1 1 78 THE EAR. 



Helmholtz has shown in a table the difference in pitch in terms of an 

 equally tempered tone (J of an octave) necessary to reduce the intensity of 

 sympathetic vibration to ^ of that produced by perfect unison (A), and the 

 number of vibrations after which the intensity of tone, in a body whose sound 

 is allowed to die out, diminishes to y 1 ^ of its original amount (B). 1 The table 

 is as follows : 



A. B. 



tone 38-00 



19-00 



9-50 



6-33 



4-75 



3-80 



(tempered minor third) . . . 3'17 



2-71 

 " (major third) 2 '37 



That is to say, suppose a body were started into vibration by an exact unison, 

 and that the resonance was reduced to y 1 ^ of its original amount, 38 vibs. 

 would be executed before the intensity of the free vibration was reduced to 

 y 1 ^ in the case of tone, and so on. Now, if the ear required 38 vibs. before 

 the sympathetic resonance was reduced y 1 ^, as the time occupied for these vibra- 

 tions for A would be ^ second, for a ^ second, for a y 1 ^ second, it is evident 

 that such a state of things would disturb musical effect, because the first note 

 would not have died away before the end of the second note. This would 

 certainly occur if we executed a shake on a piano of eight or ten notes a 

 second, so that each note would be sounded four or five times. But it is well 

 known that below A the sensation excited by such shakes is rough and 

 unpleasant, indicating, in the opinion of Helmholtz, " that the vibrating parts 

 of the ear are not damped with sufficient force and rapidity to allow of 

 successfully effecting such a rapid alternation of tone." Now, if a body is 

 thrown into vibrations by sympathetic resonance, it has the vibration frequency 

 of the exciting tone, but when the exciting tone ceases, it goes on moving at 

 the rate of its own proper tone. If, then, the ear vibrated as a single system, 

 and continued its movements for a sensible time, it would move at its own rate, 

 which, as has been shown, is independent of the rate of the exciting tone. 

 Thus two tones of a shake, either in high or low tones, would not mix, but 

 they might blend with that of a third tone, that of the ear itself, said by 

 Helmholtz to be between e"" and /", or, say, /"". 



To set this aspect of the question in a clear light, we cannot do better than 

 quote from Helmholtz. 2 " Now, if a shake of 10 notes in a second be made on 

 A, of which the vibrational number is 110, this tone would be struck every 

 \ of a second. We may justly assume that the shake would not be clear, if 

 the intensity of the expiring sound were not reduced to y 1 ^ of its original 

 amount in this i of a second. In this case, after at least 22 vibs., the parts 

 of the ear which vibrate sympathetically with A must descend to at least 

 y 1 ^ of their intensity of vibration as their tone expires, so that their power of 

 sympathetic vibration cannot be of the first degree in the table, but may belong 

 to the second, third, or some other higher degree. That the degree cannot be 

 very much higher, is shown, in the first place, by the fact that shakes and runs 

 begin to be difficult, even in tones which do not lie very low. . . . We may, on 

 the whole, assume that the parts of the ear which vibrate sympathetically have 

 an amount of damping power corresponding to the third degree of our table, 

 where the intensity of sympathetic vibration with a semitone difference of 

 pitch is only y 1 ^ of what it is for a complete unison." 



1 Helmholtz, op. cit., p. 213 ; alsoRayleigh, op .cit., vol. i. p. 52, where the mathematical 

 theory of damping is fully explained. 



2 Helmholtz, op. cit., p. 215. 



