse. The ancient Greeks, however, called the shriller sounds low, and the hoarser sounds high; probably because the hoarser sounds are generally stronger or louder, which we are also accustomed to consider as higher. In common language, a low pitch of voice means a faint sound, but in musical language it means a hoarser sound. The sound that is neither hoarse nor shrill is some ordinary pitch of voice, but without any precise criterion.

The change observed in the pitch of a violin string, when the finger is carried along the finger-board with a continued motion, is also continuous; that is, not by steps: we call it gradual, for want of a better term, although gradual properly means gradation, by degrees, steps, or starts, which are not to be distinguished in this experiment. But we may make the experiment in another way. After sounding the open string, and while the bow is yet moving across it, we may put down the finger about \( \frac{1}{2} \) inches from the nut. This will change the sound into one which is sensibly shriller than the former, and there is a manifest start from the one to the other. Or we may put down the finger \( \frac{2}{3} \) inches from the nut; the sound of the open string will change to a shriller sound, and we are sensible that this change or step is greater than the former. Moreover, we may, while drawing the bow across the string, put down one finger at \( \frac{1}{2} \) inches, and immediately after, put down another finger at \( \frac{2}{3} \) inches from the nut. We shall have three sounds in succession, each more shrill than the preceding, with two manifest steps, or sublunary changes of pitch.

Now since the last sound is the same as if the second had not been sounded, we must conceive the sum of the two successive changes as equivalent or equal to the change from the first to the third. This change seems somehow to include the other two, and to be made up of them, as a whole is made up of its parts, or as \( \frac{1}{2} \) inches are made up of \( \frac{1}{2} \) and \( \frac{1}{2} \) of an inch, or as the sum 15 is made up of 10 and 5.

Thus it happens that thinking persons conceive something like or analogous to a distance, or interval, between these sounds. It is plain, however, that there can be no real distance or space interposed between them; and it is not easy to acquire a distinct notion of the bulk or magnitude of these intervals. This conception is purely figurative and analogical; but the analogy is very good, and the observation of it, or conjecture It must now be remarked, that it is in this respect alone that sounds are susceptible of music. Nor are all sounds possessed of this quality. The smack of a whip, the explosion of a musket, the rushing of water or wind, the scream of some animals, and many other sounds, both momentary and continuous, are mere noises; and can neither be called hoarse nor shrill. But, on the other hand, many sounds, which differ in a thousand circumstances of loudness, smoothness, mellowness, &c., which make them pleasant or disagreeable, have this quality of musical pitch, and may thus be compared. The voice of a man or woman, the sound of a pipe, a bell, a string, the voice of an animal, nay, the single blow on an empty cask—may all have one pitch, or we may be sensible of the interval between them. We can, in all cases, tighten or slacken the string of a violin, till the most unformed hearer can pronounce with certainty that the pitch is the same. We are indebted to the celebrated Galileo for the discovery of that physical circumstance in all those sounds which communicates this remarkable quality to them, and even enables us to induce it on any noise whatever, and to determine, with the utmost precision, the musical pitch of the sound, and the interval between any two such sounds. Of this we shall speak fully hereafter; and at present we only observe, that two sounds, having the same pitch, are called unisons by musicians, or are said to be in unison to one another.

When two untaught men attempt to sing the same air together, they always sing in unison, unless they expressly mean to sing in different pitches of voice. Nay, it is an extremely difficult thing to do otherwise, except in a few very peculiar cases. Also, when a man and woman, wholly uninstructed in music, attempt to sing the same air, they also mean to sing the same musical notes through the whole air; and they generally imagine that they do so. But there is a manifest difference in the sounds which they utter, and the woman is said to sing more shrill, and the man more hoarse. A very plain experiment, however, will convince them that they are mistaken. N.B. We are now supposing that the performers have so much of a musical ear, and flexible voice, as to be able to sing a common ballad, or a psalm tune, with tolerable exactness, and that they can prolong or dwell upon any particular note when desired.

Let them sing the common psalm tune called St. David's, in the same way that they practise at church; and when they have done it two or three times, in order to fix their voices in tune, and to feel the general impression of the tune, let the woman hold on in the first note of the tune, which we suppose to be G, while the man sings the first three in succession, namely G, D, G. He will now perceive, that the last note sung by himself is the same with that sung by the woman, and which she thinks that she is still holding on in the first note of the tune. Let this be repeated till the performance becomes easy. They will then perceive the perfect sameness, in respect of musical pitch, of the woman's first note of this tune and the man's third note. Some difference, however, will still be perceived; but it will not be in the pitch, but in the smoothness, or clearness, or other agreeable quality of the woman's note.

When this is plainly perceived, let the man try by temperament continued steps he must raise his pitch, in orderment of the scale of accustomed to common ballad singing, he will have no great difficulty in doing this; and will find that, beginning with his own note, and fingering gradually up, there are his eighth note will be the woman's note. In short, if the two flutes be taken, one of which is twice as long as the other, and if the man sing in unison with the larger flute, the woman, while fingering, as she thinks, the same notes with the man, will be found to be fingering in unison with the smaller flute.

This is a remarkable and most important fact in the phenomena of music. This interval, comprehending and made up of seven smaller intervals, and requiring eight sounds to mark its steps, is therefore called an octave. Now, since the female performer follows the same dictates of natural ear in fingering her tune that the man follows in fingering his, and all hearers are sensible that they are singing the same tune, it necessarily follows, that the two series of notes are perfectly similar, though not the same: For there must be the same interval of an octave between any step of the lower octave and the same step of the upper one. In whatever way, therefore, we conceive one of these octaves to be parcelled out by the different steps, the partition of both must be similar. If we represent both by lines, these lines must be similarly divided. Each partial interval of the one must bear the same relation to the whole, or to any other interval, as its similar interval in the other octave bears to the whole of that octave, or to the other corresponding interval in it.

Farther, we must now observe, that although this similarity of the octaves was first observed or discovered by means of the ordinary voices of man and woman, and is a legitimate inference from the perfect satisfaction that each feels in fingering what they think the same notes, this is not the only foundation or proof of the similarity. Having acquired the knowledge of that physical circumstance, on which the pitch of musical sounds depends, we can demonstrate, with all the rigour of geometry, that the several notes in the man and woman's octave must have the same relation to their respective commencements, and that these two great intervals are similarly divided. But farther still, we can demonstrate that this similarity is not confined to these two octaves. This may even be proved, to a certain extent, by the same original experiment. Many men can sing two octaves in succession, and there are some rare examples of persons who can sing three. This is more common in the female voice. This being the case, it is plain that there will be two octaves common to both voices; and therefore four octaves in succession, all similar to each other. The same similarity may be observed in the sounds of instruments which differ only by an octave. And thus we demonstrate that all octaves are similar to each other. This similarity does not consist merely in the similarity of its division. The sound of a note and its octave are so like each other, that if the strength or loudness be properly adjusted, and there be no difference in kind, or other circumstances of clearness, smoothness, &c., the two notes, when sounded together, are indistinguishable, and appear only like a more brilliant note. They coalesce into one sound. Nay, most clear mellow notes, such as these of a fine human voice, voice, really contain each two notes, one of which is octave to the other.

We said that this resemblance of octaves is an important fact in the science of music. We now see why it is so. The whole scale of music is contained in one octave, and all the rest are only repetitions of this scale. And thus is the doctrine of the scale of melody brought within a very moderate compass, and the problem is reduced to that of the repartition of a single octave, and some attention to the junction with the similar scales of the adjoining octaves. This partition is now to be the subject of discussion.

In the infancy of society and cultivation, it is probable that the melodies or tunes, which delighted the simple inhabitants, were equally simple. Being the spontaneous effusions of individuals, perhaps only occasional, and never repeated, they would perish as fast as produced. The airs were probably connected with some of the rude rhymes, or griddles of words, which were bandied about at their festivals; or they were associated with dancing. In all these cases they must have been very short, consisting of a few favourite passages or musical phrases. This is the case with the common airs of all simple people to this day. They seldom extend beyond a short stanza of poetry, or a short movement of dancing. The artist who could compose and keep in mind a piece of considerable length, must have been a great rarity, and a minstrel fit for the entertainment of princes; and therefore much admired, and highly rewarded: his excellencies were almost incommunicable, and could not be preserved in any other way but by repeated performance to an attentive hearer, who must also be an artist, and must patiently listen, and try to imitate; or, in short, to get the tune by heart. It must have been a long time before any distinct notion was formed of the relation of the notes to each other. It was perhaps impossible to recollect today the precise notes of yesterday. There was nothing in which they were fixed till instrumental music was invented. This has been found in all nations; but it appears that long continued cultivation is necessary for raising this from a very simple and imperfect state. The most refined instrument of the Greek musicians was very far below our very ordinary instruments. And, till some method of notation was invented, we can scarcely conceive how any determined partition of the octave could be made generally known.

Accordingly, we find that it was not till after a long while, and by very rude and awkward steps, that the Greeks perceived that the whole of music was comprised in the octave. The first improved lyre had four strings, and was therefore called a tetrachord; and the first flutes had but three holes, and four notes; and when more were added to the scale, it was done by joining two lyres and two flutes together. Even this is an instructive step in the history of musical science: For the four sounds of the instrument have a natural system, and the awkward and groping attempts to extend the music, by joining two instruments, the scale of the one following, or being a continuation of that of the other, pointed out the diapason or totality of the octave, and the relation of the whole to a principal sound, which we now call the fundamental or key, it being the lowest note of our scale, and the one to which the other notes bear a continual reference. It would far exceed the limits of this Work to narrate the successive changes and additions made by the Greeks in their lyre; yet Temper would this be a very sure way of learning the natural mode of formation of our musical scale. We must refer our readers to Dr Wallis's Appendix to his edition of the Commentary of Porphyrius in Ptolemy's Harmonics, as by far the most perspicuous account that is extant of the Greek music. We shall pick out from among their different attempts such plain observations as will be obvious to the feelings of any person who can sing a common tune.

Let such a person first sing over some plain, and cheerful, or at least not mournful, tune, several times, so as to retain a lasting impression of the chief note of the tune, which is generally the last. Then let him begin, on the same note, to sing in succession the rising steps of the scale, pronouncing the syllables do, re, mi, fa, sol, la, si, do. He will perhaps observe, that this chant naturally divides itself into two parts or phrases, as the musicians term it. If he does not, of himself, make this remark, let him sing it, however, in that manner, pausing a little after the note fa. Thus, do, re, mi, fa; sol, la, si, do.—Do, re, mi, fa; sol, la, si, do.

Having done this several times, and then repeated it without a pause, he will become very sensible of the propriety of the pause, and of this natural division of the octave. He will even observe a considerable similarity between these two musical phrases, without being able, at first, to say in what it consists.

Let him now study each phrase apart, and try to compare the magnitude of the changes of sound or steps of the scale which he makes in rising from do to re, from re to mi, are unequal, and from mi to fa. We apprehend that he will have no difficulty in perceiving, after a few trials, that the trachords do re, and re mi, are sensibly greater than the step are simi fa. We feel the last step as a sort of slide; as an attempt to make as little change of pitch as we can. Once this is perceived, it will never be forgotten. This will be still more clearly perceived, if, instead of these syllables, he use only the vowel a, pronounced as in the word hall, and if he sing the steps, sliding or flurring from the one to the other. Taking this method, he cannot fail to notice the smallness of the third step.

Let the finger farther consider, whether he does not feel this phrase musical or agreeable, making a sort of tune or chant, and ending or closing agreeably after this slide of a small, or, as it were, half step. It is generally thought so; and is therefore called a close, a cadence, when we end with a half step ascending.

Let the finger next resume the whole scale, fingering the four last notes sol, la, si, do, louder than the other four, and calling off his attention from the low phrase, and fixing it on the upper one. He will now be able to perceive that this, like the other, has two considerable steps; namely, sol la and la si, and then a smaller step, si do. A few repetitions will make this clear, and he will then be sensible of the nature of the similarity between these two phrases, and the propriety of this great division of the scale into the intervals do, fa, and sol, do, with an interval fa, sol between them.

This was the foundation of the tetrachords, or lyres of four strings, of the Greeks. Their earliest music or modulation seems to have extended no farther than this phrase. It pleased them, as a ring of four bells pleases many country parishes.

The finger will perceive the same satisfaction with close the close of this second phrase as with that of the former; and if he now sing them both, in immediate succession, succession, with a slight pause between, we imagine that we will think the close or cadence on the upper do even more satisfactory than that on the fa. It seems to us to complete a tune. And this impression will be great- ly heightened, if another person, or an instrument, should found the lower do, while he chooses on the up- per do its octave. Do seems to be expected, or looked for, or sought after. We take fa as a step to do, and there we rest.

Thus does the octave appear to be naturally com- posed of seven steps, of which the first, second, fourth, fifth, and sixth, are more considerable, and the third and seventh very feebly smaller. Having no direct measures of their quantity, nor even a very distinct no- tion of what we mean by their quantity, magnitude, or bulk, we cannot pronounce, with any certainty, whe- ther the greater steps are equal or unequal; and we presume them to be equal. Nor have we any distinct notion of the proportion between the larger and smaller steps. In a loose way we call them half notes, or sup- pose the rise from mi to fa, or from fa to do, to be one- half of that from do to re, or from re to mi.

Accordingly, this seems to have been all the musical science attained by the Greek artists, or those who did not profess to speak philosophically on the subject. And even after Pythagoras published the discovery which he had made, or more probably had picked up among the Chaldeans or Egyptians, by which it appeared, that accurate measures of sounds, in respect to gravity and acuteness, were attainable, it was affirmed by Aritho- nus, a scholar of Aristotle, and other eminent philoso- phers, that these measures were altogether artificial, had no connection with music, and that the ear alone was the judge of musical intervals. The artist had no other guide in tuning his instrument; because the ra- tios, which were said to be inherent in the sounds (though no person could say how), were never percep- ted by the ear. The justice of this opinion is abun- dantly confirmed by the awkward attempt of the Greeks to improve the lyre by means of these boasted ratios. Instead of illustrating the subject, they seem rather to have brought an additional obscurity upon it, and threw it into such confusion, that although many voluminous dissertations were written on it, and on the composition of their musical scale, the account is so perplexed and confused, that the first mathematicians and artists of Europe acknowledge, that the whole is an impene- trable mystery. Had the philosophers never meddled with it, had they allowed the practical musicians to con- struct and tune their instruments in their own way, so as to please their ear, it is scarcely possible that they should not have hit on what they wanted, without all the embarrassment of the chromatic and enharmonic scales of the lyre. It is scarcely possible to contrive a more cumbersome method of extending the simple scale of Nature to every case that could occur in their musi- cal compositions, than what arose from the employment of the musical ratios. This seems a bold assertion; but we apprehend that it will appear to be just as we pro- ceed.

The practical musicians could not be long of finding the want of something more than the mere diatonic scale made use of in their instruments. As they were always ac- companied by the voice, it would often happen that a lyre or flute, perfectly tuned, was too low or too high for the voice that was to accompany it. A singer can pitch his tune on any sound as a key; and if this be too high for the singer who is to accompany him, he can take it on a lower note. But a lyrist cannot do this. Suppose his instrument two notes too low, and that his accompanist can only sing it on the key which is the fa of the lyre. Should the lyrist begin it on that key, his very first step is wrong, being but a half step, whereas it should be a whole one. In short, all the steps but one will be found wrong, and the lyrist and singer will be perpetually jarring. This is an evident consequence of the inequality of the fourth and seventh steps to the rest. And if the other steps, which we imagine to be equal, be not exactly so, the discordance will be still greater.

The method of remedying this is very obvious. If the intervals mi fa and fa do, are half notes, we need the Pytha- gorian division of the octave, which gives us seven equal steps, each of the whole notes; and then, in place of seven un- equal steps, we shall have twelve equal ones, or twelve musical ra- tions, each of them equal to a semitone. The lyre thus constructed will now suit any voice whatever. It will perfectly resemble our keyed instruments, the harp, chord, or organ, which have twelve seemingly equal in- tervals in the octave. Accordingly, it appears that such additions were practised by the musicians of Greece, and approved of by Aristoxenus, and by all those who referred every thing to the judgment of the ear. And we are confident that this method would have been ad- opted, if the philosophers had had less influence, and if the Greeks had not borrowed their religious cere- monies along with their musical science. Both of these came from the same quarter; they came united; and it was sacrilegious to attempt innovations. The doc- trine of musical ratios was an occupation only for the refined, the philosophers; and by subjecting music to this mysterious science, it became mysterious also, and so much the more venerable. The philosophers knew, that there was in Nature a certain inextricable con- nection between mathematical ratios and those intervals which the ear relished and required in melody; but they were ignorant of the nature and extent of this con- nection.

What is this connection, or what is meant when we speak of the ratios of sounds? Simply this.—Pytha- goras is said to have found, that if two musical cords were strained by equal weights, and one of them be twice the length of the other, the short one will found the octave to the note of the other. If it be two-thirds of the length of the long string, it will found the fifth to it. If the long string found do, the short one will found fa. If it be three-fourths of the length, it will found the fourth or fa. Thus the ratio of 2 : 1 was called the ratio of the diapason; that of 3 : 2 was called the diapente; and that of 4 : 3 the diatessaron.

Moreover, if we now take all the four strings, and make that which founds the gravest note, and is the longest, twelve inches in length; the short or octave string must be six inches long, or one-half of twelve; the diapente must be eight inches, or two-thirds of twelve; and the diatessaron must be nine inches, which is three-fourths of twelve. If we now compare the diapente, not with the gravelest string, but with the octave of six inches, we see that they are in the ratio of 4 to 3, or the ratio of diatessaron. And if we compare the diatessaron with Temperance, we see that their ratio is that of 9:6, or scale of music.

Thus is the octave divided into a fifth and a fourth, do, sol, and fa, do, in succession. Also the fourth do, fa, and the fifth fa, do, make up the octave. The note which stands as a fifth to one of the extreme sounds of the octave, stands as a fourth to the other. And, lastly, the two fourths do fa, and sol do, leave an interval fa sol between them; which is also determined by nature, and the ratio cor- responding to it is evidently that of 9 to 8.

This is all that was known of the connection of mu- sic with mathematical ratios. It is indeed said by Tam- blichus, that Pythagoras did not make this discovery by means of strings, but by the sounds made by the ham- mers on the anvil in a smith's shop. He observed the sounds to be the key, the diatessaron, and the dissonance of music; and he found, that the weights of the hammers were in this proportion; and as soon as he went home, he tried the sounds made by cords, when weights, in the proportions above-mentioned, were ap- plied to them. But the whole story has the air of a fable, and of ignorance. The sounds given by a smith's anvil have little or no dependence on the weight of the hammers; and the weights which are in the proportions of the numbers mentioned above will by no means pro- duce the sounds alleged. It requires four times the weight to make a string sound the octave, and twice and a quarter will produce the dissonance, and one and seven- ninths will produce the diatessaron. It is plain, there- fore, that they knew not of what they were speaking: yet, on this flight foundation, they erected a vast fab- ric of speculation; and in the course of their researches, these ratios were found to contain all that was ex- cellent. The attributes of the Divinity, the symme- try of the universe, and the principles of morality, were all resolvable into the harmonic ratios.

In the attempts to explain, by means of the myste- rious properties of the ratios 2:1, 3:2, 4:3, and 9:8, which were thus defined by Nature, it was ob- served, that their favourite lyres of four strings could be combined in two principal manners, so as to produce an extensive scale. One lyre may contain the notes do, re, mi, fa; and the acuter lyre may contain the notes sol, la, si, do; and, being let in succession, having the interval fa sol between the highest note of the one and the lowest of the other, they make a complete octave. These were called disjunct tetrachords. Again, a third tetrachord may be joined with the upper tetrachord last mentioned, in such sort, that the lowest note of the third tetrachord may be the same with the highest of the second. These were called conjunct tetrachords (A).

By thus considering the scale as made up of tetra- chords, the tuning of the lyre was reduced to great simplicity. The musician had only to make himself perfect in the short chant do, re, mi, fa, or to get it by heart, and to sing it exactly. This intonation would apply equally to the other sol, la, si, do. We are well informed that this was really the practice. The direc- tions given by Aristothenes, Nicander, and others, for varying the tuning, according to certain occasional ac- commodations, show distinctly that they did not tune as we do, founding the two strings together, except in the case of the diapason or octave. It was all done by the judgment of the ear in melody. The most valu- able circumstance in the discovery of Pythagoras was the determination of the interval between the fourth and the fifth, by which the tetrachords were separated. The filling up of each tetrachord was left entirely to the ear; and when the doctrine of the mathematical ra- tios showed that the large intervals do re, re mi, fa sol, sol la, la si, should not be precisely equal, Aristothenes refused the authority of the reasons alleged for this in- equality, because the ear perceived none of the ratios as ratios, and could judge only of sounds. He farther af- firmed, that the inequalities which the Pythagoreans en- joined, were so trifling, that no ear could possibly per- ceive them. And accordingly, the theorists disputed about the respective situations of the greater and smaller tones (for they named the great steps) so much spoken of, and had different systems on the subject.

But the strongest proof of the indistinct notion that And by the theorists entertained about the influence of these ratios in music is, that they would admit no more but those introduced by Pythagoras; and their reasons for the rejection of the ratio of 5 to 4, and of 6 to 5, were either the most whimsical fancies about the perfections of the sacred ratios, or assumptions expressly founded on the supposition, that the ear perceives and judges of the ratios as ratios; than which nothing can be more false. Had they admitted the ratio of 5 to 4, they would have obtained the third note of the scale, and would at once have gotten the whole scale of our music. The ratios of 6:5, and 16:15, follow of course; and every sound of the tetrachords would have been determined. For 5:4 being the ratio of the ma- jor third, which is perfectly pleasing to the ear, as the mi to the note do, and 3:2 being the ratio of the fifth do sol, there is another interval mi sol determined; and this ratio, being the difference between do sol and do mi, or between 3:2 and 5:4, is evidently 6:5. In like manner, the interval mi fa is determined, and its ratio, being 4:3 — 5:4, is 10:15.

But farther; we shall find, upon trial, that if we put in a sound above sol, having the relation 5:4 to fa, it will be perfectly satisfactory to the ear if sung as the note la. And if, in like manner, we put in a note a- bove la, having the relation 5:4 to fa, we find it satis- factory to the ear when used as si. If we now exa- mine the ratios of these artificial notes, we shall find the ratio of the notes sol la to be 10:9, and that of la si to be 9:8, the same with that fa sol; also si do will appear to be 16:15, like that of mi fa.

We have no remains of the music of the Greeks, by which we can learn what were their favourite passages or musical phrases; and we cannot see what caused them to prefer the fourth to the major third. Few musicians of our times think the fourth in any degree comparable with the major third for melodiousness; and fewer for harmoniousness. The piece or tune pub- lished by Kircher from Alypius is very suspicious, as

(A) This is the principle, but not the precise form, of the disjunct and conjunct tetrachords. The Greeks did not begin the tetrachord with what we make the first note of our chant of four notes, but began one of them with mi, and the other with fa; to which they afterwards added a note below. This beginning seems to have been directed by some of their favourite cadences; but it would be tedious to explain it. no other person has seen the MS.; and the collection found at Bada is too much disfigured, and probably of too late a date, to give us any solid help. In all probability, the common melodies of the Greeks abounded in caly leaps up and down on the third and fifth, and on the fourth and sixth, just as we observe in the airs for dancing among all simple people. Their accomplished performers had certainly great powers both of invention and execution; and the chromatic and enharmonic divisions of the scale were certainly practised by them, and not merely the speculations of mathematicians. To us, the enharmonic scale appears the most jarring discord; but this is certainly owing to our not seeing any pieces of the music so composed, and because we cannot in the least judge by harmony what the effect of enharmonic melody would be. But we have sufficient evidence, from the writings of the ancient Greeks, that the enharmonic music fell into disuse even before the time of Ptolemy, and was totally and irrecoverably lost before the 5th century. Even the chromatic was little practised, and was chiefly employed for extending the common scale to keys which were seldom used. The uncertainties respecting even the common scale remained the same as ever; and although Ptolemy gives (among others) the very name that is now admitted as the only perfect one, namely, his diatonicum intemum, his reasons of preference, though good, are not urged with strong marks of his confidence in them, nor do they seem to have prevailed.

These observations show clearly, that the perception of melody alone is not sufficiently precise for enabling us to acquire exact conceptions of the scale of music. The whole of the practicable science of the ancients seems to amount to no more than this, that the octave contained five greater and two smaller intervals, which the voice employed, and the ear relished. The greater intervals seemed all of one magnitude; and the smaller intervals appeared also equal, but the ear cannot judge what proportion they bear to the larger ones. The musicians thought them larger than one-half of the great intervals (and indeed the ratio 16:9 of the artificial mi fa and si do, is greater than the half of 9:8 or 10:9). Therefore they allowed the theorists to call them limma instead of homitona; but they, as well as the theorists, differed exceedingly in the magnitudes which they assigned them.

The best way that we can think of for expressing the scale of the octave is, by dividing the circumference of a circle in the points C, D, E, F, G, A, and B (fig. 1.), in the proportion we think most suitable to the natural scale of melody. According to the practical notion now under our consideration, the arches CD, DE, FG, GA, and AB, are equal, containing nearly 59°; and the arches EF and BC are also equal, but smaller than the others, containing about 33°. Now, suppose another circle, on a piece of card-paper, divided in the same manner, to move round their common centre, but instead of having its points of division marked C, D, E, &c., let them be marked do, re, mi, fa, sol, la, si. It is plain, that to whatever point of the outer circle we set the point do of the inner one, the other points of the outer circle will show the common notes which are fit for those steps of the scale. The similarity of all octaves makes this simple octave equivalent to a rectilineal scale similarly divided, and repeated as often as we please. Fig. 1. represents this instrument, and will be often referred to. A sort of symmetry may be observed in it. The point D seems to occupy the middle of the scale, and re seems to be the middle note of the octave. The opposite arch GA, and the corresponding interval sol la, seems to be the middle interval of the octave. The other notes and intervals are similarly disposed on each side of these. This circumstance seems to have been observed by the Greeks, by the inhabitants of India, by the Chinese, and even by the Mexicans. The note re, and the interval sol la, have got distinguished situations in their instruments and scales of music.

With respect to the division of the circles, we shall only observe at present, that the dotted lines are conformable to the principles of Aristoxenus, the whole octave being portioned out into five larger and equal intervals, and two smaller, also equal. The larger are called mean or medium tones; and the smaller are called limmas or semitones. The full lines, to which the letters and names are affixed, divide the octave into the artificial portions, determined by means of the musical ratios, the arches being made proportional to the measures of those ratios. Thus the arches CD, FG, AB, are proportional to the measure or logarithm of the ratio 9:8; GA and DE are proportional to the logarithm of 10:9; and the arches EF and BC are proportional to the logarithm of 16:15. We have already mentioned the way in which those ratios were applied, and the authority on which they were selected. We shall have occasion to return to this again. The only farther remark that is to be made with propriety in this place is, that the division on the Aristoxenean principles, which is expressed in this figure, is one of an indefinite number of the same kind. The only principle adopted in it is, that there shall be five mean tones, and two small equal semitones; but the magnitude of these is arbitrary. We have chosen such, that two mean tones are exactly equal to the arch CE, determined by the ratio 5:4. The reasons for this preference will appear as we proceed (n).

By this little instrument (the invention, we believe, of a Mr D'Ormeillon, about the beginning of last century), we see clearly the insufficiency of the seven notes of the octave for performing music on different keys. Set the flower de luce at the Aristoxenean B, and we shall see that E is the only note of our lyre which will do for one of the steps of the octave in which we intend to sing and accompany. We have no founds in the lyre for re, mi, sol, la, si. The remedy is clearly pointed out. Let a set of strings be made, having the same relation to si, which those of the present lyre have to do, and insert them in the places pointed out by the Aristoxenean divisions of the moveable octave. We need only five of them, because the si and fa of the present lyre will answer. These new founds are marked by a +.

But it was soon found, that these new notes gave but indifferent melody, and that either the ear could not perfectly determine

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(n) We shall be abundantly exact, if we make CD = 61°72; CE = 115°9; CF = 149°42; CG = 210°58; CA = 263°3; and CB = 326°43. Temperament determine the quality of the tones and semitones exact- ment of the lyre enough, or that no such partition of the octave would Scale of answer. The Pythagoreans, or partisans of the musi- cal ratios, had told them this before. But they were in no better condition themselves; for they found, that if a series of sounds, in perfect relation to the octave, be inserted in the manner proposed, the melody will be no better. They put the matter to a very fair trial. It is easy to see, that no system of mean tones and limmas will give the same music on every key, unless the tones be increased, and the limmas diminished, till the limma becomes just half a tone. Then all the intervals will be perfectly equal. The mathematicians computed the ratios which would produce this equality, and desired the Aristoxeneans to pronounce on the music. It is said, that they allowed it to be very bad in all their most favourite passages. Nothing now remained to the Aristoxeneans but to attempt occasional methods of tuning. They saw clearly, that they were making the notes unequal which Nature made equal. The Py- thagoreans, in like manner, pointed out many altera- tions or corrections of intervals which suited one tetra- chord, or one part of the octave, but did not suit an- other. Both parties saw that they were obliged to de- viate from what they thought natural and perfect: therefore they called these alterations of the natural or perfect scale a temperament.

The accomplished performers were the best judges of the whole matter, and they derived very little affini- tude, finesse, and would, in general, be better translated Tem- by symmetry. But we cannot conceive that they paid any marked attention to the effect of simultaneous sounds, so as to enjoy the pleasure of certain confor- mances, and employ them in their compositions. We judge in this way from the rank which they gave them in their scale. To prefer the fourth to the major third seems to us to be impossible, if it be meant of simulta- neous sounds. And the reason which is assigned for the preference can have no value in the opinion of a musician. It is because the ratio of $4:3$ is simpler than that of $5:4$. For the same reason, the fifth is prefe- red to both, and the octave to all the three, and union to every other consonance. They would not allow the major third $5:4$ to be a concord at all. We have made numberless trials of the different concords with persons altogether ignorant of music. We never saw, an instance of one who thought that mere union gave any positive pleasure. None of all whom we examined had much pleasure from an octave. All, without ex- ception, were delighted with a fifth, and with a major third; and many of them preferred the latter. All of them agreed in calling the pleasure from the fifth a sweetness, and that from the major third a cheerfulness, or smartness, or by names of similar import. The great- er part preferred even the major sixth to the fourth, and some felt no pleasure at all from the fourth. Few had much pleasure from the minor third or minor sixth.

N.B. Care was taken to sound these concords with- out any preparation—merely as sounds—but not as mak- ing part of any musical passage. This circumstance has a great effect on the mind. When the minor third and sixth were heard as making part of the minor mode, all were delighted with it, and called it sweet and mournful. In like manner, the chord $5$ never fail- ed to give pleasure. Nothing can be a stronger proof of the ignorance of the ancients of the pleasures of harmony.

We do not profess to know when this was discov- ered. We think it not unlikely that the Greeks and Ita- lians got it from some of the northern nations whom they called Barbarians. We cannot otherwise account for its prevalence through the whole of the Ruffian em- pire—the ancient Slavi had little commerce with the empire of Rome or of Constantinople; yet they sung in parts in the most remote periods of their history of which we have any account; and to this day, the most uncultivated boor in the Ruffian empire would be affec- ted to sing in unison. He listens a little while to a new tune, holding his chin to his breast; and as soon as he has got a notion of it, he bursts out in concert, throwing in the harmonic notes by a certain rule which he feels, but cannot explain. His harmonics are gener- ally altered major and minor thirds, and he seldom misses the proper cadences on the fifth and key. Per- haps the invention of the organ produced the discovery. We know that this was as early as the second centu- ry (c). It was hardly possible to make much use of that instrument without perceiving the pleasure of con- cordant sounds.

(c) It is said that the Chinese had an instrument of this kind long before the Europeans. Cauteus says, that it was brought from China by a native, and was so small as to be carried in the hand. It is certain that the Emperor Constantine Copronymus sent one to Pepin king of France in 757, and that his son Charlemagne got another from the Emperor Michael Palaeologus. But they appear to have been known in the English churches before that time. The discovery of the pleasures of harmony occasioned a total change in the science of music. During the dark ages of Europe, it was cultivated chiefly by the monks; the organ was soon introduced into the churches, and the choral service was their chief and almost their only occupation. The very construction of this instrument must have contributed to the improvement of music, and instructed men in the nature of the scale.

The pipes are all tuned by their lengths; and these lengths are in the ratios of the strings which give the same notes, when all are equally stretched. This must have revived the study of the musical ratios. The tuning of the organ was performed by consonance, and no longer depended on the nice judgment of sounds in succession. The dullest ear, even with total ignorance of music, can judge, without the smallest error, of an exact octave, fifth, third, or other concord; and a very mean musician could now tune an organ more accurately than Timotheus could tune his lyre. Other keyed instruments, resembling our harpsichord, were invented, and instruments with fretted fingerboards. These soon supplanted the lyres and harps, being much more commodious, and allowing a much greater variety and rapidity of modulation. All these instruments were the fruits of harmony, in the modern scale of that word.

The deficiencies of the old diatonic scale were now more apparent, and the necessity of a number of intercalary notes. The finger-board of an organ or harpsichord, running through a series of octaves, and admitting much more than the accompaniment of one note, pointed out new sources of musical pleasure arising from the fulness of the harmony; and, above all, the practice of choral singing suggested the possibility of a pleasure altogether new. While a certain number of the choir performed the Cantus or Air of the music, it was irksome to the others to utter mere sounds, supporting or composing the harmony of the Cæcilius, without any melody or air in their own parts. It was thought probable that the harmonic notes might be so portioned out among the rest of the choir, that the succession of sounds uttered by each individual might also constitute a melody not unpleasant, and perhaps highly grateful.

On trial, it was found very practicable. Canons, motets, fugues, and other harmonies, were composed, where the airs performed by the different parts were not inferior in beauty to the principal. The notes which could not be thrown into this agreeable succession, were left to the organist, and by him thrown into the bass.

By all these practices, the imperfections of the scale of fixed sounds became every day more sensible, especially in full harmony. Scientific music, or the properties of the ratios, now recovered the high estimation in which they were held by the ancient theorists; and as the musicians were now very frequently men of letters, chiefly monks, of sober characters and decent manners, music again became a respectable study. The organist was generally a man of science, as well as a performer. At the first revival of learning in Europe, we find music studied and honoured with degrees in the universities, and very soon we have learned and excellent dissertations on the principles of the science. The inventions of Guido, and the dissertations of Salmas, Zarlino, and Xoni, are among the most valuable publications that are extant on music. The improvements introduced by Guido are founded on a very refined examination of the scale; and the temperaments proposed by the other two have scarcely been improved by any labours of modern date. Both these authors had studied the Greek writers with great care, and their improvements proceed on a complete knowledge of the doctrines of Pythagoras and Ptolemy.

At last the celebrated Galileo Galilei put the finishing hand to the doctrines of those ancient philosophers, by the discovery of the connection which subsists in nature between the ratios of numbers and the musical intervals of sounds. He discovered, that these numbers freely express the frequency of the recurring pulses or undulations of air which excite in us the sensation of sound. He demonstrated that if two strings, of the same tension, diameter, and thickness, be stretched by equal weights, and be twanged or pinched so as to vibrate, the times of their vibrations will be as their lengths, and the frequency or number of oscillations made in a given time will be inversely as their lengths. The frequency of the sonorous undulations of the air is therefore inversely as the length of the string. When therefore we say that 2 : 1 is the ratio of the octave, we mean, that the undulations which produce the upper sound of this interval are twice as frequent as those which produce its fundamental sound. And the ratio 3 : 2 of the diapente or fifth, indicates, that in the same time that the ear receives three undulations from the upper sound, it receives only two from the lower. Here we have a natural connection, not peculiar to the sounds produced by strings; for we are now able to demonstrate, that the sounds produced by bells are regulated by the same law. Nay, the improvements which have been made in the science of motion since the days of Galileo, shew us that the undulations of the air in pipes, where the air is the only substance moved, is regulated by the same law. It seems to be the general property of sounds which renders them susceptible of musical pitch, of acuteness, or gravity; and that a certain frequency of the sonorous undulations gives a determined and unalterable musical note. The writer of this article has verified this by many experiments. He finds, that any noise whatever, if repeated 240 times in a second, at equal intervals, produces the note C sol fa ut of the Gondorin gamut. If it be repeated 360 times, it produces the G sol re ut, &c. It was imagined, that only certain regular agitations of the air, such as are produced by the tremor or vibration of elastic bodies, are fitted for exciting in us the sensation of a musical note. But he found, by the most distinct experiments, that any noise whatever will have the same effect, if repeated with due frequency, not less than 30 or 40 times in a second. Nothing surely can have less attention to the name of a musical sound than the solitary snap which a quill makes when drawn from one tooth of a comb to another; but when the quill is held to the teeth of a wheel, whirling at such a rate, that 720 teeth pass under it in a second, the sound of g in air, is heard most distinctly; and if the rate of the wheel's motion be varied in any proportion, the noise made by the quill is mixed in the most distinct manner with the musical note corresponding to the frequency of the snaps. The kind of the original noise determines the kind of the continuous sound produced by it, making it harsh and fretful, or smooth and mellow, according as the original noise is abrupt or gradual; but even the most abrupt noise Temperament produces a tolerably smooth sound when sufficient- Scale of Music.

noise produces a tolerably smooth sound when sufficient- Scale of Music.

This frequency is expressed by the musical ratios of Pythagoras.

Concord, Discord, are properties of particular ratios of frequency.

proach nearer and nearer to perfect concord, these coincidences become rarer and rarer; and if it be infinitely near to perfect concord, the coincidences of vibration will be infinitely distant from each other. This, and many other irrefragable arguments, demonstrate that coalescence of sound, which makes the pleasing harmony of a fifth, for example, does not arise from the coincidence of vibrations; and the only thing which we can demonstrate to obtain in all the cases where we enjoy this pleasure, is a certain arrangement of the component pulses, and a certain law of succession of the dislocations or intervals between the non-coinciding pulses. We are perfectly able to demonstrate that when, by continually screwing up one of the notes of a consonance, we render the real coincidence of pulses less frequent; the dislocations, or deviations from perfect coincidence, approach nearer and nearer to a certain definable law of succession; and that this law obtains completely, when the perfect ratio of the duration of the pulse is attained, although perhaps at that time not one pulse of the one sound coincides with a pulse of the other. Suppose two organ pipes, sounding the note C folia ut, at the distance of ten feet from each other, and that their pulses begin and end at the same instants, making the most perfect coincidence of pulses—there is no doubt but that there will be the most perfect harmony; and we learn by experience that this harmony is perfectly the same, from whatever part of the room we hear it. This is an unquestionable fact. A person situated exactly in the middle between them will receive coincident pulses. But let him approach one foot nearer to one of the pipes, it is now demonstrable that the pulses, at their arrival at his ear, will be the most distant from coincidence that is possible; for every pulse of one pipe will affect the pulse from the other; but the law of succession of the deviations from coincidence will then obtain in the most perfect manner. A musical sound is the sensation of a certain form of the aerial undulation which agitates the auditory organ. The perception of harmonious sound is the sensation produced by another definite form of the agitation. This is the composition of two other agitations; but it is the compound agitation only that affects the ear, and it is its form or kind which determines the sensation, making it pleasant or unpleasant.

Our knowledge of mechanics enables us to describe this form, and every circumstance in which one agitation can differ from another, and to discover general mathematical features or circumstances of resemblance, which, in fact, accompany all perceptions of harmony. We are surely justified in saying that these circumstances are sure tellers of harmony; and that when we have ensured their presence, we have ensured the hearing of harmony in the adjusted sounds. We can even go farther in some cases; We can explain some appearances which accompany imperfect harmony, and perceive the connection between certain distinct results of imperfect coincidences, and the magnitude of the deviations from perfect harmony, which are then heard. Thus, we can make use of these phenomena, in order to ascertain and measure those deviations; and if any rules of temperament should require a certain determinate deviation from perfect harmony in the tuning of an instrument, we can secure the appearance of that phenomenon which corresponds to the deviation, and thus can produce the precise tempera- temperament suggested by our rules. We can, for example, destroy the perfect harmony of the fifth C₅, and flatten the note g till it deviates from a perfect fifth in the exact ratio of 320 to 321, which the musicians call the one-fourth of a comma. The most exquisite ear for melody is almost insensible of a deviation four times greater than this; and yet a person who has no musical ear at all, can execute this temperament by the rules of harmony without the error of the fortieth part of a comma.

For this most valuable piece of knowledge we are indebted to the late Dr Robert Smith of Cambridge, a very eminent geometer and philosopher, and a good judge of music, and very pleasing performer on the organ and harpsichords. This gentleman, in his Dissertation on the Principles of Harmonics, published for the first time in 1749, has paid particular attention to a phenomenon in coincident sounds, called a beating. This is an alternate enforcement and diminution of the strength of sound, something like what is called a close flake, but differing from it in having no variation in the pitch of the sounds. It is a sort of undulation of the sound, in which it becomes alternately louder and fainter. It may be often perceived in the sound of bells and musical glasses, and also in the sounds of particular strings. It is produced in this way: Suppose two unisons quite perfect; the vibrations of each are either perfectly coincident, or each pulse of one sound is interposed in the same situation between each pulse of the other. In either case they succeed each other with such rapidity, that we cannot perceive them, and the whole appears an uniform sound. But suppose that one of the sounds has 240 pulses in a second, which is the undulation that is produced in a pipe of 24 inches long; suppose that the other pipe is only 23 inches and 3/8ths long. It will give 243 pulses in a second. Therefore the 1st, the 80th, the 160th, and the 240th pulse of the first pipe will coincide with the 1st, the 8th, the 16th, and the 243rd pulse of the other. In the instants of coincidence, the agitation produced by one pulse is increased by that produced by the other. The commencement of the next two pulses is separated a little, and that of the next is separated still more, and so on continually: the dilatations of the pulses, or their deviations from perfect coincidence, continually increasing, till we come to the 40th pulse of the one pipe, which will commence in the middle of the 41st pulse of the other pipe; and the pulses will now bifect each other, so that the agitations of the one will counteract or weaken those of the other. Thus the compounded sound will be stronger at the coincidences of the pulses, and fainter when they bifect each other. This reinforcement of sound will therefore recur thrice in every second. The frequency of the pulses are in the ratio of a comma, or 81 : 80. Therefore this constitutes an unison imperfect by a comma. If therefore any circumstance should require that these two pulses should form an unison imperfect by a comma, we have only to alter one of the pipes, till the two, when founded together, beat thrice in a second. Nothing can be plainer than this. Now let us suppose a third pipe tuned an exact fifth to the first of these two. There will be no beating observable; because the recurrence of coincident pulses is so rapid as to appear a continued sound. They recur at every second vibration of the bass, or 120 times in a second.

But now, instead of founding the third pipe along with the first, let it found along with the second. Dr Smith demonstrates, that they will beat in the same manner as the unisons did, but thrice as often, or nine times in a second. When therefore the fifth C₅ beats nine times in a second, we know that it is too sharp or too flat (very nearly) by a comma.

Dr Smith shows, in the same manner, what number of pulses are made in any given time by any concord, in order to perfect or tempered, in any assigned degree. We humbly think that the most attentive person must be sensible of the very great value of this discovery. We are obliged to call it his discovery. Mercatus, indeed, had great effect taken particular notice of this undulation of imperfect pulses of consonances, and had offered conjectures as to their cause; conjectures not unworthy of his great ingenuity. Mr Sauvage also takes a still more particular notice of this phenomenon *, and makes a most ingenious use of it for the solution of a very important musical problem; namely, to determine the precise number of pulses which produce any given note of the gamut. His method is indeed operose and delicate, even as simplified and improved by Dr Smith. The following may be substituted for it, founded on the mechanism of founding cords. Let a violin, guitar, or any such instrument, be fixed up against a wall, with the fingerboard downward, and in such a manner, that a violin string, strained by a weight, may press on the bridge, but hang free of the lower end of the fingerboard. Let another string be strained by one of the tuning pins till it be in unison with some note (suppose C) of the harpsichord. Then hang weights on the other string, till upon drawing the bow across both strings, at a small distance below the bridge, they are perfect unisons, without the smallest beating or undulation, and taking care that the pressure of the bow on that string which is tuned by the pin be so moderate as not to affect its tension sensibly. Note exactly the weight that is now appended to it. Now increase this weight in the proportion of the square of 80 to the square of 81; that is, add to it its 4th part very nearly. Now draw the bow again across the strings with the same caution as before. The sounds will now beat remarkably; for the vibrations of the loaded string are now accelerated in the proportion of 80 to 81. Count the number of undulations made in some small number (suppose 10) of seconds. This will give the number of beats in a second; 80 times this number are the single pulses of the lowest found; and 81 times the same number gives the pulses of the highest of these imperfect unisons.

If this experiment be tried for the C in the middle of our harpsichords, it will be found to contain 240 pulses very nearly; for the strings will beat thrice in a second. The beats are best counted by means of a little ball hung to a thread, and made to keep time with the beats.

Here, then, is a phenomenon of the most easy observation, and requiring no skill in music, by which the lord exact pitch of any sound, and the imperfection of any concord may be discovered with the utmost precision; and the temperaments by this method may concordant sounds be produced, which are absolutely perfect in their harmony, or harmonizing any degree of imperfection or temperament that we please. An instrument may generally be tuned to perfect harmony, in some of its notes, without any difficulty. Temperament, as we see done by every blind Crowder. But if a certain determinate degree of imperfection, different perhaps in the different concords, be necessary for the proper performance of musical compositions on instruments of fixed sounds, such as those of the organ or harpsichord kind, we do not see how it can be disputed, that Dr Smith's theory of the beating of imperfect consonances is one of the most important discoveries, both for the practice and the science of music, that have been offered to the public. We are inclined to consider it as the most important that has been made since the days of Galileo. The only rivals are Dr Brook Taylor's mechanical demonstration of the vibrations of an elastic cord, and its companion, and of the undulations of the air in an organ pipe, and the beautiful investigations of Daniel Bernoulli of the harmonic sounds which frequently accompany the fundamental note. The musical theory of Rameau we consider as a mere whim, not founded in any natural law; and the theory of the grave harmonies by Tartini or Romieu is included in Dr Smith's theory of the beating of imperfect consonances. This theory enables us to execute any harmonic system of temperament with precision, and certainty, and ease, and to decide on its merit when done.

We are therefore surprised to see this work of Dr Smith greatly undervalued, by a most ingenious gentleman in the Philosophical Transactions for 1800, and called a large and obscure volume, which leaves the matter just as it was, and its results useless and impracticable. We are sorry to see this; because we have great expectations from the future labours of this gentleman in the field of harmonics, and his late work is rich in refined and valuable matter. We presume humbly to recommend him attention to his own admonitions to a very young and ingenious gentleman, who, he thinks, proceeded too far in animadverting on the writings of Newton, Barrow, and other eminent mathematicians. We also beg his leave to observe, that Dr Smith's application of his theory may be very erroneous (we do not say that it is perfect); in consequence of his notion of the proportional effects produced on the general harmony by equal temperaments of the different concords. But the theory is untouched by this improper use, and stands as firmly as any proposition in Euclid's Elements.

We are bound to add to these remarks, that we have often heard music performed on the harpsichord described in the second edition of Dr Smith's Harmonics, both before it was sent home by the maker (the first in his profession), and afterwards by the author himself, who was a very pleasing performer, and we thought its harmony the finest we ever heard. Mr Watt, the celebrated engineer, and not less eminent philosopher, built a handsome organ for a public society, and, without the least ear or relish for music, tuned three octaves of the open diapason by one of Dr Smith's tables of beats, with the help of a variable pendulum. Signor Doria, leader of the Edinburgh concert, tried it in presence of the writer of this article, and said, "Bellissima—topra modo bellissima!" Signor Doria attempted to sing along with it, but would not continue, declaring it impossible, because the organ was ill tuned. The truth was, that, on the major key of E♭, the tuning was exceedingly different from what he was accustomed to, and he would not try another key. We mention this particular, to shew how accurately Mr Watt had been able to execute the temperament he intended.

This theory is valuable, therefore, by giving us the management of a phenomenon intimately connected with harmony, and affording us precise and practicable measures of all deviations from it. It bids fair, for this reason, to give us a method of executing any system of temperament which we may find reason to prefer. But we have another ground of estimation of this theory. By its assistance, we are able to ascertain with certainty and precision the true untempered scale of music, which eluded all the attempts of the ingenious Greeks; and we determine it in a way suited to the favourite music of modern times, of which almost all the excellencies and pleasures are derived from harmony. We do not say that this total innovation in the principle of musical pleasure is unexceptionable; we rather think it very defective, believing that the thrilling pleasures of music depend more upon the melody or air. We appeal even to instructed musicians, whether the heart and affections are not more affected (and with much more distinct variety of emotion) by a fine melody, supported, but not obscured, by harmonies judiciously chosen? It appears to us that the effect of harmony, always filled up, is more uniformly the same, and less touching to the soul, than some simple air sung or played by a performer of flexibility and powers of utterance. We do not wonder, then, that the ingenious Greeks deduced all their rules from this department of music, nor at their being so satisfied with the pleasures which it yielded, that they were not solicitous of the additional support of harmony. We see that melody has suffered by the change in every country. There is no Scotchman, Irishman, Pole, or Ruffian, who does not lament that the skill in composing heart-touching airs is degenerated in his respective nation; and all admire the productions of their mute of "the days that are past." They are "pleasant and mournful to the soul."

But we still prefer the harmonical method of forming the scale, on account of its precision and facility; and we prefer the theory of beats, because it also gives us the most satisfactory scale of melody; and thus, not by repeated corrections and recorrections, but by a direct process. By a table of beats, every note may be fixed at once, and we have no occasion to return to it and try new combinations; for the beatings of the different concords to one bass being once determined, every beating of any one note with any other is also fixed.

We therefore request the reader's patient attention to the experiment which we have now to propose. This experiment is made with two organ pipes equally voiced, and pitched to the note C in the middle of our harpsichords. Let one of them at least be a stopped pipe, its piston being made extremely accurate, and at the same time easily moved along the pipe. Let the flank of it be divided into 240 equal parts. The advantage of this form of the experiment is, that the sounds can be continued, with perfect uniformity, for any length of time, if the bellows be properly constructed. In default of this apparatus, the experiment may be made with two harpsichord wires in perfect union, and touched by a wheel rubbed with rosin instead of a bow, in the way the sounds of the vielle or hurdy-gurdy are produced. This contrivance also will continue the sounds uniformly at pleasure. A scale of 240 parts must must be adapted to one string, and numbered from that end of the string where the wheel or bow is applied to it. Great care must be taken that the shifting of the moveable bridge do not alter the strain on the wire. We may even do pretty well with a bow in place of the wheel; but the sound cannot be long held on in any pitch. In defining the phenomena, we shall rather abide by the string, because the numbers of the scale, or length of the sounding part of the wire, correspond, in fact, much more exactly with the sounds.

The deviations of the scale of the pipe do not in the least affect the conclusions we mean to draw, but would require to be mentioned in every instance, which would greatly complicate the process.

Having brought the two open strings into perfect unison, so that no beating whatever is observed in the consonance, slide the moveable bridge slowly along the string while the wheel is turning, beginning the motion from the end most remote from the bow. All the notes of the octave, and all kinds of concords and discords, will be heard; each of the concords being preceded and followed by a ruffling beating, and that succeeded by a grating discord. After this general view of the whole, let the particular harmonious relations of the bridge be more carefully examined as follows:

I. Shift the moveable bridge to the division 120. If it has been exactly placed, we shall hear a perfect octave, and tave without any beating. It is, however, seldom so exactly set, and we generally hear some beating. By gently shifting the bridge to either side, this beating becomes more or less rapid; and when we have found in which direction the bridge must be moved, we can then slide it along till the beating ceases entirely, and the sounds coalesce into one sound. We can scarcely hear the treble or octave note as distinguishable from the bass or fundamental afforded by the other string. If the notes are duly proportioned in loudness, we cannot hear the two as distinct sounds, but a note seemingly the same with the fundamental, only more brilliant. (N.B. It would be a great improvement of the apparatus to have a micrometer screw for producing those small motions of the bridge.)

Having thus produced a fine octave, we can now perceive that, as we continue to shift the bridge from its proper place, in either direction, the beating becomes more and more rapid, changes to a violent rattling flutter, and then degenerates into a most disagreeable jar. This phenomenon is observed in the deviation of every concord whatever from perfect harmony, and must be carefully kept in remembrance.

Before we quit this concord, the octave, produced by the biflection of the pipe or string, we must observe, that, with respect to ourselves, the octave must beat almost twice in a second, before we can observe clearly any mis-tune in it; by founding the notes in succession, or as steps in the scale of melody. We never knew any ear so nice as to discover a mis-tuning when it beats but once in three seconds. We think ourselves intitled therefore to say, that we are infallible of a temperament in melody amounting to one third of a comma; and we never knew a person sensible of a temperament half this bulk.

When the imperfection of the octave is clearly sensible by founding the notes in succession, it is extremely disagreeable, feeling like a struggle or endeavour to attain a certain note, and a failure in the attempt. This seems owing to the familiar similarity of octaves, in the habit of talking and singing of men and women together. But when the notes are founded together, although we are not much more sensible of the imperfection of the harmony directly, as a failure in the sweetness of the concord, we are very sensible of this phenomenon of beating; and any person who can distinguish a weak sound from a stronger one, can easily perceive, in this indirect manner, any fraction of a comma, however minute. This makes the tuning by harmony much more exact than by melody alone. It is also much more accommodated to the genius of modern music. The ancients had favourite passages, which were frequently introduced into their airs, and they were solicitous to have these in good tune. It appears from passages in the writings of Galen, that different performers excelled chiefly in their skill in making those occasional temperaments which their music required. Our music is much more refined, by reason of our harmonic accompaniments, which are an abominable noise when mis-tuned in a degree, which would have passed with the ancients for very good melody. Arithoxenus says, that the ear cannot discover the error of a comma. This would now be intolerable.

But another advantage attends our method. We obtain, by its assistance, the most perfect scale of melody; perfect in a degree attainable only by chance by the Greeks. This is now to be our business to unfold.

II. Set the moveable bridge at 158, and found the two strings. They will beat very disagreeably, being plainly out of tune. Slide it gradually toward 160, and the beats will grow slower and slower; will change to a gentle and not unpleasant undulation; and at last, when the bridge is at 160, will vanish entirely, and the two sounds will coalesce into one sweet concord, in which neither of the component sounds can be distinguished. If the sound given by the short string be now examined as a step in the scale of melody, it will be found a fifth to the sound of the long string or fundamental note, perfectly satisfactory to the nicest ear. Thus one step of the scale has been ascertained.

III. Slide the bridge slowly along the string. The beating will recommence, will become a flutter, and then a jarring noise; and will again change to an angry flutter, beating about eight times in a second, when the bridge stands at 169 nearly. Pushing it still on, but very slowly, the flutter will become an indistinct jarring noise; which, by continuing the motion, will again become a flutter, or beat about six in the second. The bridge is now about 171.

IV. Still continuing the motion, the flutter becomes a jarring noise, which continues till the bridge is near the 180, when the rapid flutter will again be heard. This will become slower and slower as we approach to 180; and when the bridge reaches that point, all beating vanishes, and we have a soft and agreeable concord, but far inferior to the former concord in that cheering sweetness which characterizes the fifth. When this note is compared with that of the fundamental string as a step in the scale of melody, it is found to correspond to the note fa, or the fourth step in the scale, and in that employment to give complete satisfaction to the ear.

V. Still advancing the moveable bridge toward the nut, VI. As we move the bridge from 192 to 200, we hear again the same beatings, which, in the immediate vicinity to 192, have a peevish fretful expression; instead of the angry waspish expression before mentioned. When the bridge has passed that situation which produces only grating discordance, we hear the beatings again, and they become flower, and cease altogether when the bridge arrives at 200. Here we have another consonance, which must be called a concord, because it is rather agreeable than otherwise, but strongly marked by a mournful melancholy in the expression. In the scale of melody, it forms the third step in those airs which express lamentation or grief. It is called the minor third, to distinguish it from the last enlivening concord, which, being a larger interval, is called the major third.

It is well known, that these two thirds give the distinguishing characters to the only two modes of melodious composition that are admitted into modern music. The series containing the major third is called the major, and that containing the minor third is called the minor mode. It is worthy of remark, that the fanatical preachers, in their conventicles and field sermons, affect this mode in their harangues, which are often distinctly musical, modulating entirely by musical intervals, and keeping the whole of their chant in subordination to a fundamental or key note. This is not unnatural, when we consider the general scope of their discourses, namely, to inspire melancholy and humiliating thoughts, awakening sorrow, and the like. It is not to cay to account for the usual whine of a beggar, who generally craves charity in the major third: This is the case, at least, in the northern parts of this island.

If we continue to shift the bridge still nearer to the end of the string, we shall hear nothing but a succession of vile discordant noises, somewhat less offensive when the bridge is about the divisions 213 and 216, but even there very unpleasant.

VII. Let us therefore change our manner of proceeding a little, and again place the bridge at 160, which will give us the pleasing concord of the fifth. Instead of pushing it from that place toward the nut, let it be moved toward the wheel or bow. Without repeating what we have said of the reappearance of the beatings, their acceleration, and their degenerating into a jarring discord, to be afterwards succeeded by another beating, &c. &c. we shall only observe, that when we place the bridge at 150, we have no beatings, and we hear a consonance, which is in a slight degree pleasant, and may therefore be called a concord. It has the other marks of a concord which we have been making so much use of; for the beatings recommence when we shift the bridge to either side of 150. This note makes the sixth step in the descending scale of mournful melody; that is, when we are passing from the acute to the graver notes, with the intention of putting an emphasis on the third and the fundamental. Although not eminent as a concord with the fundamental alone, it has a most pleasing effect when listened to in subordination to the whole series, or when founded along with other proper accompaniments of the fundamental.

VIII. Placing the bridge at 144, we obtain another very pleasing concord, differing in its expression from any of the foregoing. We find it difficult to express its character. It is greatly inferior to the fifth in sweetness, and to the major third in gravity, but seems to possess, in a lower degree, both of those qualities. In the scale of cheerful melody, it is the fifth note, which we have distinguished by the syllable la. It is also used even in mournful melody, when we are ascending, with the intention of closing with the octave.

In shifting the bridge from 144 to 120, we obtain nothing but discordant, or at least disagreeable conso-uperances. And, lastly, if we move the bridge beyond 120, to divisions which are respectively the halves of those numbers which produced the concords already treated of, we obtain the same steps in the scale of the upper octave. Thus if the bridge be at 80, we have the fifth to the octave note, or twelfth to the fundamental. If it be at 60, we obtain the double octave, &c. &c. &c.

We have perhaps been rash in affixing certain moral or sentimental characters to certain concords; for we have seen instances of persons who gave them different denominations; but these were never contradictory to ours, but always expressed some sentiment allied to that which we have assigned. We never met with an instance of a person capable of a little discriminating reflection, who did not acknowledge a manifest sentimental distinction among the different concords which could not be confounded. We doubt not but that the Greeks, a people of exquisite sensibility to all the beauties of taste and sentiment, paid much attention to these characters, and availed themselves of them in their compositions. We do not think it at all unlikely, that greater effects have been produced by their music, which was studied with this express view, than have ever been produced by the modern music, with all the addition of harmony. We have allowed too great a share of our attention to mere harmony. Our great authors are much less solicitous to compose an enchanting air, than to construct a full score of rich and well conducted harmony. We do not profess to be nice judges in musical composition, but we may tell what we ourselves experience. We find our minds worked up by a continuance of fine harmony into a general sensibility; into a frame of mind which would prepare and fit us for receiving strong impressions of moral sentiment, if these were distinctly made. But we have seldom felt any distinct emotions excited by mere instrumental music. And when the harmonies have been merely to support the performance of a voice, the words have been either fo frittered by musical divisions, as to become in some measure ludicrous—or have been fo indistinct, and made fo trifling a part of the music, that there was nothing done to give a particular shape to the moral impression on our mind. We have generally been strongly affected by some of the anthems which were in vogue in former times; and we think that we perceived the cause of this difference: There was a great simplicity in the voice parts; the syllables were not drawn out into long musical phrases, but pronounced nearly according to their proper quantities; so that the sentiment of the speaker was expressed with all the force of good declamation, and the harmony of the accompaniment then strengthened the appropriate effect of the melody. We mean not to offer these observations as of much authority, but merely to mention some facts, and to assign what we felt to be their causes, in order to promote, in some degree, however insignificant, the cultivation of musical science. With this view, we venture to say, that some of the best compositions of Knapp of York uniformly affect us more than the more admired anthems of Bird and Tallis. A cadence, which Knapp gives almost entirely to the melody, is laboured by Bird or Tallis with all the rules of art; and you have its characters of perfect or imperfect, full or disappointed, cadences, and such an apparatus of preparation and resolution of discords, that you foresee it at the distance of several bars, and then the part assigned to the voice seems a very trifle, and merely to fill up a blank in the harmony. Such compositions smell of the lamp, and fall of their purpose, that of charming the learned ear. But enough of this digression.

Thus have we found a natural relation between certain sounds strongly marked by very precise characters. The concordance of sound is marked by the absence of all undulation, and the deviations from this harmony are shewn to be measurable by the frequency of those undulations. We have also found, that the notes, which are thus harmonious along with the fundamental, are steps in the scale of natural music (for we must acknowledge melody to be the primitive music, dictated by nature). We have got the notes do—mi, fa, sol, la—do, ascertained in a way that can no longer be mistaken.

Let us now examine what physical or mechanical relations these sounds stand into each other. Our monochord gives us the lengths of the strings; and the discovery of Galileo shows us, that there are also the durations of the aerial pulses which produce the sensations of musical notes. Their ratios may therefore be truly called the ratios of the sounds. Now we see that the strings which produce the sounds do sol are 240 and 160. These are in the ratio of 3 to 2. In this manner, we may state all the ratios observed in our experiment, viz:

Do : mi have the ratio of 240 to 192, or 5 to 4. Do : fa 240 : 180 = 4 : 3 Do : sol 240 : 160 = 3 : 2 Do : la 240 : 144 = 5 : 3 Mi : sol 192 : 160 = 6 : 5 = do : mi Fa : sol 180 : 160 = 9 : 8 Sol : la 160 : 144 = 10 : 9 Mi : fa 192 : 180 = 16 : 15

Here we get the sight of all the ratios which the ingenious and unwearyed speculations of the Greek mathematicians enlisted into the service of music, without being able to give a good reason why. The ratio 5 : 4, which their fallacious metaphysicians rejected, and which others wished to introduce from motives of mere necessity to fill up a blank, is pointed out to us by one of the finest concords. The interval between the fourth and the fifth is, very fortunately, a step of the scale.

The next step sol la is more important. For the ear for melody would have been very well satisfied with an interval equal to fa sol, or 9 : 8; but if the moveable bridge be set at the division 142, corresponding to such a step, we should have a very offensive fluttering. It is reasonable therefore to conclude, from analogy, that the interval sol la does not correspond to the ratio 9 : 8; and that 10 : 9, which is, at least, equally satisfactory to the ear, is the proper step, even in the scale of melody. If we consider what may be called the scale of harmony, there is no room left for doubt. To enjoy the greatest possible pleasure of harmony, we must not only take each note as it is related to the fundamental, but also as it is related to other notes of the scale. It may chance to be convenient to assume, for the fundamental of our occasional scale of modulation, the string of the lyre which is tuned as fa to its proper fundamental; or it may increase the harmony (and we know that it does), if we accompany the note do with both of the notes fa and la. To have the fine concord of the major third, it is necessary that the interval fa la be equivalent to the ratio 5 : 4. Now fa is 180, and 5 : 4 = 180 : 144. Therefore, by making the step sol la equal to 9 : 8, we should lose this agreeable concord, and get discord in its place.

And thus is evinced, in opposition to Arithoxenus, the propriety of having both a major and a minor tone; the first expressed by 9 : 8, and the last by 10 : 9. The difference between these steps is the ratio 81 : 80, called a comma by the Greek theorists.

We shall want two steps of the scale, and two sounds determined or notes corresponding to them, namely re and fa; and of these we wish to establish them on the same authority with Vitruvius. We see that this cannot be done by a concordance with the fundamental do. The ear sufficiently informs us that the steps do re and la fa must be tones, and not semitones, like mi fa. The sensible familiarity of the two tetrachords do re mi fa and sol la fa do, also teaches us that the step fa do should be a semitone like mi fa. This seems to be all that mere melody can teach us. But we have little information whether we shall make la fa a major or a minor tone. If we copy the tetrachord do re mi fa exactly, we shall make the step fa do like mi fa, and equivalent to the ratio 16 : 15. This requires the moveable bridge to be placed at 128. The sound produced by this division is perfectly satisfactory to the ear as a step of the scale of melody. Moreover, our satisfaction is not confined to the comparison of it with the note do, into which we slide by this gentle step. It makes agreeable melody when used as the third to the note sol. If we examine it mathematically, we find it a perfect major third to sol; for sol requires the 160th division. Now 160 : 128 = 5 : 4, which is the ratio of the pulses of a major third. All these reasons seem enough to make us adopt this determination of the note fa.

It remains to consider how we shall divide the interval do—mi. It is a perfect major third. So is fa la, of the same length, and so is sol fa. But in the first of these two, we have liked to see that it must be composed of a major tone with a minor tone above it; and in the second we have a minor tone followed by a major tone above. We are left uncertain therefore whether do re shall resemble fa la or sol fa in the position of its two parts. Arithoxenus and his followers declared the ear to be equally pleased with both. Ptolemy's Systema Diatonicum Inteenum makes do re a major tone, and other systems make it a minor. Temperament in modern times it has been considered as uncertain; and the only reason which we have to offer for a preference of the major tone for the first step is, that, so far as we can judge by our own feelings, the sounds in the relation of \(9:8\) are less discordant than sounds in the relation of \(10:9\), and because all the other steps have been determined by means of concords with the key. We refer, for a more particular examination of the principles on which these arrangements are valued, to Dr Smith's Harmonics, Prop. I where he shows how one is preferable to another, in proportion as it affords a greater number of perfect concords among the neighbouring notes, which is the favourite object in all modern music. Upon this principle our arrangement is by far the best, because it admits five more concords in the octave than the other. But we have considered the subject in a different manner, merely to avail ourselves of the phenomenon by which all the steps, except one, seem to be naturally ascertained, and by which the connection between harmony and melody seems to be pointed out to us.

It will be convenient to represent the tones major and minor and the hemitone, by the symbols \(T, t,\) and \(H\). Also to mark the notes by the Roman numerals, or by cyphers, according as they are the extremes of major or minor intervals. By this notation the octave may be represented thus:

\[ \begin{array}{ccccccc} C & D & E & F & G & A & B \\ \end{array} \]

The reader will remark, that the primary divisions which we assigned to the representation of an octave in fig. 1, by the circumference of a circle, are in conformity to this Ptolemaic partition of the octave. He will also be sensible, that the division into five equal mean tones and two equal hemitones, which is expressed by the dotted lines, agreeing with the Ptolemaic division only at C and E, is effected by bisecting the arch CE; and therefore the deviation of the sound substituted for the Ptolemaic D is half the difference of CD and DE, that is, half a comma. The deviations therefore at F, G, A, and B, are each a quarter of a comma.

It is well known, that if the logarithm of the length of one string be subtracted from that of another, the difference is a measure of the ratio between them. Therefore \(30103\) is the measure of the musical interval called the octave, and then the measures of the

| Comma | 540 or 54 | |-------|-----------| | Hemitone | 2803 280 | | Minor tone | 4576 458 | | Major tone | 5115 512 | | 3rd | 7918 792 | | IIId | 9691 969 | | 4th | 12494 1249 | | Vth | 17609 1761 | | 6th | 20412 2041 | | VIIth | 22185 2219 | | VIIith | 27300 2730 | | VIIIth | 30103 3010 |

This is a very convenient circumstance. If we take only the four first figures as integers, and make the octave consist of 3010 parts, we have a scale more exact than the nicest harmony requires. The circumference of a circle may be so divided into 301 degrees, and the moveable circle have a nonius, subdividing each into 10. Or it may be divided into 558 degrees, each of which will be a comma. Either of these divisions will make it a most convenient instrument for expeditiously examining all temperaments of the scale that can be proposed. 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 shall see many useful employments of these instruments by and by.

Having thus endeavoured to communicate some plain notion of the formation and singular nature of that gradation of sounds which produces all the pleasures of music, and of the manner of obtaining the steps of this gradation with certainty and precision, we proceed to consider how those musical passages may be performed on such keyed instruments as the organs and harpsichords, as they are now constructed. These instruments have twelve sounds and intervals in every octave, in order that an air may be performed in any pitch; that is, taking any one of the sounds as a key note. It is plain that this cannot be done with accuracy; for we have now seen that the interval mi fa is bigger than half of do re or re mi, &c. and therefore the intercalary sound formerly mentioned to be inserted between C and D, D and E, &c. will not do indiscriminately for the sharp of the sound below and the flat of the sound above it. When the tones are reduced to a mean size, the ear is scarcely sensible of the change in melody, and the harmony of the fifths and fourths is not greatly hurt. But when the half notes are inserted, and employed to make up harmonious intervals, as recommended by Zarlino, the harmony is very coarse indeed.

But we must make the reader sensible of the necessity of some temperament, even independent of those artificial notes. Therefore

Let the scholar tune upwards the four Vths \(e, g, c, f\), \(d, a, a, e\), all perfect, admitting no beating whatever. This is easily done, either with the organ or the wheel monochord already described. Then tune downwards the perfect octaves \(e, e, e\). Now examine the IIId \(c\) which results from this process. If the instrument be of the pitch hitherto supposed (\(c\) making 240 pulses in a second), this IIId will be heard beating 15 times in a second, which is a discordance altogether intolerable, the note \(c\) being too sharp in the ratio of 81 to 80, which makes a comma. It is easily found, by calculation, that \(c\) makes 303\(\frac{1}{2}\) pulses, instead of 300, required for the IIId to \(c\).

N.B. It may not be amiss to inform our readers, that if any concord, whose perfect ratio is \(\frac{m}{n}\) (\(m\) being the greatest term of the smallest integers expressing that ratio), be tempered sharp by the fraction \(\frac{p}{q}\) of a comma, and if \(M\) and \(N\) be the pulses made by the acute and grave notes of the concord during any number of seconds, the number \(b\) of beats made in the same time by this concord will be \(\frac{2qm}{161p-q}\) or \(\frac{2qm}{161p+q}\); and It is impossible, therefore, to have perfect Vths and perfect IIId at the same time. And it will be found, that the 3d c.g resulting from this process, and the VIth e.a., are still more discordant, rattling at an intolerable rate. Now the major and minor thirds, alternately succeeding each other, form the greatest part of our harmonies; and the VIth is also a very frequent accompaniment. It is necessary therefore to sacrifice somewhat of the perfect harmony of the Vths, in order that we may not be disgusted with the discord of those other harmonies; and it is this mutual accommodation, and not the changes made necessary by the introduction of intercalary notes, which is properly called TEMPERAMENT. It will greatly assist us in understanding the effects of the temperaments of the different concords, if we examine all the divisions of the circular representation of the octave and musical scale given in fig. 1, by placing the index of the moveable circle on that note of the outer circle for which we want the proper harmonies, or accompaniments, which are either the IIId and Vth, or the 4th and VIth. We shall thus learn, in the first place, the deviations of the different perfect notes of the scale from the notes required for this new fundamental; and we must then study what effect the same temperament produces on the agreeableness of the harmony of different concords having the same bass or the same treble, taking it for granted that the hurt to the harmony of any individual concord is proportional to its temperament.

It is in this delicate department of musical science that we think the great merit of Dr Smith's work consists. We see that the deviation from perfect harmony is always accompanied with beats, and increases when they increase in frequency—whether it increases in the same proportion may be a question. We think that Dr Smith's determination of the equality of imperfect harmony in his 13th proposition includes every mathematical or physical circumstance that appears to have any concern in it. What relates immediately to our sensations is, as yet, an impenetrable secret. The theory of beats, as delivered by this author, affords very easy, though sometimes tedious, methods of measuring and of ensuring all the varieties which can obtain in the beating of imperfect consonances. It appears to us therefore very unjust to say, with the late writer in the Philosophical Transactions, that this obscure volume has left the matter where it found it. The author has given us effective principles, although he may have been mistaken in the application; which however we are far from affirming. Our limits will not allow us to give any account of that theory; and indeed our chief aim in the present article is to give a method of temperament which requires no scientific knowledge of the subject. But we could not think of losing the opportunity of communicating, by the way, to unlearned persons, some more distinct notions of the scale of musical sounds, and of its foundation in nature, than scholars usually receive from the greater number of mere music masters. The acknowledged connection of the musical ratios with the pleasures of harmony and melody, has (we hope) been employed in an easy and not obscure manner; and the phenomena which we have faithfully narrated, shew plainly that, by diminishing the rattling undulations of tempered concords, we are certain of improving the harmony of our instruments. We shall proceed therefore on this principle for the use of the mere performer, but at the same time introducing some very simple deductions from Smith's theory, for which we expect the thanks of all such readers as wish to see a little of the reasons on which they are to proceed.

The experiment, of which we have just now given an Method in account, shews that four consecutive fifths compose a practice greater interval than two octaves and a major third. Yet, in the construction of our musical instruments of fixed sounds, they must be considered as of equal extent; since we have 7 half intervals in the Vth, and 12 in the octave, and four in the IIId, four Vths contain 28, and two octaves contain 24; and these, with the four which compose a IIId, make also 28. It is plain, therefore, that whatever we do with the IIIds, we must lessen the Vths. If therefore we keep the IIId perfect, we must lessen each of the Vths by 4th of a comma; for we learned, by the beating of the imperfect IIId c.e, that the whole excess of the four Vths was a comma. Therefore the Vth c.g must be flattened 4th of a comma. But how is this to be done with accuracy? Recollect the formula given a little ago, where the number of beats b in any number of seconds is \( \frac{2gmN}{161p+q} \). In the present case \( q = 1, m = 3, N = 240 \) per second, and \( p = 4 \). Therefore the formula is \( \frac{2 \times 3 \times 240}{161 \times 4 + 1} = \frac{1440}{645} = 2.25 \) in a second, or 9 beats in four seconds very nearly.

In like manner, the next Vth g.d must be flattened 4th of a comma, by making it beat half as fast again, or 13½ beats in four seconds (because in this Vth \( N = 360 \)). But as this beating is rather too quick to be easily counted, it will be better to tune downwards the perfect octave g.G, which will reduce N to 180 for the Vth G.d. This will give us 1,68 per second, or 10 beats in 6 seconds very nearly.

There is another way of avoiding the employment of too quick beats. Instead of tuning the octave g.G, make c.G beat as often as c.g. This is even more exactly an octave to g than can be estimated by a good ear. Dr Smith has demonstrated, that when a note makes a minor concord with another note below it, and therefore a major concord with the octave to that note, it beats equally with both; but if the major concord be below, it beats twice as fast with the octave above. Now, in the present case, c.g is a Vth, and c.G a 4th. For the same reason c.f would beat twice as fast as c.F.

In the next place, the Vth d.a must be made to beat flat 15 times in 6 seconds.

In like manner, instead of tuning upward the Vth a.e, tune downward the octave a.a, and then tune upward the Vth a.e, and flatten it till it beat 15 times in 8 seconds.

If we take 15 seconds for the common period of all these beats, we shall have The beats of \( c g = 34 \) \( G d = 25 \) \( d a = 37 \frac{1}{2} \) \( a e = 28 \)

We shall now find \( e e \) to be a fine IIIId, without any sensible beating; and then we proceed in the same way, always tuning upward a perfect Vth; and when this would lead us too high, and therefore produce too quick beating, we should tune downward an octave.

Do this till we reach \( b \), which should be the same with \( e \), or a perfect octave above \( e \). This will be a full proof of our accurate performance. But the best process of tuning is to flop when we get to \( g \). Then we tune this downward from \( e \), and octaves upward when the Vths would lead us too low. Thus we get \( f F, f f, f b, b b, b b, b b, b b \), and thus complete the tuning of an octave. We take this method, instead of proceeding upwards to \( b \); because those notes marked sharp or flat are, when tuned in this way, in the best relation to those with which they are most frequently used as IIIda.

This process of temperament will be greatly expedited by employing a little pendulum, made of a ball of about two ounces weight, sliding on a light deal rod, having at one end a pin hole through it. To prepare this rod, hang it up on a pin stuck into the wall-panelling, and slide the ball downward, till it makes 20 vibrations in 15", by comparing it with a house clock. In this condition mark the rod at the upper edge of the ball. In like manner, adjust it for 24, 28, 32, 36, 40, 44, 48 vibrations, making marks for each, and dividing the spaces between them by the eye, noticing their gradual diminution. Then, having calculated the beats of the different Vths, set the ball at the mark suited to the particular concord, and temper the sound till the beats keep pace exactly with the pendulum.

But, previous to all this, we must know the number of pulses made in a second by the C of our instrument. For this purpose we must learn the pulses of our tuning fork. To learn this, a harpsichord wire must beretched by a weight till it be unison or octave below our fork; then, by adding \( \frac{1}{2} \) th of the weight to what is now appended, it will be tempered by a comma, and will beat, when it is founded along with the fork; and we must multiply the beats by 80: The product is the number of pulses required. And hence we calculate the pulses of the C of our instrument when it is tuned in perfect concord with the fork.

The usual concert pitch and the tuning forks are so nearly consonant to 240 pulses for C, that this process is scarcely necessary, a quarter of a tone never occasioning the change of an entire beat in any of our numbers.

The intelligent reader cannot but observe, that this system of tuning with perfect IIIda, which is preferred to all others by many great masters, is the one represented by our circular figure of the octave. The IIIda is there perfect, and the Vth CG is deficient by a quarter of a comma. We cannot here omit taking notice of a most valuable observation of Dr Smith's on this temperament, and, in general, on any division of the octave into mean tones and equal limmas.

The octave being made up of five mean tones and two limmas, it is plain that, by enlarging the tones, we diminish the limmas, and that the increment of the tone is two fifths of the contemporaneous diminution of the limma. If, therefore, we employ the symbol \( v \) to express any minute variation of this temperament, and make the increment of a mean tone \( = 2 v \), the contemporaneous variation which this induces on a limma will be \( = -5 v \); and if the tone be diminished by the same quantity \( = 2 v \), the limma will increase by the quantity \( 5 v \). Let us see what are the contemporaneous changes made on all the intervals of the octave when the tone is diminished by \( 2 v \).

1. A Vth is made up of three tones and a limma. Therefore the variation of its temperament is \( = -6 v + 5 v \), or \( = -v \). That is, the Vth is flattened from its former temperament, whatever that may have been, by the quantity \( = v \). Consequently the 4th, which is always the complement of the Vth to the octave, has its temperament sharpened by the quantity \( v \).

2. A IIId, being a tone distant from the fundamental, has its temperament changed by \( = -2 v \). Therefore a minor 7th is raised by \( = 2 v \).

3. A minor 3rd is made up of a tone and a limma; therefore its variation is \( = -2 v + 5 v \), or \( = 3 v \). Therefore a major VIth (its complement) loses \( = 3 v \).

4. A maj. IIId, or two tones, has its variation \( = -4 v \). Therefore a minor 6th has its variation \( = 4 v \).

5. A maj. VIth, the complement of a limma has \( = 5 v \).

6. A tritone, or IVth, must have the variation \( = -6 v \). Therefore the false 5th must have \( = 6 v \).

From this observation, Dr Smith deduces the following simple mathematical construction: In the straight line CE (fig. 2.) take the six equal parts C, g, d, da, a, E, founded on the fix parallel lines gG, dD, &c. Let these lines represent so many scales of the octave, so placed that the points C, g, d, &c. may represent the points C, g, d, &c. of the circular scale in fig. 1, where it is cut by the dotted lines representing the system of mean tones and limmas. Then, if we take a certain length dG on the first line, to the right hand of the line CE, to represent a quarter of a comma. G will mark the place of the perfect Vth, while g represents that of the mean or tempered Vth. \( \frac{1}{2} \) th, Set off dD, double of gG, in like manner, to the right hand on the second parallel. This will be the place of the perfect IIId to the key note C. Also let off aA, on the third parallel, to the left-hand, equal to \( \frac{1}{2} \) G. This will mark the place of A, the VIth to the key note C. \( \frac{1}{2} \) th, Place E on the point e, because, in the system of mean tones represented in fig. 1, the IIIda were kept perfect. \( \frac{1}{2} \) th, Make B, to the right hand on the fifth line, equal to gG, to mark the place of the perfect VIth to the key note C. And, \( \frac{1}{2} \) th, make T, to the right hand on the sixth line, equal to twice gG. This will serve for showing the contemporaneous temperament of the tritone, or IVth, contained between F and B, as also of its complement, the false 5th in fig. 1.

It is evident that the temperament of all the notes of the octave, according to the above mentioned system, are properly represented in this figure. The Vth is tempered flat by the quarter comma Gg; the IIId is tempered flat by the half comma Dd; the VIth is tempered sharp by a quarter comma Aa; the IIId is perfect; the VIth is flat by a quarter comma Bb; and the 4th is sharp by a quarter comma Gg.

Now, let any other straight line C be drawn from the point C across these parallels. This will mark, by the inter- This gives the beats made in 16 seconds by the notes Temperament of the Scale of Music.

The third scale should have \( gG \) divided into 60 parts, for the beats made by the notes \( e, e' \), or the notes \( c, d \).

The fourth scale should have \( gG \) divided into 72 parts. This gives the beats made by the key note \( C \), with its minor third \( e \).

The fifth scale should have \( gG \) divided into 48 parts, for the beats made by the notes \( e, f \).

The sixth scale should have \( gG \) divided into 89 parts, on which \( Aa' \) is measured, to get the beats of the subordinate concord formed by \( g \) and \( e \) in the harmony of \( K - III - V \).

And, lastly, \( gG \), divided into 80 parts, will give the beats made by \( j \) and \( a \) in the harmony of \( K - 4 - VI \).

We are ignorant of the immediate efficient causes of Harmony; the pleasure we receive from certain consonances, and onefiftieth, should therefore receive, with satisfaction, anything that can help us to approximate to a measure of its degrees. We know that, in fact, the pleasantness of any individual concord increases as the undulations called beats diminish in frequency. It is probable that we shall not deviate very far from the truth, if we suppose the harmoniousness of an individual tempered concord to be proportional to the flowness of these undulations. But it by no means follows, that a tempered Vth and a IIId are equally pleasant, each in its kind, when they beat equally slow. There is a difference in kind in the pleasures of these concords; and this must arise from the peculiar manner in which the component pulses of each concord divide each other. We are certain that this is all the difference that obtains between them in Nature. But the harmoniousness here spoken of is the arrangement which produces this pleasure. We are entitled to say, that this is equal in two given instances, when the arrangements are precisely similar; and when the things arranged are the same, nothing seems to remain in which the instances can differ.

At any rate, it is of consequence to be able to proportion and distribute these undulations at pleasure. They are unpleasant; and when reinforced by uniting, must be more so. The theory puts it in our power to prevent this union; perhaps by making them very unequal; or, if this should give a chance of periodical accumulation, we may find it better to make them all equal. Surely to have all this in our power is very desirable; and this is obtained by the theory of the beats of imperfect consonances.

But we are forgetting the process of tuning, and have only tuned three or four notes of our octave. We must tune the rest by considering their relation to notes whole already tuned. Thus, if \( gc \) makes 36 beats in 16 seconds, \( Fe \) should make one third less, or about 24 in the same time; because \( N \) in the formula is now 160 instead of 240. Proceeding in this way, we shall tune the octave \( Cc \) most accurately as a system of mean tones with perfect IIIds, by making the notes beat as follows. A point is put over the note that is to be tuned from the other, and \( a+ \), or \( a- \), means that the concord is to be tempered sharp or flat. Thus \( gc \) is tuned from \( e \).

Make \( gc \) beat — 36 times in 16 seconds

\( Gc \) + 36

\( Gd \) — 27, i.e., \( \frac{3}{4} \)ths of \( gc \)

\( ef \) — 48

\( 402 \) Make Make \( c \) beat +60 times in 16 seconds

\( c \) o, i.e., a perfect IIIId

\( d \) o

\( e \) o

\( f \) downward — 24, i.e., \( \frac{8}{9} \)ths of \( c \)

\( g \) o, i.e., a perfect octave

\( h \) downward — 43, i.e., \( \frac{8}{9} \)ths of \( c \)

\( C \) o an octave.

Other processes may be followed, and perhaps some of them better than the process here proposed. Thus, \( b \) and \( c \) may be tuned as perfect IIIIds to \( d \) and \( g \) downwards. Also, as we proceed in tuning, we can prove the notes, by comparing them with other notes already tuned, &c. &c. &c.

We have directed to tune the two notes \( b \) and \( c \) by taking the leading Vth downwards. We should have come at the same pipes in the character of \( a \) and \( d \) in the process of tuning upwards by Vths. But this would not have produced precisely the same sounds, although, in our imperfect instruments, one key must serve for \( a \) and \( b \). By tuning them as here directed, they are better fitted for the places in which they will be most frequently employed in our usual modulations.

It may reasonably be asked, Why so much is sacrificed in order to preserve the IIIIds perfect? Were they allowed to retain some part of the sharp temperament that is necessary for preserving the Vths perfect, we should perhaps improve the harmony. And since enlarging the Vth makes the tone greater, and therefore the limma \( mi \) \( fa \) much smaller, it will bring it nearer to the magnitude of a half tone; and this will be better suited for its double service of the sharp of the note below, and the flat of the note above. Accordingly, such a temperament is in great repute, and indeed is generally practised, although the VIIIs and the subordinate chords of full harmony are evidently hurt by it. Even Dr. Smith recommends it as well suited to our defective instruments, and gives an extremely easy method of executing it by means of the beats. His method is to make the Vth and IIIId beat equally fast, along with the key, the Vth flat, and the IIIId sharp. He demonstrates (on another occasion), that concords beat equally fast with the same balls when their temperaments are invertedly as the major terms of their perfect ratios. Therefore draw \( E \)G, and divide it in \( p \), so that \( E \)p may be to \( p \)G as 3 to 5. Then draw \( C \)p, cutting \( g \)O in \( g' \), and \( E \)K in \( e' \); and this temperer will produce the temperament we want. It will be found, that \( E \) and \( G \) are each of them \( \frac{3}{2} \) of their respective scales.

Therefore make \( e \)g beat 32 times in 16 seconds

\( G \)c 32

\( G \)d 24

\( G \)b 24, and tune \( b \)B

\( d \)a 36, and tune \( a \)A

\( df \) 36

\( ac \) 27

\( ec \) 27

\( eb \) 40\(\frac{1}{2}\), proving \( b \)B

\( eg \) 40\(\frac{1}{2}\)

\( Fc \) 21\(\frac{1}{2}\), and tune \( Pf \)

\( Fa \) beat 21\(\frac{1}{2}\), proving \( a \)

\( bb \)f 28\(\frac{1}{2}\), and tune \( bb \)B

\( eb \) 38\(\frac{1}{2}\)

\( cc \) o

It may be proper to add to all these instructions a caution about the manner of counting the clock while the tuner is counting the beats. If this is to continue for 16 seconds, let the person who counts the clock say one at the beat he begins with, and then telling them over to himself, let him say done instead of 17. Thus 16 intervals will elapse while the tuner is counting the beats. Were he to begin to count at one, and stop when he hears sixteen, he would get the number of beats in 15 seconds only.

We do not hesitate to say, that this method of tuning by beats is incomparably more exact than by the mere judgment of the ear. We cannot mistake more than one beat. This mistake in the concord of the Vth amounts to no more than \( \frac{1}{10} \)th of a comma; and in the IIIId it is only \( \frac{1}{25} \).

It may be objected that it is fit only for the organ-pipes and instruments of continued sounds, but will not do for the quickly perishing sounds of the harpsichord. True, it is the only method worthy of that noble instrument, and this alone is a title to high regard. But farther; the accuracy attainable by it, renders it the only method fit for the examination of systems of temperament. Even for the harpsichord it is much more exact, and more certain in its process, than any other. It does not proceed, by a random trial of a flattened series of Vths, and a comparison with the resulting IIIIds, and a second trial, if the first be unsatisfactory. It says at once, let the Vth beat so many times in 16 seconds. Even in the second method, without counting, and merely by the equality of the beats of the Vth and IIIId, the progress is easy. Both are tuned perfect. The Vth is then flattened a little, and the IIIId sharpened;—if the Vth beat faster than the IIIId, alter it first.

All difficulty is obviated by the simple contrivance of a variable pendulum, already described. This may be made exact by any person that will take a little pains; and when once made, will serve for every trial. When the ball is set to the proper number, and the pendulum set a swinging, we can come very near the truth by a very few trials.

N.B. In tuning a piano forte, which has always two strings to a key, we must never attempt tuning them both at once; the back union of both notes of the concord must be damped, by flicking in a bit of soft paper behind it.

We hope that the instructions now given, and the application of them to two very respectable systems of temperament, are sufficient for enabling the attentive reader to put this method of tuning successfully in practice, and that he perceives the efficiency of it for attaining the desired end. But before we take leave of it, we beg leave to mention another circumstance, which evinces the just value of the general theory of the beats of imperfect consonances as delivered by Dr. Smith.

These reinforcements of sound, which are called beats, origin of ing, are noises. If any noise whatever be repeated, the ear, with sufficient frequency, at equal intervals, it becomes musical, or a musical note, of a certain determinate pitch. If it recur 60 times in a second, it becomes the note \( C \)fa \( w \), or the double octave below the middle \( C \) of our harp. harpsichords, or the note of an open pipe eight feet long. Now there is a similar (we may call it the very same) reinforcement of sound in every concord. Where the pulse of one found of the concord bifurcates the pulse of the other, the two sounds are more uniformly spread; but where they coincide, or almost coincide, the condensation of one undulation combines with that of the other, and there comes on the ear a stronger condensation, and a louder sound. This may be called a noise; and the equable and frequent recurrence of this noise should produce a musical note. If, for instance, c and a are founded together: There is this noise at every third pulse of c, and every fifth pulse of a; that is, 80 times in a second. This should produce a note which is a 12th below c, and a 17th major below a; that is, the double octave below f, which makes 320 vibrations in a second. That is to say, along with the two notes c and a of the concord, and the compound found, which we call the concord of the Vith, we should hear a third note FF in the bass. Now this is known to be a fact, and it is the grave harmonic observed by Rameau and Tartini about the year 1754, and verified by all musicians since that time. Tartini prized this observation as a most important discovery, and considered it as affording a foundation for the whole science of music.

We see that it is all included in the theory of beats published five years before, namely, in 1749; and every one of these grave harmonics, or Tartini's sounds, as they have been called, are immediate consequences of this theory. The system of harmonious composition which Tartini has, with wonderful labour and address, founded on it, has therefore no solidity. It is, however, preferable to Rameau's, because it proceeds on a fact founded on the nature of musical sounds; whereas Rameau's is a mere whim, proceeding on a false assumption; namely, "that a musical sound is essentially accompanied by its octave, 12th, and 17th in alto."

This is not true, though such accompaniment be very frequent, and it be very difficult to prevent it. Mr Rameau ought to have seen this. Are these acute harmonies 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; which is certainly absurd. We shall have a better occasion for considering this point when we describe the Trumpet Marigai in a future article.

We have taken notice of only two systems of temperament; both of them are systems of mean tones, and are in good repute as practicable methods. It would be almost an endless task to mention all the systems of temperament which have been proposed. Dr Smith, after having, with great ingenuity, appreciated the changes of harmoniousnesses that are induced on the different concords by the same temperament, and having assigned that proportion of temperament which renders them equally harmonious, each in its kind, gives a system of temperament, which he calls equal harmony. Each concord (excepting the octave) is tempered in the inverse proportion of the product of the terms of its perfect ratio. It is very nearly equivalent to a division of the octave into 50 equal parts. We do not give any farther account of it here, although we think its harmony preferable to anything that we have ever heard. We heard it, as executed for him, and under his inspection, by the celebrated harpsichord-maker Kirk-

mann, both when the instrument was yet in the hands of the maker, and afterwards by the ingenious author. We have also heard some excellent musicians declare, that the organ of Trinity college chapel at Cambridge was greatly improved in its harmony by the change made on its temperament under the inspection of Dr Smith. When we name Stanley, we presume that the authority will not be disputed. We mention this, because the writer in the Philosophical Transactions speaks of this system, with flattened major thirds, as of no value. But we do not give any farther account of it, because it is not suited to our instruments, which have but twelve founds in the octave.

The reader will please to recollect, that the great object of temperament is twofold. First, to enable us to transpose music from one pitch to another, so that we may make any note of the organ the fundamental of the piece. This undoubtedly requires a system approaching to one of mean tones, because the harmony must be the same in every key. This requires temperament, because a sound must be occasionally considered, either as the sharp of the note below it, or the flat of the one above. This cannot produce perfect harmony, because the limma of the perfect diatonic scale is greater than a half tone. Thus a temperament is necessary merely for the sake of the melody. But, secondly, the nature of modern music requires every note to be accompanied, or considered as accompanied, with full harmony. This is, in fact, the same thing with modulating on every different note as a fundamental; but it requires a much closer attention to the perfection of the intervals, because a defect or excess in an interval that would scarcely offend the ear, if the notes were heard in succession, is quite intolerable when they are founded together. Here the difference between the major and minor tone is of almost as great moment as the difference of the limma from a semitone. The second object, therefore, is to obtain, in the compass of three octaves, as many good concords of full harmony; that is, consisting of a fundamental with its major third and its fifth, erect or inverted, as possible. There is no other harmony, although our notes have frequently a different situation and appearance.

It is no wonder that, in a subject where we are yet to seek for a principle, the attempts to attain this object have been very various, and very gratuitous. The most very mathematicians, even in modern times, have allowed themselves to be led away by fancies about the simplicity and consequent perfection of ratios; and having no clear principle, it is no wonder that some of their deductions are contrary to experience. According to Euler, those ratios which are most perfect, that is, most simple, admit of least temperament. The octave is therefore infinitely perfect; for it is allowed by all, that it must not have the smallest temperament. A Vith must be less tempered than a IIIrd. Even the practical musician thinks that he has tempered these two concords equally, when the offensive quality of each is made equally so; but in this case it is demonstrable, that the Vith has been much more tempered than the IIIrd. But this could not be discovered till we got the theory of beats.

Most of the mathematical musicians adhered to systems of mean tones; or, which are equivalent to such systems, giving similar harmonies on every key of the Temperament of the harpsichord. This is surely the most natural, and is peculiarly suggested by the transposing of music from one pitch to another; but they differ exceedingly, and without giving any convincing arguments, in their estimation of the effects of the same temperament on different concords. Much of this, we apprehend, arises from disposition. Persons of a gay disposition relish the harmony of the third, and prefer a sharp to a flat temperament of this concord. Persons of a more pensive disposition, prefer such temperaments as allow the minor thirds to be more perfect.

But there are many, eminent both as performers and as theorists, who reject any system which gives the same harmonies on every note of the octave. They observe, that in the progress of the cultivation of music in Europe, the melodies of all nations have gradually approached to a certain uniformity. Certain cadences, cloches, strains, and phrases, are becoming every day more common; and even in the conduct of a considerable piece of music, and the gradual but slow passage of the modulation from one key into another, there is a certain regularity. Nay, they add, that this cannot be greatly deviated from without becoming very offensive. We may remain ignorant of the cause of this uniformity; but its existence seems to prove that it arises from some natural principle; and therefore it ought to be complied with, and our temperaments should be accommodated to it. The result of this uniformity in the music of our times is, that the modulation on some keys is much less frequent than on others, and this frequency decreases in a certain order. Supposing that we begin on C. A piece of plain music seldom goes farther than G and F. A little more fancy and refinement leads the composer into D, or into B, &c. &c. It would therefore be desirable to adjust our temperaments so, that the harmonies in C shall be the best possible, and gradually less perfect in the order of modulation. Thus we shall, in our general practice, have finer harmony than if it were made equal throughout the octave; because the unavoidable imperfections are thrown into the least frequented places of the scale. The practical musicians add to this, that by such a temperament the different keys acquire characters, which fit each of them more particularly for the expression of different sentiments, and for exciting different emotions. This is very perceptible in our harpsichords as they are generally tuned. The major key of A is remarkably brilliant; that of F is as remarkably simple, &c.

We cannot say that we are altogether convinced by these arguments. The violin is unquestionably the instrument of the greatest powers. A concert of instruments of this kind, unembarrassed by the harpsichord, or any instruments incapable of occasional temperament, is the finest music we have. The performers make no such degradations of harmony, but keep it as perfect as possible throughout; and a violin performer is sensible of violence and constraint when he accompanies a keyed instrument into these unfrequented paths. Let him play the same music alone, and he will play it quite differently, and much more to his own satisfaction. We imagine, too, that much of the uniformity (spoken of) is the result of imitation and fashion, and even of the temperaments that we have preferred. There is an evident distinction in the native music of different nations. An experienced musician will know, from a few bars, whether an air is Irish, Scotch, or Polish. This distinction is in the modulation; which, in those nations, follows most of the different courses, and should therefore, on the same principle, lead to different temperaments.

With respect to the variety of characters given to the different keys, we must acknowledge the fact. We have tuned a piano forte in the usual manner; but instead of beginning the process with C, we began it with D. An excellent performer of voluntaries sat down to the instrument, and began to indulge his rich fancy; but he was confounded at every step: he thought the instrument quite out of tune. But when he was informed how it had been tuned, and then tried a known plain air on it, he declared it to be perfectly in tune. It is still very doubtful, however, whether we should not have much finer music, by equalising the harmony in the different keys, and trifling for the different expression to much spoken of to a judicious mixture of other notes called discords.

After all, the great uncertainty about the most proper temperament has remained to long undetermined, this uncertainty being caused by the want of any method of executing with certainty any temperament that was offered to the public. What signifies it on what principle it may be proper to flat? Dr Smith's ten a Vth one-fifth of a comma, and sharpen a VIth theory, one-seventh of a comma, unless we are able to do both the one and the other? Till Dr Smith published the theory of beats, the monochord was the only affluence we had: but however nicely it may be divided, it is scarcely possible to make the moveable bridge so steady and so accurate in its motion, that it will not sensibly derange the tension of the string. We have seen some very nice and costly monochords; but not one of them could be depended on to one eighth of a comma. Even if perfect, they give but momentary sounds by pinching. The bow cannot be trusted, because its pressure changes the tension. Mr Watt's experiments with his monochord continued found showed this evidently. A pitch-pipe with a sliding piston promises the greatest accuracy; but we are fairly disappointed, because the graduation of the piston cannot be performed by any mathematical rule. It must be pushed more than half way down to produce the octave, more than one-third to produce the Vth, &c., and this without any rule yet discovered. Thanks to Dr Smith we can now produce an instrument tuned exactly, according to any proposed system, and then submit it to the fair examination of musicians. Even the speculatil may now form a pretty just opinion of the merits of a system, by calculating, or measuring by such scales as we have proposed, the beats produced by the tempered concords in all parts of the octave. No one who has listened with attention to the rattling beats of a full organ, with its twelfth and fifteenth flats all sounding, will deny that they are hostile to all harmony or good music. We cannot be much mistaken in preferring any temperament in proportion as it diminishes the number of those beats. We should therefore examine them on this principle alone; attending more particularly to the beats of the third major, because there are in fact the loudest and most disagreeable; and we must not content ourselves with the beats of each concord with the fundamental of the full harmony, whether K—III—V, or K—4—VI, or K—3—V, or K—4—6, which sometimes occurs. We must attend equally to the beats of the two notes of This examination is neither difficult nor tedious.

1. Write down, in one column, the lengths of the strings or divisions of the monochord; in another write their logarithms; in a third the remainders, after subtracting each from the logarithm of the fundamental. 3. Have at hand a similar table for the perfect diatonic scale.

4. Compare these, one by one, and note the difference, + or −, in a 4th column. These are the temperaments of each note of the scale. 5. Compare every couple of notes which will compose a major or minor third, or a fifth, by subtracting the logarithm of the one note from that of the other. The differences are the intervals tempered. 6. Compare these with the perfect intervals of the diatonic scale, and note the differences, + or −, and set them down in a fifth column. These are all the temperaments in the system.

7. If we have used logarithms consisting of five decimal places, which is even more than sufficient, consider these numeral temperaments as the q of the formula given in n°6; for calculating the beats, and then p is always = 540. Or we may make another column, in which the temperaments are reduced to some easy fraction of a comma.

We shall content ourselves with giving one example; the temperament proposed by Mr Young in the Philosophical Transactions for 1800. It is contained in the following table:

| 1. | 2. | 3. | 4. | 5. | |----|----|----|----|----| | C | 10000 | 5.00000 | IIIds upward on | C | 135 | | C# | 94723 | 4.97645 | G. F. | 190 | | D | 89304 | 4.95087 | D. Bb | 245 | | Eb | 83810 | 4.92330 | A. Eb | 346 | | E | 79752 | 4.90174 | E. Ab | 448 | | F | 74971 | 4.87461 | B. C# | 494 | | F# | 71041 | 4.85151 | 13490 | FX | 540 | | G | 68822 | 4.82492 | 17308 | 3ds upward on | E. E. | 236 | | G# | 63148 | 4.80366 | 19964 | D. B. | 291 | | A | 59676 | 4.77580 | 22420 | D. B. | 346 | | Bb | 55131 | 4.74921 | 25079 | G. FX | 391 | | B | 52324 | 4.72610 | 27390 | C. C# | 448 | | C | 50000 | 4.69897 | 30103 | F. G# | 494 |

Vths upward on

F. G#. C#. F# perfect

E. Bb. E. B. 46 Flat.

C. G. D. A. 116

Interval of a comma - 540 minor third - 7918 major third - 9691 fifth - 17009

The first column of the above table contains the ordinary designations of the notes. The second contains the corresponding lengths of the monochord. The third contains the logarithms of column second. The fourth contains the difference of each logarithm from the first. The next column contains, first, the temperaments of all the major thirds, having for their lowest note the found corresponding to the letter. Thus 494, or 494 of a comma, is the temperament of the IIId, B — D#, and Temperament C# — F. Secondly, it contains all the minor thirds formed of the notes represented by the letters. The column below contains the temperaments of the Vths. Templars.

N.B. These temperaments are calculated by the author. We have found some of them a little different. Thus we make the temperament of C — G only 108. Below this we have set down the measures of the perfect intervals, which are to be compared with the differences of the logarithms in column third.

We presume not to decide on the merits of this system of temperament: Only we think that the temperaments of Kirnberger, several thirds, which occur very frequently, are much too great; and many instances of the 6th, which is frequent in the flat key, are still more strongly tempered. A temperament, however, which very nearly coincides with Dr Young's, has great reputation on the continent. This is the temperament by Mr Kirnberger, published at Berlin in 1774, in his book called Die Kunst des reinen Satzes in der Musik. The eminent mathematician Major Templehoff has made some important observations on this temperament, and on the subject in general, in an essay published in 1775, Berlin. Dr Young's is certainly preferable.

The monochord is thus divided by Kirnberger:

| C | 10000 | F = 7500 | Bb = 5025 | | C# | 9492 | F# = 7111 | B = 5313 | | D | 8889 | G = 6667 | 5000 | | Eb | 8437 | G# = 6328 | | E | 8000 | A = 5963 |

We conclude this article (perhaps too long) by earnestly recommending to persons who are not mathematically disposed, the sliding scales, either circular or rectilineal, containing the octave divided into 301 parts, and a drawing of fig. 2. on card paper, of proper size, having the quarter comma about two inches, and a series of scales corresponding to it. This will save almost the whole of the calculation that is required for calculating the beats, and for examining temperaments by this test. To readers of more information, we earnestly recommend a careful perusal of Smith's Harmonics, second edition. We acknowledge a great partiality for this work, having got more information from it than from all our patient study of the most celebrated writings of Ptolemy, Huyghens, Euler, &c. It is our duty also to say, that we have got more information concerning the music of the Greeks from Dr Wallis's appendix to his edition of Porphyrius's Commentary on Ptolemy's Harmonics, than from any other work.