468 HARMONY degree of unpleasantness, however, . depends on the number of the beats or flashes per second, and also varies with the pitch of the sound or the color of the light. But when the beats have reached a certain number in a second, they no longer produce intermittent effects on the nerves; for the action produced by one beat lasts, without perceptible diminu- tion, until the arrival of the following one, and the sensation becomes continuous ; in other words, when the beats follow with sufficient rapidity, they blend together and form a smooth, sonorous effect, like a simple musical sound. This relation between discontinuous and continuous impressions on the nerves, and unpleasant and pleasant sensations, is at the foundation of Helmholtz's theory of musical harmony. We must now consider the effects resulting when, instead of producing only simple sounds together, as above, we simul- taneously produce composite sounds differing slightly in pitch. If we sound two tuning forks, each giving the middle C of the piano, we shall have two simple sounds in unison. Now gradually elevate one of them in pitch and observe the changing sensations. The harshness increases until they are separated about a tone; then the disagreeable sensation diminishes, and entirely vanishes when the notes have been separated by an interval equal to a minor third. But if, instead of sounding the forks, we use two reed pipes giving the same notes, we observe that the slightest de- parture from unison at once causes a very un- pleasant sensation ; the reason of this is, that besides the beats of the fundamental simple sounds of the pipes, we have the sensations produced by the beating of some 20 harmon- ics of their fundamentals. Therefore the tuning of reed pipes is difficult, but their in- tervals are defined with an extraordinary de- gree of sharpness. It is here also to be re- marked that the number of beats per second given by any pair of harmonics is directly as their height in the harmonic series. Tims if the fundamental or first harmonics give 3 beats per second, the sixth harmonics will give 18 beats per second. Therefore, in sounding two such pipes, each giving 20 harmonics, we should have produced on the ear 632 beats per sec- ond, 3 belonging to the first pair of harmonics, and 60 to the 20th pair. Helmholtz's discov- ery consists in the demonstration of the fact that the degree of smoothness or consonance of any given chord depends entirely on the number of elementary harmonics and resultant tones which beat together in the given notes, on the intensities of these beats, and on the number per second of beats produced by each dissonant pair of harmonics. This fact* he proved by nearly every means known to mod- ern science, and thus established a real physi- cal cause for the harmonious or dissonant sensa- tions we experience on combining various notes. We can best illustrate the truth of Ilelmholtz's theory and show his main results by giving in musical notation the principal intervals of fun- damental notes, indicated in minims, with their accompanying harmonics written over them in crotchets. Only the first six harmonics are in- dicated, because those of higher order are gen- erally either absent from a musical sound, or exist with such feeble intensity as not greatly to affect the degree of consonance. The respec- tive harmonics which beat we have connected together by straight lines, so that at a glance one can approximately determine the degree of consonance of a given interval. The inter- vals here given are the true intervals of the natural scale, and not the false intervals of the tempered scale. On the latter scale the only consonant interval is the octave. The intervals we have selected are the octave, the fifth, the fourth, the major third, the major sixth, and the minor seventh ; the ratios of the vibra- tions giving the notes of these intervals are respectively as 1 : 2, 2 : 3, 3 : 4, 4 : 5, 3 : 5, and 9 : 16. THE OCTAVE. No dissonance here occurs because the har- monics of both notes are in unison. We have here two pairs in unison, 3-2 and 6-4 ; but a slight departure from perfect smoothness of effect is caused by the third harmonic of the higher note beating with the fourth and fifth of the lower. If the vibra- tions of the two fundamental notes of this in- terval are not rigorously 'as 2 : 3, there will be discord. Hence, on all instruments of fixed equal-tempered scales, as the organ or piano, even the interval of the fifth is slightly discor- dant, only the octave intervals being in tune. THE FOURTH.
