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. The theory was first put forward by the famous physicist, Von Helmholtz, that the reason certain musical intervals are not agreeable to the trained ear is because the difference between the frequencies of the constituent fundamental tones _or the harmonics present in them_ give rise to beats, approximately of 30 to 40 per second. In order to simplify our explanations we will deal with two cases only, viz. that of the _octave_ interval and that of the _seventh_. The first is a perfect concord, and the second, at least on stringed instruments, is a discord. It has already been explained that when a string vibrates it does so not only as a whole, but also in sections, giving out a fundamental note with superposed harmonics. Suppose we consider the octave of notes lying between the frequencies 264 and 528, which correspond to the notes C and C^1 forming the middle octave on a piano. The frequencies and differences of the eight tones in this octave are as follows:— FREQUENCIES OF THE NOTES OF THE MIDDLE OCTAVE OF A PIANO. Notes. Frequency. Difference.