Waves and ripples in water, air, and æther

by Sir J. A. Fleming

Several of these intervals or ratios of frequencies have received names. Thus the interval C to E, = 4:5, is called a major third, and the interval E to G, = 5:6, is called a minor third; the interval C to G, = 2:3, is called a fifth, and that of C to C^¹, = 1:2, is called an octave. For the purposes of music it has been found necessary to introduce other notes between the seven notes of the octave. If a note is introduced which has a frequency greater than any one of the seven in the ratio of 25 to 24, that is called a sharpened note; thus the note of which the frequency is ³⁄₂n × ²⁵⁄₂₄ would be called G sharp, and written G♯. In the same way, if the frequency of any note is lowered in the ratio of 24 to 25, it is said to be flattened. Then the note whose frequency is ³⁄₂n × ²⁴⁄₂₅ would be called G flat, and written G♭.

It is obvious that if we were to introduce flats and sharps to all the eight notes we should have twenty-four notes in the octave, and the various intervals would become too numerous and confusing for memory or performance. Hence in keyed instruments the difficulty has been overcome by employing a scale of equal temperament, made as follows: The interval of an octave is divided into twelve parts by introducing eleven notes, the ratio of the frequency of each note to its neighbours on either side being the same, and equal to the ratio 1 to 1·05946.

The scale thus formed is called the chromatic scale, and by this means a number of the flats and sharps become identical; thus, for instance, C♯ and D♭ become the same note. The octave has therefore twelve notes, which are the seven white keys, and the five black ones of the octave of the keyboard of a piano or organ.

Every one not entirely destitute of a musical ear is aware that certain of these musical intervals, such as the fifth, the octave, or the major third, produce an agreeable impression on the ear when the notes forming them are sounded together. On the other hand, some intervals, such as the seventh, are not pleasant. The former we call concords, and the latter discords. The question then arises—What is the reason for this difference in the effect of the air-vibrations on the ear? This leads us to consider the nature of simple and complex air vibrations or waves.

Let us consider, in the first place, the effect of sending out into the air two sets of air waves of slightly different wave lengths. These waves both travel at the same rate, hence we shall not affect the combined effects of the waves upon the air if we consider both sets of waves to stand still. For the sake of simplicity, we will consider that the wave-length of one train is 20 inches, and that of the other is 21. Moreover, let the two wave-trains be so placed relatively to one another that they both start from one point in the same phase of movement; that is, let their zero points, or their humps or hollows, coincide. Then if we draw two wavy lines ([see Fig. 56]) to represent these two trains, it will be evident that, since the wave-length of one is 1 inch longer than that of the other—that is, a distance equal to twenty wave-lengths—one wave-train will have gained a whole wave-length upon the other, and in a distance equal to ten wave-lengths, one wave train will have gained half a wave-length upon the other. If we therefore imagine the two wave-trains superimposed, we shall find, on looking along the line of propagation, an alternate doubling or destruction of wave-effect at regular intervals. In other words, the effect of superimposing two trains of waves of slightly different wave-lengths is to produce a resultant wave-train in which the wave-amplitude increases up to a certain point, and then dies away again nearly to nothing, as shown in the lowest of the three wave-lines in [Fig. 56].

Fig. 56.—The formation of beats by two wave-trains.

We must, then, determine how far apart these points of maximum wave-amplitude or points of no wave effect lie. If the wave-length of one train is, as stated, 20 inches, then a length of ten wave-lengths is 200 inches, and this must be, therefore, the distance from a place of maximum combined wave-effect to a place of zero wave-effect. Accordingly, the distance between two places where the two wave trains help one another must be 400 inches, and this must also be the distance between two adjacent places of wave-destruction. If, therefore, we look along the wavy line representing the resultant wave, every 400 inches we shall find a maximum wave-amplitude, and every 400 inches a place where the waves have destroyed each other. We may call this distance a wave-train length, and it is obviously equal to the product of the constituent wave-lengths divided by the difference of the two constituent wave-lengths.

It follows from this that if we suppose the two wave-trains to move forward with equal speed, the number of maximum points or zero points which will pass any place in the unit of time will be equal to the difference between the frequencies of the constituents. Let us now reduce this to an experiment. Here are two organ-pipes exactly tuned to unison, and when both are sounded together we have two identical wave-trains sent out into the air. We can, however, slightly lengthen one of the pipes, and so put them out of tune. When this is done you can no longer hear the smooth sound, but a sort of waxing and waning in the sound, and this alternate increase and diminution in loudness is called a beat. We can easily take count of the number of beats per second, and by the reasoning given above we see that the number of beats per second must be equal to the difference between the frequencies of the two sets of waves. Thus if one organ-pipe is giving 100 vibrations per second to the air, and the other 102, we hear two beats per second.

Now, up to a certain point we can count these beats, but when they come quicker than about 10 per second, we cease to be able to hear them separately. When they come at the rate of about 30 per second they communicate to the combined sound a peculiar rasping and unpleasant effect which we call a discord. If they come much more quickly than 70 per second we cease to be conscious of their presence by any discordant effect in the sound.