shown on the principles of Mechanics, that if the string is shortened by being stopped at any point, the vibrations of the shortened string (or those movements backwards and forwards which, being quicker than the eye can distinguish, cause the string to have that fuzzy appearance which children view with interest), will be more numerous in a given time (as for instance in the time between two ticks of a watch) than those of the whole or longer string; and further, that they are more numerous in the same proportion that the string is shorter. As for instance, if the string is shortened to the half which is 12 inches, then the vibrations of the shortened string will be twice as many as those of the long; if it is shortened to 16 inches, which is two-thirds, the shortened string will make three vibrations in the time the longer makes two; and in like manner for any other lengths. After which the child will willingly receive, and in fact be delighted to be told, that the reason why the string shortened to the half, sounds what is called the Octave, most probably is its vibrations being twice as quick as before. At all events what is certain is, that when the vibrations are made twice as quick, the Octave comes, and when not, not. From which the child will readily admit the conclusion, that the reason it can sing with its father as well as if they made the same sounds (though it is clear one voice is shrill and the other is deep) is because the vibrations of its own sounds are always twice as quick. A child will eagerly seize on all this, as what makes the subject easy; it is only grown people who fancy it what they call hard.

10. At this point the pupil will have no difficulty in perceiving, that since the shrillness of the sound depends on the degree in which the string is shortened, the readiest way of expressing how much a sound is shriller than the sound of the open string, is to write down the length of the string which will make it, or, which comes to the same thing, the proportion of this length to the length of the open string. And the two numbers which express this proportion in its lowest terms, are called the ratio of the sound. Thus the readiest way of describing the Octave, is to say that its ratio is that of 1 to 2, or as usually written, 1: 2; which means that it is the sound made by shortening the original string in the proportion of 1 to 2, or by the half. And the like in other cases.

11. The child having acquired this knowledge on the subject of the Octave, must be encouraged to make similar discoveries with respect to the other sounds, as for instance the Fifth. Let it therefore, as before, run up the notes with its voice from c to the Fifth which is g, using the sounds of the piano forte in aid if it chooses; and when it has got a clear impression of the sound it wants, let it feel about as before for the place where the string must be stopped to make it. And it will not be long in discovering, that the place which makes the Fifth is to be found by dividing the string into three equal parts and shortening it by one of them, as will be done by stopping it at 16 inches from the bottom; and consequently the ratio of the Fifth may be set down as being that of 2: 3. And the pupil will find it as easy to believe as before, that the reason why the string stopped at two-thirds of its length produces the sound which is called the Fifth, and which is manifestly one of the sounds the practised musician makes on the violin and the baby does not, is because the shortened string makes three vibrations in the time the open string makes two.

12. Let the child be next asked to find the place of the Fourth; and it will speedily discover that it makes it by shortening the string by one

fourth, or making the sounding part 18 inches, thus pointing out the ratio of the Fourth as being that of 3: 4. And if asked next to find the places of the Major and the Minor Third, it will probably venture a guess, and try shortening the string by the fifth part and by the sixth, which will be found right; making the sounding part for the Major Third 19 inches, and its ratio that of 4: 5; and for the Minor Third, the sounding part 20 inches, and its ratio that of 5 : 6.

13. Here then the secret is out. The child has by its own efforts arrived at the knowledge, that the peculiarity in the sounds which bring music out of the string, is that there is a short and easy proportion between the numbers of the vibrations in the same time, in them and in the sound of the original string. Two lengths of string taken at hazard, would have some proportion or other between the numbers of their vibrations; as for instance one might make a hundred vibrations while the other made ninetynine. But this is not a short and easy proportion; and therefore two such lengths of string would not be among those which make music. They make such sounds as a baby might produce on a violin, but not such as would be made by a musician who knew how to play. And for the principle which has enabled us to arrive at the knowledge of what makes this difference, the child will not be sorry to know that we are indebted to Pythagoras; for unless he or somebody else had made the observation, nobody would at this hour have been able to tell with exactness what was meant by an Octave, a Fifth, or any other of the names in use in music, or what was a just one and what was not.

14. If the child, pursuing the clue obtained, were to shorten the string by the seventh part and by the eighth, it would find this did not produce more of the sounds it had learned to sing. The reason of which is, that the sounds so produced make no Fifths, Fourths, or Major or Minor Thirds, with any of the sounds in other ways established, whereas the others do; as will be better understood after completing the other sounds, and seeing how they make combinations of this kind with one another (§§ 23 to 40.)

15. The places of the Major and Minor Sixth are still to look for; and it is plain they must lie between those of the Fifth and Octave. And when the child has found them, with a little help from the tutor, it will be able to ascertain, that the first is made by dividing the string into five equal parts and shortening it by two of them, which makes the length of the sounding part 142 inches; and the other by dividing the string into eight equal parts and shortening it by three of them, which makes the length of the sounding part 15 inches. Or to express it by the method of ratios, the ratio of the Major Sixth is that of 3 : 5, and of the Minor Sixth 5: 8. In which it is not difficult to see that 3 to 5 is the shortest and easiest proportion it is possible to make or imagine, which shall fall on a place lying between the places of the Fifth and Octave; and that (skipping over the proportions of 4 to 7 and 5 to 7 for the same reason as those of 6 to 7 and 7 to 8) 5 to 8 is the next. And what has been here called a short and easy proportion, is in mathematical language called a simple ratio.

16. An Interval is the difference in shrillness or deepness between one musical sound and another; and is described by naming the ratio of the two sounds, or of the lengths of string which make them. And two intervals are said to be equal, when the proportions between the lengths of string are the same, though the sounds themselves are not the same, but both of them different. And that interval is said to be the greatest, where

the disproportion between the lengths of string is greatest. Thus the interval between the Fifth and the Minor Third is said to be equal to the interval between the Major Sixth and the Fourth; for they are both made by lengths of string having to one another the proportion of 4 to 5. And the interval between the Minor Sixth and the Minor Third is said to be greater than either of these; for it is made by lengths of string having to one another the proportion of 3 to 4, which is a greater disproportion than that of 4 to 5, inasmuch as three-fourths of anything is less than four-fifths. 17. For convenience, the two numbers which express the ratio of any interval are divided the less by the greater, and the vulgar fraction resulting is called the Measure of the interval. Or when preferred, the Measure is given in decimals. And this Measure always expresses the portion of a string (as the half, two-thirds, three-fourths, &c.) which must be taken to produce the shriller of the two sounds, while the whole string produces the other.

18. An interval may either be presented between two sounds sounded simultaneously, or in succession. For distinction the first may be called a simultaneous interval, and the other a successional. The same sounds which produce the effect the moderns call harmony when heard together, produce the effect they call melody when heard in succession. Which looks as if melody was a kind of retrospective harmony, depending on a perception of harmonious relation to sounds which have gone before. Arpeggio passages are an illustration of this sort of connexion; their sounds being those of a harmonious chord spread out to meet the incapacity of the instrument for producing sustained sounds.

19. The sounds hitherto described are called the Consonances; from the agreeable effect they produce when sounded along with the Key-note. And the combinations they make with the Key-note when sounded along with it, or the combinations of any other sounds making the same interval, are called Concords; as are also the intervals when one is sounded after the other.

20. The Measures of the Consonances as taken from the marks on the string, are consequently as follow. A line over one or more figures, indicates that such figures are to form a perpetually recurring decimal. Key-note, or Open String

1.
or .833

or .8

or .75

Fifth
Minor Sixth
Major Sixth
Octave

or .66

or .625 or '6

or ⚫5

21. If the pupil should ask why two of the Consonances are called Thirds and two Sixths, it is because the two are not ordinarily found in the same passage or air, but in some the sharper or Major set is found employed in the places of Third and Sixth, and in others the flatter or Minor. Out of which arise two distinct kinds or characters of music, and the two are distinguished by the one being said to be in the Major series, and the other in the Minor.

22. The Key-note and any of its repetitions in the shape of successive Octaves above and below, is called the Tonic. Which is a useful term when more than one are intended to be included. Also the Major and Minor series of the same Key, with reference to one another, are called the Tonic Major and Minor, as being on the same Tonic. The successsive Octaves to any sound, either above or below are called its Replicates.

23. The next thing is to show the pupil that the Consonances make exact concords with one another; with reservation only that the interval between them is not less than makes a concord at all, and that the Third and Sixth shall be both Major or both Minor as is but reasonable.

24. And here it is a misfortune that the moderns have not separate names for Consonances and for concords, as to a great extent the ancients had (See Appendix, § III.). For it is impossible a learner should not be puzzled with such phrases as the interval between the Major Third and Major Sixth being "a Fourth," or "equal to a Fourth." But what is meant is, that the interval the Major Sixth makes with the Major Third, is equal to the interval the Fourth makes with the Key-note, or is made by the same proportion of lengths of string, viz. 3 to 4. And in like manner the term "Octave" is found used to express both the sound which is the Octave to the Key-note, and the interval between two sounds in any part of the music which are such that one of them is the Octave to the other; besides which there is a third use of the term, which is when the division of the string is extended so as to reach to two or more Octaves one after another, and sounds intermediate to these are described as being "in the first, second, &c., octave,' or "in the octave above what some other sound is in, or "the octave below." In which a good deal may be done for clearness, by settling that "the Fourth, Fifth, &c." shall always mean the sound of that name in the existing Key; and " a Fourth, Fifth, &c." shall mean an interval between two sounds equal to that which the Fourth, Fifth, &c. makes with the Key-note, meaning always a Fourth, Fifth, &c. above unless the contrary is expressed. And further, that the word "Octave" shall be printed with a capital letter when it means a sound, and with a small letter when it means anything else. All of which has been adhered to in the present work.

25. To begin with the Minor Third, as being the nearest to the Key-note. The length of string which makes the exact Minor Third is five-sixths of the open string, or 20 inches; and the length which makes the exact Fifth is two-thirds or 16 inches. But 16 is four-fifths of 20; which is the proportion that makes the exact Major Third from the Key-note, and consequently the interval between the Minor Third and the Fifth is exactly equal to the interval between the Key-note and the Major Third, which is expressed by saying it is an exact Major Third. In like manner may be shown that the interval between the Minor Third and Minor Sixth, is an exact Fourth; between the Major Third and Fifth, an exact Minor Third; between the Major Third and Major Sixth, an exact Fourth; between the Fourth and Minor Sixth, an exact Minor Third; between the Fourth and Major Sixth, an exact Major Third; and that between any of the Consonances in one octave and any in the octave above or below, when the interval is not greater than an octave, exact concords will be made, which for brevity are not specified. And in the same manner may be ascertained, that the interval between the Minor Third and the Octave is an exact Major Sixth; between the Major Third and the Octave, an exact Minor Sixth; between the Fourth and the Octave, an exact Fifth; between the Fifth and the Octave, an exact Fourth; between the Minor Sixth and the Octave, an exact Major Third; between the Major Sixth and the Octave, an exact Minor Third. All very curious instances of the properties of numbers, and what nobody could have made if it had not been made for us.

26. To take (as in the last §) first the exact Minor Third to the open string, and then the exact Fifth to the open string, and find what is the interval between that Minor Third and that Fifth, is called subtracting a Minor Third from a Fifth, or finding their difference. Which was shown to be an exact Major Third. And so in other cases.

27. Conversely, to take first the exact Minor Third to the open string, and then the sound which shall make an exact Major Third to that, and find what is the sound on the string so arrived at, is called, adding a Major Third to a Minor Third, or finding their sum. Which was shown to be an exact Fifth. And so in other cases.

28. To any person accustomed to the use of vulgar fractions it can hardly fail to have occurred, that there was an easier way of performing the preceding operations, and that for expressing in a vulgar fraction the Measure of the sum of two intervals, nothing is required but to multiply their Measures together, and for expressing the measure of the difference of two intervals, divide the Measure of the greater interval (which will be the smaller of the two Measures) by the Measure of the other, which will be done by inverting the latter and multiplying by it.

29. The interval between any named sound and the Octave is called the complement of the first sound; and the sound which bears the same name as this interval is called the complemental. A consequence of which is, that the interval between the complemental and the Octave, is always equal to the interval between the original sound and the Key-note. Thus the complement of the exact Major Third is an exact Minor Sixth, and the Minor Sixth is the complemental to the Major Third; and vice versá. Hence, from what has preceded, to find the complement of any sound, double the Measure and invert it, keeeping everything in its lowest terms. Which leads to the observation, that the sounds peculiar to the Minor series are all the complementals of sounds in the Major and vice versâ, to the minutest particular; the Fourth and Fifth being common to both series, and complementals to each other. Which is a source of great facilities in the sequel. All this has commonly been conveyed under the term "inversion;" which is not so significant as "complement."

THE DISSONANCES; WHAT THEY ARE; AND THAT THEY ARE DOUBLE.

30. The pupil will readily take notice, that as yet nothing has been said of any but sounds towards the middle of the octave, viz. from the Minor Third to the Major Sixth, and that there is a large interval at each end, viz. between the Key-note and the Minor Third, and between the Major Sixth and the Octave, equal to each other, and looking very invitingly for the insertion of something to he called the Major and the Minor Second at one end, and the Minor and the Major Seventh at the other (as, whether with more or less of accuracy, is seen done on the pianoforte), and which though not making concords with the Key-note, shall make them with some of the sounds established under the title of Consonances.

31. To begin with the note the pupil is most familiar with, which is the Major Second. It is clear that a sound hereabouts, may be so contrived as to make concords with some of the Consonances already determined. A sound, for instance, may be arranged, which shall be an exact Fourth below the Fifth of the Key; or one may be arranged which shall be an exact Minor Third below the Fourth, and an exact Fifth below the Major Sixth. And here comes to light the fact, that for both these purposes no