being the same. The octave, being thus closely related to the fundamental sound, appears to the ear only as a repetition of that sound at a different height, and thus forms a close to one series of sounds and a starting-point for another.

The next most simple ratio within these limits is that of 2 to 3 ; that is, when for every two vibrations of the first sound there are three of the second. This ratio gives us the fifth sound of the scale. The first sound and the octave may be sung together; but, as they appear so nearly alike, they are not regarded as forming harmony. The fifth sound appears as a decidedly different sound, but, from the intimacy of its relation to the first, forms with it the most perfect concord that we have in music. It may be as well to state, for the information of the unmathematical reader, that those ratios are the most simple which can be expressed in the smallest whole numbers.

The next ratio is that of 3 to 4, which gives us the fourth of the scale. This sound is at the same time the fifth below the eighth or octave, as the fifth is the fourth below it.

*

We have now obtained the four principal notes of the scale. The next ratio, that of 4 to 5, gives us the third sound of the scale. This comes as a subordinate sound between the first and the fifth, and, with them, forms a complete harmonic combination, or a common chord or major triad. This chord, then, is produced by the ratios 4 to 6 and 4 to 5, or 4, 5, 6. The next ratio, 3 to 5, gives us the sixth, which similarly completes a major triad on the fourth. Lastly, by applying the same harmonic ratios to the fifth, we get the second and the seventh sounds of the scale; but these last two sounds are related to the fundamental sound only in a secondary degree, and the ratios which they form with it are consequently less simple, being 8 to 9 for the second, and 8 to 15 for the seventh. We have now all the sounds of the scale. We have seen that they are derived from the fundamental sound or key-note in the following order (Fig. 1):

FIG. 1.-ORDER IN WHICH SOUNDS OF SCALE ARise.

Do Do Sol Fa Mi La Re Si

The five sounds immediately succeeding the key-note have all exceedingly simple ratios, and form concords with it; whence they are called the Consonances. The last two sounds have complex ratios, and do not accord with the key-note; whence they are called the Dissonances.

Now, as the pleasure derived from musical sounds arises from the apprehension of their relations, so, the more simple and intelligible the relation, the greater the amount of pleasure derived. But the more readily the ear apprehends the relations, the more easily also will the voice be able to imitate the sounds. Therefore, in order to secure a just intonation of the notes of the scale, which is the first object to be attained in teaching singing, the natural course is to train the ear to the perception of the various relations, in the order of their simplicity. The several sounds must in every case be practised first in their primary relations, and not until afterwards in their secondary and more remote relations.

All the consonances are primarily related to the key-note, and these must be practised first in connection with it. In order the better to

attune the ears of the pupils to the ratios, the sounds should be sung first in combination by two divisions of the class, and afterwards in succession by the whole class, thus (Fig. 2) :—

FIG. 2.-THE CONSONANCES IN CONNECTION WITH THE KEY-NOTE.

But, as we have seen, the consonances and the key-note group themselves into two complete harmonic combinations or full chords. These, in their primary form, exhibit three relations. They should now be sung in this form, first as harmony, then as melody; thus (Fig. 3):

The Dissonances must be practised next. They are derived from the fifth of the scale, on the model of the two triads which have just been practised, and may therefore be learned at once by singing a triad on the fifth in the same manner; thus (Fig. 4)]:

FIG. 4.-TRIAD ON FIFTH SOUND.

Ꮎ

These three triads are the materials of the major scale, and the foundation of the whole structure of music. As they are all formed by the same ratios (4, 5, 6), the major triad may be considered as the primary elemental form of music. We do not misapply Longfellow's lines, when we say—

"These are the three great chords of might,

And he whose ear is tuned aright,

Will hear no discord in the three,

But the most perfect harmony."

The gradual succession of the sounds of the scale is determined by the progressions of these chords. These progressions must therefore be practised next. The mere singing of one after another in the manner in which that has already been done, is indeed a progression between the two as wholes; but we have now to practise the progression between the individual sounds composing the chords. For this purpose, we must put the chords into such positions, that the sounds between which a passage is to be made, that is, those which are most nearly related in pitch, shall be brought into contiguity. We must therefore first practise the three chords in their three positions, as follows (Fig. 5) :— FIG. 5.-THREE ESSENTIAL CHORDS IN THEIR THREE POSITIONS.

The fundamental principle on which a progression from one chord to another depends is, that the two chords shall have one sound in common. This common tie establishes their relationship; and, to make their relationship the more perceptible to the ear, the sound in question may be held on during the progression. According to this principle, the progressions

to be practised are those between the tonic chord and the sub-dominant and dominant chords respectively. In each case, the progression should start from and return to the fundamental harmony, so that it may leave a complete impression on the ear. The chords should be sung first by the whole class as melody, then as harmony by the class in three divisions. The progression of the chords of the tonic and sub-dominant may be arranged in the three following ways, according to their three positions (Fig. 6) :—

FIG. 6.-PROGRESSION OF TONIC AND SUB-DOMINANT HARMONIES.

This progression will impress on the ear the relations among themselves of the four consonances which come in gradual succession, including the two which form the lower semitone.

The progression of the chords of the tonic and dominant may similarly be arranged in the three following ways (Fig. 7)

FIG. 7.-PROGRESSION OF TONIC AND DOMINANT HARMONIES.

This progression will impress the respective relations between the first, third, and eighth sounds and the two dissonances, including that between the two notes forming the upper semitone.

The union of these two progressions in the direct order gives us the harmonic structure of the upper tetrachord of the scale, which, ending on the fundamental harmony, has the most complete effect, and should be practised first (Fig. 8):

FIG. 8.-HARMONIC STRUCTURE OF UPPER TETRACHORD.

The union of the same progressions in the reverse order produces the harmonic structure of the lower tetrachord (Fig. 9) :

FIG. 9.-HARMONIC STRUCTURE OF LOWER TETRACHORD.

The pupils will hear that this progression is incomplete, and will immediately perceive the necessity for following it by the other tetrachord, and the beauty of the enlarged, complete, and symmetrical progression which is thus produced. The tetrachords may be combined in each of the three positions of the chords. We need only give them in the first position as an example (Fig. 10):

FIG. 10.-COMBINATION OF TWO TETRACHORDS.

We shall, by this time, have given the pupils a conception of the

scale as a complete harmonic structure, and shall have impressed on their memories the principal relations of the various sounds composing this structure to the key-note, and among themselves. We shall thus have secured, in all probability, a tolerably just intonation of the several sounds; and we may now allow the pupils to sing them separately in the order of their height.

When the sounds of the scale are sung without this previous practice of their primary harmonic relations, they are measured, not from their primes, with which they form simple and definite proportions, but from one another in their order of gradual succession, each from the one which immediately preceded it. The ratios which the sounds form with one another, when taken in this order, are as follows (Fig. 11):

FIG. 11.-RATIOS OF VIBRATIONS OF SOUNDS OF SCALE.

89

10

15 16 8 9 9 10 8 9 15 16

It will thus be seen that the voice has to execute a series of comparatively complex ratios, and that these ratios differ among one another. We have, first of all, five large intervals (tones) and two smaller ones (semitones), the former having the more simple ratios, and being therefore the more easy to sing. But, further, the tones are not all alike in size. It will also be observed, that we pass to sounds having a direct and near relation to the key-note, through others having an indirect and remote relation to it. The fact is, that to sing the scale with purity of intonation without hearing the accompanying harmonies, and without having an impression of them fresh in the mind, is a task at which even a great public singer may fail. The difficulty of getting pupils to sing the semitones correctly is notorious among teachers of singing. We believe that correctness in singing these important intervals (for they determine the structure of the scale as melody) can be attained only by the plan which we have recommended, whether adopted sooner or later in the instruction.

The principle which we have here endeavoured to establish is recognized directly by M. Weber in the theoretical part of his method, although he has not carried it out in the practical part. M. Weber makes the following excellent remarks, which will not only enforce but continue our own :

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Experience teaches us that the ratios which produce the sounds three and four, and seven and eight, are not usually performed with sufficient exactness, and that, in this way, the key-note from which the singers started gets lost. The defective performance of the ratios must become perceptible as early in the scale as the third and fourth sounds, because they ought to have the most agreement with the key-note. So soon as it happens that these sounds are not sung pure, it becomes impossible to sing the fifth pure, as, under these circumstances, the key-note no longer stands firm. But the moment the key-note begins to waver, it also becomes impossible to sing the octave pure: it is derived from the new key-note, which is generally lower than the former one. There is another peculiarity in regard to the, fourth even if the harmonic main points, three and five, are properly sustained, still the fourth will generally fail. This is natural. The fourth forms with the third

the most remote ratio, 15-16, which is the most difficult to hit, if the fourth does not occur as an harmonic main point, which, however, it cannot be simultaneously with the third and fifth; and therefore a pupil who is not acquainted with the different harmonic relations of the sounds most nearly connected with the key-note, will be much more likely to sing the third twice, and then pass on to the fifth, than to give the fourth even an approximation to its proper height. On the other band, so soon as the fourth is made to occur as an harmonic main point, purity in singing it may be attained with ease.'

The principle is recognized indirectly by Dr. Marx in his admirable work, "General Musical Instruction," now brought within the reach of the general English public by the professional zeal and commercial enterprise of Mr. Novello; and by Mr. Curwen, in the first lessons in his "Grammar of Vocal Music," which, in our opinion, are exceedingly good, and show Mr. Curwen to be himself a good elementary teacher.

The truth of the principle has also been felt by M. Wilhem, who, although he begins the practice of interval with the second, and ends with the octave, yet teaches the tonic chord of the scale at the commencement of the whole. The harmonic relations of the scale are not given, in his method, until the middle of the second part, and then they are given incorrectly, the progression being carried out of the key in the case of the descending upper tetrachord. The "twaddling thirds" produced by singing the scale in canon, as is done in his exercises on seconds over and over again ad nauseam, is certainly not calculated to give a very correct impression of its fundamental harmonic relations. M. Wilhem also teaches the two tetrachords after he has taught the whole scale. But, indeed, in this method, everything seems to be done backwards.

J. T.

(To be continued.)

AFFECTION OF TEACHER.- "This affectionate earnestness, therefore, it is plain, should be sought for by all who possess it not, and should be made still more influential in the case of those in whom it dwells. The three great rules which have been laid down for ministers are, on this point, equally applicable to teachers; namely, that they should get their subject into their minds-throw themselves into their subject and pour both themselves and their subject into the bosoms of those whom they address. As has been well said, 'There is most of the heart where there is most of the will; and there is the most of the will where there is most endeavour: and where there is most endeavour, there is generally most success; so that endeavour must prove the truth of our desire, and success will generally prove the sincerity of our endeavour."-British Messenger.