to exhibit these phenomena to several persons at once, the forks should first be mounted on their resonant-boxes, and, after the pellets have been attached, stroked with a resined bow, care being taken to produce tones as nearly as possible equal in intensity. Slow beats may also be obtained from any instrument capable of producing tones whose vibration-numbers differ by a sufficiently small amount. Thus, if the strings corresponding to a single note of the pianoforte are not strictly in unison, such beats are heard on striking the note. If the tuning is perfect, a wax pellet attached to one of the wires will lower its pitch sufficiently to produce the desired effect. Beats not too fast to be readily counted arise between adjacent low notes on the harmonium, or, still more conspicuously, on large organs. They are also frequently to be heard in the sounds of church bells, or in those emitted by the telegraph wires when vibrating in a strong wind. In order to observe them in the last instance, it is best to press one ear against a telegraph-post and close the other: the beats then come out with remarkable distinctness. It should be noticed that, when we are dealing with two composite sounds, several sets of beats may be heard at the same time, if pairs of partial-tones are in relative positions suited to produce them. Thus, suppose that two

clangs coexist, each of which contains the first six partial-tones of the series audibly developed. Since the second, third, &c. partial-tones of each clang make twice, three times, &c. as many vibrations per second as their respective fundamental-tones, [§ 43] it follows that the differences between the vibration-numbers of successive pairs of partial-tones belonging to the two clangs will be twice, three times, &c., the difference between the vibration-numbers of the two fundamental-tones. Accordingly, if the fundamentaltones give rise to beats, we may hear, in addition to the series so accounted for, five other sets of beats, respectively twice, three, four, five and six times as rapid as they. In order to determine the number of beats per second for any such set, we need only multiply the number of the fundamental beats by the order of the partial-tones concerned. The beats of two simple tones necessarily become more rapid if the higher tone be sharpened, or the lower flattened; i.e. if the interval they form with each other be widened. The beats may, however, also be accelerated without altering the interval, by merely placing it in a higher part of the scale. Greater vibration-numbers are thus obtained with a proportionately larger difference between them, though their ratio to each other remains what it was before. Thus the rapidity of the beats due to

an assigned interval depends jointly on two elements, the width of the interval and its position in the musical scale; in other words, on both the relative and absolute pitch of the tones which form it.

CHAPTER VIII.

ON CONCORD AND DISCORD.

79. A question of fundamental importance now presents itself, viz. What becomes of beats when they are so rapid that they can no longer be separately perceived by the ear? In order to answer it, the best plan is to take two unison-forks of medium pitch, mounted on their resonance-boxes, attach a small pellet of wax to a prong of one of them, and then gradually increase the quantity of wax. At first very slow beats are heard, and as long as their number does not exceed four or five in a second, the ear can follow and count them without difficulty. As they become more rapid the difficulty of counting them augments, until at last they cease to be separately recognisable. Even then, however, the ear retains the conviction that the sound it hears is a series of rapid alternations, and not a continuous tone. Its intermittent character is not lost, although the intermittances themselves pass by too rapidly

for individual recognition. Exactly the same thing may be observed in the roll of a side drum, which no one is in danger of mistaking for a continuous sound.

Rapid beats produce a decidedly harsh and grating effect on the ear; and this is quite what the analogy of our other senses would lead us to expect. The disagreeable impressions excited in the organs of sight by a flickering unsteady light, and in those of touch by tickling or scratching, are familiar to every one. The effect of rapid beats is, in fact, identical with the sensation to which we commonly attach the name of dissonance. Let us examine, in somewhat greater detail, the conditions necessary for its production between two simple tones. If we take a pair of middle-C forks, and gradually throw them more and more out of unison with each other in the way already described, the roughness due to their beats reaches its maximum when the interval between them is about a half-Tone: for a whole Tone it is decidedly less marked, and when the interval amounts to a Minor Third, scarcely a trace of it remains. Hence, in order that dissonance may arise between two simple tones, they must form with each other a narrower interval than a Minor Third. If we call this interval the beating-distance for two such tones, we may express the above condition thus. Dissonance can arise directly between two simple