THE SENSE OF HEARING. 831 



sound becomes higher in pitch ; for it is obvious that as we ascend the scale, 

 and the waves of sound become progressively shorter, spaces would be left 

 between the individual waves unless their number were proportionately 

 increased. 



Harmony and Discord. Tones are concordant, or harmonize, when they 

 produce no beats on being sounded together ; they are discordant when beats 

 are produced, and the painful sense of dissonance increases in intensity up to 

 about 33 beats per second. Perfect concord is obtained by blending notes 

 whose vibrations are to each other as small whole numbers. 



Thus, in the major cord c E G c 



the vibration-numbers are 132 165 198 264 



their ratios are 4568 



If notes the ratios of whose vibration-rates can be represented only by large 

 whole numbers are combined, a discord is formed, for the reason that their 

 upper partials interfere with one another and cause beats ; there is no especial 

 virtue in the small integer. 1 



Thus, in the discord ODE 



the vibration-numbers are 132 148.5 165 



which are not reducible to small whole numbers. 2 



Combinational Tones. When two tones are sounded together, there is 

 produced a new, usually weaker, tone, whose vibration-number is the numerical 

 difference between the vibration-rates of the original tones. It is therefore 

 known as a differential tone. Such tones may arise from upper partials as well 

 as from the fundamentals ; they do not appear to be formed, as might be sup- 

 posed, by the fusion of beats. Other "combinational" tones of more intricate 

 relations, as well as beats, arise from the interaction of vibrations when many 

 different notes, as those of an orchestra, are sounded together. To calculate 

 the physical result of the combination of these impulses, which it is the duty 

 of the tympanic membrane to transmit, is a problem of exceeding complexity. 



Resume. To sum up the subject, musical sounds are distinguished in sen- 

 sation by the three factors, loudness, pitch, and quality, sometimes called color 

 or timbre. These sensations depend in turn on definite physical characters of 

 air-waves : their amplitude, or the extent of motion of the air-molecules ; their 

 frequency, or rate of succession of the waves; their form, which is deter- 

 mined by the pitch and relative predominance of the upper partials combined 

 with the fundamental tone. 



Fatigue. That the ear is subject to fatigue toward a note that has been 

 sounded is easily demonstrated in the following way : Strike a single note of, 

 say, a major chord on the piano, and immediately afterward sound the full 

 chord; the quality of the latter will be altered from its normal character, 

 owing to the lessened prominence of the note which had been struck. 3 We 

 may therefore not improperly speak of a successive contrast in auditory sensa- 



1 Tyndall : Sound. * Waller : Human Physiology, 1891. 



3 Foster : Text-book of Physiology, 5th ed., 1891. 



