CHAPTER V.

1. MUSICAL SOUNDS-2. LAW OF SIMPLE RATIO—3. UNISON, INTERFERENCE -4. BEATS-5. THEIR EXPLANATION-6. RESULTANT NOTES-7. OCTAVES AND OTHER HARMONICS-8. CONSONANT CHORDS AND THEIR LIMITS-9. THe major fifth, FOURTH, SIXTH, AND THIRD; THE MINOR THIRD AND SIXTH-10. THE SEVENTH HARMONIC.

1. It has been shown in the last chapter that all the sounds in nature are not musical sounds, properly so called. In order that a sound may acquire a musical character, it must satisfy the essential condition of being agreeable to the ear. It is on this account that all the sounds produced by imperfect instruments must be rejected, whatever may be their pitch. All those, also, which are too high or too low must be rejected as either disagreeable or insignificant. There remain, therefore, the notes comprised between about 27 and 4000 vibrations per second, which form an interval of a little more than seven octaves, between which limits the music of all countries and all nations is written.

But it would be a grave error to suppose that between the limits hinted at, all the notes can be used arbitrarily or at hazard. Experience shows that any one of these notes may be chosen in executing or beginning a piece

of music. But when once this note is selected, all the others that are to follow or accompany it are limited, and we move in a very restricted circle. This is not only the case in our modern music, but also holds good for the music of every epoch. There is no instance known of a musical system, however barbarous it may be, in which the choice of the notes is left to the fancy of the composer or performer. The history of music, on the contrary, teaches us that it has always been sought to select, from the enormous number of possible notes, an infinitely more restricted number, according to certain established rules, in which musical instinct was at times influenced by scientific theories of greater or less value, giving the preference to one and sometimes to another of such theories. We will consider later on the different conceptions which instinctively or rationally have guided different nations in the historical development of music. For the present I will content myself by saying that in our modern music, art has outstripped science with rapid strides, and it is only quite recently that the latter has been able to give a complete and rational explanation of what the former has effected by means of delicate æsthetic feeling.

2. It may be established as one of the fundamental principles of our music, that the ear can only endure notes, be they simultaneous or successive, on this condition-namely, that they should bear simple ratios to each other in respect of the number of their vibrations per second;

that is to say, that the ratio of the number of vibrations per second of the notes should be expressed by low numbers. All the bearings of this simple principle will be pointed out in a later chapter, and since, thanks to the great researches of Helmholtz, it has acquired of late years, notwithstanding its simplicity, an even simpler and wider significance, I will for the present content myself with indicating its more important consequences.

It is not without some hesitation that I enter upon such a subject. I shall have to go through a series of figures, and indeed to argue entirely upon figures. The road is rather a rough and thorny one, but I trust that, like the traveller who courageously climbs the steep and rugged sides of a mountain in order to enjoy at last a vast and magnificent panorama, so from the highest peak of this argument a vast horizon will open out before the reader, in which he will discover the synthesis of one of the grandest creations of the imagination—a creation that in itself forms one of the most brilliant pages in the history of human culture.

3. The most simple ratio that can be imagined between the vibrations per second of two notes is that in which both are represented by the same number of vibrations. The two notes are then said to be in unison. If they be sounded one after the other, they only form one more prolonged note; if they be sounded together, they only give one note of double loudness. It sometimes happens, however, that two equal notes, instead of supporting each

other, are enfeebled in their effects. Cases of this kind are due to what is called interference. This happens

whenever the vibrations of the two notes are made in the

reverse way—that is to say, when the vibrating body of the first note makes a movement in one given direction, whilst the other makes a precisely contrary movement. It is evident that such opposing vibratory movements must destroy each other's effect, when superposed in the air in which they are propagated; as a particle of air which ought to move at the same time and with the same force in two opposite directions, not being able to follow either, remains at rest.

Fig. 27.

The apparatus represented in fig. 27 enables us to produce interference at will. It is composed of a vibrating plate in which a Chladni's figure is formed by the vibrating segments A B, A' B'. The vibrations in two

contiguous segments, as A' and B', are contrary, or reverse, inasmuch as when the particles at A' fall, those at B' rise, and vice versa; they are similar in two opposite segments, as at A and A'. DCE is a bifurcated pipe which gives by itself the same note as the plate, and is closed at the top by a paper membrane, which serves, when sprinkled with sand, to indicate the vibrations in the pipe. If now the plate be caused to vibrate, and some sand be sprinkled on it to indicate its mode of vibration, two points having similar vibrations, as A and A', may be selected, and the branches of the pipe placed over them, without, however, allowing them to touch; the sand on the membrane will then dance about and dispose itself regularly, which shows that the air in the pipe vibrates, because the vibrations of A and A' support each other in producing this effect. Again, two points having an opposite motion, as A' and B', may be selected and the branches of the pipe placed over them; the effect will then be nil, and the sand will not move.

From all this we may conclude that when two equal and simultaneous vibratory movements are superimposed, they support each other, but that, on the other hand, their effect is destroyed if they are equal and opposite.

4. The question is interesting as to what happens when the two notes produced are almost though not quite identical, and have not therefore quite the same number of vibrations per second. A new phenomenon then appears, known by the name of beats.