THE ANALYTIC PROPERTIES OF THE EAR. 1175 



that of an interval with a certain relation of phases, but after a short 

 period the phase relationship will change and will pass through every 

 possible value. The cycle is audible ; there are beats on the imperfect 

 harmony. Now it might be said that these beats were produced 

 between 510 and the first partial of ut 2 , namely, w 3 =512, but as the 

 beats are always most distinct when ul 2 is sounded very feebly, this 

 appears to be out of the question. In like manner, beats may be heard 

 on the approximate harmonies, 2 : 3, 3 : 4, 4 : 5, 5 : 6, 6 : 7, 7 : 8, 1 : 3, 3 : 5, 

 and they all " fulfil the condition of having the whole period of the 

 imperfection, and not any submultiple of it, for their period." It is also 

 easy to produce beats on a major chord in which one of the forks is 

 slightly flattened. For example, sound ut 3 , mi s , and sol 3 = 4:5:6; then 

 slightly flatten sol 3 or mi 3 , and a peculiar beat will be heard as if a wheel 

 were being turned against a surface, one small part of which was rougher 

 than the rest ; the effect is always loudest when the tone of the forks 

 is allowed to almost die away, and one of the forks is then gently touched 

 with the bow. 



From the point of view of these investigations, the question under 

 discussion assumes an aspect different from that present to the mind of 

 Helmholtz. A harmony (by which we mean an octave, fifth, major 

 chord, etc.) is a sound which, without being a simple tone, has the 

 variations of air pressure strictly periodic, and, according to Fourier's 

 harmonic analysis (analysis of periodic variations), any periodically vary- 

 ing quantity may be looked on as the sum of quantities, varying separately 

 according to the simple harmonic law, in periods respectively equal to 

 the main period, half the main period, a third of the main period, a 

 fourth of the main period, and so on. Thus, the variation of the air 

 pressure of a harmony is the sum of the variations of simple tones, one 

 having a period equal to the period of the harmony, a second one-half, 

 a third one-third, and so on. A harmony is therefore composed of 

 simple tones, if produced by such instruments as well-bowed tuning- 

 forks. On listening to such a complex tone or sound, a tone may pre- 

 dominate in the sensory impression, and the pitch of the sound is 

 referred to the main period of this tone, the other tones merely giving 

 a special character to the sound. In musical harmony, however, one 

 tone does not thus predominate, and a sound of two, three, or four, or 

 more tones, having commensurable periods, is heard, and the period of 

 the harmony is the least multiple of the period of its constituent tones. 

 The harmonic number of a tone in a harmony is denoted by the number 

 of times that the period of the harmony contains this particular tone. 

 Again, the quality or character of a harmony depends on the amplitudes 

 of the tones forming it, the frequency period of these tones, and the 

 relation of their phases. It is most important to observe the effect of 

 phase. Thus the instants of maximum pressure or of minimum pressure 

 may coincide, or there may be a coincidence of the maximum pressure 

 of one tone with the minimum pressure of the other. If, now, the tone 

 of a perfect binary harmony be very slightly sharpened or flattened, and 

 if the two tones are sustained so as to secure perfect uniformity as to 

 pitch and intensity, the effect is an imperfect harmony, with a slow 

 change of phase relation through a cycle, and there is a variation of the 

 quality of the tone recurring periodically on the imperfect harmony. 

 This is the beat in the imperfect harmony, and it indicates that the ear 

 does distinguish between an increase of pressure on the drum-head and 



