446 Notices respecting New Boohs. 



differences in quality due to phase can be shown. Koenig has, 1 

 believe, attacked the supposed law in this way. And it can un- 

 doubtedly be shown by very simple experiments that apparent 

 differences of quality do arise from the differences of phase. Take 

 two notes sounding an octave, and put them very slightly out of 

 tune ; then there will be a slow heavy beat consisting of variations 

 of intensity of the lower note (if the tones are pure) . By putting 

 the notes in tune again the effect at any instant may be arrested 

 and maintained ; and thus we may obtain either a combination with 

 a heavy lower note, or one with a light lower note, according to 

 the relative phase. Whether we explain this phenomenon by 

 difference tones or by Koenig's hypothesis of beat notes is a 

 question on which I do not enter. But I point out the fact that 

 this supposed law of non-dependence of quality on phase is 

 certainly not true in fact ; for the combinations with heavy and 

 light lower notes obtained as above, by varying the phase, present 

 quite different qualities in the resulting compound, as might be 

 expected. 



And yet I have no doubt that the above law of Helmholtz 

 divines correctly what would happen if we could get rid of the 

 difference tones ; that is, if we could reduce the case to one of 

 infinitely small displacements, in which terms of the second order 

 could be entirely neglected. But I suppose that those who take 

 Koenig's view would not admit this. 



While on this point I may mention that the absence of formulae 

 accompanying the little tables which deal with the mathematics of 

 sympathy makes it very difficult to find out how they were 

 calculated, notwithstanding the explanations in the Appendix. In 

 the table of Resonance and Phase for instance (p. 125) it would be 

 convenient to add, " If x be the number in the column for resonance, 

 and t the angle, then ar=cos r." According to the notation in the 

 Appendix (p. 401), the amplitude is taken to represent the reso- 

 nance (evidently by a slip of the pen), whereas the square is 

 intended. 



This whole question of representing intensity of sound by the 

 square of the amplitude of a stationary vibration seems very 

 doubtful. There is perhaps hardly any other mode of treatment 

 actually ready for use ; but there can be no doubt that a correct 

 treatment of the question of loudness of sound must (1) deal with 

 the portion of sound that reaches the ear, and not merely with the 

 vibration of the vibrating body (which may and generally does give 

 to the surrounding medium vibrations very much less than its own), 

 and (2) must base the theory of perception of loudness on assump- 

 tion consistent with Fechner's law, such as the arrangement of 

 sounds in magnitudes, in a manuer analogous to that practised in 

 the case of stars. 



I have long thought that the best basis for the hypothesis as to 

 the constitution of the resonant mechanism of the ear is afforded 

 by the complete explanation of the musical phenomena thus attain- 

 able. No doubt, however, great interest attaches to the connection 



