reason is, scientifically speaking, that the number 7 is not sufficiently small for a consonant harmony, and that being great, it has the fault of being a prime number. Since, even in discords, it is of the greatest importance not to make too much use of such numbers, numbers greater than 7, but divisible by 2, 3, 4, or 5, have a great numerical advantage over it.* And this is the true and principal reason why no use is made of it in music.

Without wishing to push too far forward, and to prophesy what will happen in the future, it may be observed that the systematic introduction of the seventh harmonic into music would produce in it a very deep and almost incalculable revolution-a revolution which does not seem justifiable, because for our magnificent musical system another would be substituted, perhaps as magnificent, but certainly not better, and probably worse, and at any rate more artificial. This does not, however, exclude the possibility that a secondary part in the musical system will hereafter be assigned to the seventh harmonic. To certain chords-for example, the chord of the diminished seventh -and discords it is well suited, and may sometimes be substituted with advantage for some in present use.

However this may be, it is certain that for us the seventh harmonic represents the great line of demarcation between consonant harmonies and discords. Below it is

* Euler had remarked the importance of the numbers 2, 3, and 5, and established upon them a rule for the development of our musical system.

consonance, and above it dissonance, and between these is a great hiatus. We have thus the following harmonies. in the interval of one octave :

This demonstration may be terminated by representing in musical notation the harmonies examined above, with the resultant notes of the first order which arise from them. The harmonies will be found on the upper line, in the treble clef; the resultant notes on the lower line, in the bass clef.

This mode of illustration will serve to make what has been explained above more clear.

CHAPTER VI.

1. HELMHOLTZ'S DOUBLE SIREN-2. APPLICATION OF THE LAW OF SIMPLE RATIO TO THREE OR MORE NOTES-3. PERFECT MAJOR AND MINOR CHORDS, THEIR NATURE-4. THEIR INVERSION.

1. THE laws explained in the preceding chapter can be demonstrated by means of the siren constructed by Helmholtz, which is called the double siren (fig. 28). It is composed of two complete sirens, ao, a1, placed one over the other, so that their revolving discs face each other. These are attached to the same arbor k, and therefore turn together with the same velocity. In the middle of this arbor is the counter (not represented in the figure), which is intended to measure the number of turns when absolute measurement is required. Each disc carries four concentric circles of holes, according to an idea already realised by Dove; and by means of four buttons i any one of the circles of holes, or all simultaneously, can be set in action. There are thus eight notes, which can be produced at will, at the disposition of the experimenter.

A strong current of air, which can be caused to enter either siren at go or g1, produces, as in the simple siren, the rotation of the discs and the formation of the

notes.

In the upper disc the circles have successively 9, 12, 15, 16, in the lower disc, 8, 10, 12, 18 holes. There are thus many possible combinations of notes, the vibrations of which have simple ratios to each other.

The revolving discs are covered by cylindrical boxes ho, ho, h1, hɩ, which serve to reinforce the notes produced, and to make them clearer.

For the study of beats there is an arrangement by which these ratios can be slightly altered. In fact, the box of the upper siren can be turned independently of the rotatory movement of the discs. This is attained by means of a handle d, which turns the toothed wheel e, which gears into one on the upper siren, and is so arranged that for each three turns there is one complete revolution of the siren itself. Under the handle is a graduated circle divided into sixteenths, whence each division corresponds to of a complete turn of the siren. So when the siren is turned, it follows the holes in the revolving disc, or moves in a contrary direction to them, according to the direction of the rotation.

The effect in the first case is to somewhat lower the note, in the second to raise it. The reason of this is, that the note depends on the number of puffs of air which are produced each time that the holes of the revolving disc coincide with the holes of the siren; these puffs will be more or fewer in number, according as the siren is caused to move in a direction contrary to, or the same as that of the revolving disc. Very varied experiments