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 tones only when they are within beating-distance of each other1. It follows from what was shown at the end of Chapter VII. that the same beating interval will give rise to very different numbers of beats per second according as the tones which form it occupy a low or high position in the scale. Such an interval, e.g. a whole Tone, becomes palpably less dissonant as it is successively raised in pitch. Accordingly the beating-distance, which for tones of medium pitch we have roughly fixed at a Minor Third, must be understood to expand somewhat in low, and correspondingly contract in high, regions of the scale. The general partial-tone series consists of simple tones which, up to the seventh, are mutually out of beating-distance: above the seventh they close in rapidly upon each other. In the neighbourhood of 10, the interval between adjacent partial-tones is about a whole Tone; near 16, a semi-Tone; higher in the series they come to still closer quarters. These high partial-tones are, therefore, so situated as to produce harsh dissonances with each other. Where they are strongly developed in a clang, there 1 It will be shown in the sequel [§§ 89-92] that dissonance may, in the case of simple tones forming intervals wider than a Minor Third, arise from beats other than those here under consideration. will therefore be a certain inevitable roughness in its quality. This is the cause of the harsh character of trumpet or trombone notes, and also of the shrill tremulous sounds sometimes observed in the human voice (§ 68). In fact we may regard the partialtone series above the seventh tone, when fully represented, as contributing mere noise to the clang. This explains why it is advantageous that the lower partial-tones of organ reeds should be more strengthened by resonance than the higher ones [see § 66]. 80. It has been shown that, when two simple tones are simultaneously sustained, beats can arise directly between them only under one condition, viz. that the tones shall differ in pitch by less than a Minor Third, or thereabouts. When, however, the two co-existing sounds are no longer simple tones, but composite clangs each consisting of a series of well-developed partial-tones, the case becomes altogether different. Let us examine the state of things which then presents itself. The sounds of most musical instruments practically contain only the first six partial-tones; we will, therefore, assume this to be the case with the clangs before us. No beats can then arise between partial-tones of the same clang, since no two of them are written a Minor Third of each other. Dissonance due to beats will, however, be produced |