Notices respecting New Books. 321 



Simple tones are soft and pleasant, but dull. Musical tones, ac- 

 companied by a moderately loud series of the low upper partials, 

 are comparatively rich. If the even partials are wanting, the tone 

 is hollow and nasal. When the prime tone is weak compared with 

 the upper partials, the quality of tone is poor. 



The vowel-sounds are tones produced by the vocal chords, with 

 a resonance-chamber capable of altering in length, width, and re- 

 sonant pitch, and therefore capable at different times of reinforcing 

 different partials of the compound tone. By means of his reso- 

 nators, Prof. Helmholtz has examined the upper partials of the 

 different vowels, and has been able, by means of a combination of 

 tuning-forks and resonant chambers, to construct some of the 

 vowels, as U 6. 



The last part of this division of the work is occupied with the 

 Physiology of the Ear, which is very fully discussed, and seems to 

 justify us in concluding that in the basilar membrane with Corti's 

 arches, we have a stringed instrument capable of performing all 

 the analysis accomplished by the finest ears. 



The second division of the work concerns itself with the nature 

 of combinational tones and beats, as well as the manner in which 

 they distinguish Consonance and Dissonance. 



Combinational tones arise when the aerial displacements are so 

 large that they can no longer be taken proportional to the forces of 

 restitution. 



Suppose a particle of mass m to be impinged on by two sonorous 

 waves, with forces fsinpt and (/sin (qt + c) in the same direction, 

 and that the force due to its displacement x is ax-\- bx 2 , we have 

 for its equation of motion 



d 2 v 

 — m — - = ax -f bx 2 + f sin pt + g sin (at + c). 

 at 



Integrating this equation by successive approximations, we find 

 two vibrations of the respective periods p — q and p + q> The 

 former are the differential tones of Tartini ; the latter, discovered 

 by Prof. Helmholtz, he calls summational tones. The summational 

 tones are much weaker than the differential tones ; and both, by 

 their combination with the prime tones, give rise to weaker com- 

 binational tones. 



These tones are objective, and can be reinforced by resonators 

 tuned to be in unison with them. 



Beats, again, are subjective ; they are produced by two tones 

 nearly of the same pitch j and if Prof. ITelmholtz's theory of the 

 function of Corti's arches is correct, they are due to the same 

 Corti's arches being set in vibration by the two tones. The num- 

 ber of beats per second is equal to the difference of the vibrational 

 numbers of the interfering tones, this being the number of times 

 that the tones reinforce each other. The disagreeable character of 

 beats is to be ascribed to the fatigue which they cause in the ear 

 by the intermittent excitement they produce. Dissonance, then, 

 arises from the upper partials of two tones being so near in pitch 



Phil May. S. 4. Vol. 50. No. 331. Oct. 1875. Y 



