or the Division of the Octave. 169 



T. Perronet Thompson. Their examination would involve much 

 technical detail. They both represent forms of just intonation. 



I need not enter into any detail as to Helmholtz's work in 

 this Magazine, more especially as it is now accessible in an 

 English translation. But I must allude to a serious defect in 

 TyndalFs exposition of Helmholtz's theory of beats in the ear, 

 to which my attention was drawn by Professor Mayer's paper in 

 the 'American Journal of Science/ October 1874, though I be- 

 lieve it had been commented on before by Mr. Sedley Taylor. 



In ' Tyndall on Sound ' (2nd edit. p. 296) we find the fol- 

 lowing statement as the reason why no beats are audible in the 

 octave c,-c 2 : — 



" Here our rates of vibration are 512, 256; difference =256. 

 It is plain that in this case we can have no "beats, the difference 

 being too high to admit of them." 



This is not Helmholtz's position. His theory is that notes 

 an octave apart affect different portions of the nervous mecha- 

 nism of the ear, and consequently no beats ever take place 

 between these sounds at all when they are received in the ear. 



Again, Helmholtz does not say, as Tyndall makes him, that 

 beats blend always into a continuous sound when they attain 

 the limit of 132 per second. He says (3rd edit. p. 270) of the 

 132 beats per second produced by the interval b 3 c 4 , iC and these 

 are really audible in the same manner as the 33 beats of b 1 c 2 , 

 although they sound somewhat weaker in the higher position." 



An account of some repetitions of the experiments of Pro- 

 fessor Mayer contained in the paper above referred to appeared 

 recently in the Philosophical Magazine. They consist of deter- 

 minations of the limiting rates at which beats cease to be distin- 

 guished for all parts of the musical scale. 



Mr. Ellis's paper in the ( Proceedings of the Royal Society' 

 for 1874 deals almost exclusively with just intonation. The 

 remark I would make with reference to this is, that the chief 

 difficulty in applying just or approximately just systems to ordi- 

 nary music lies in the music itself; this does not conform to 

 certain rules which must be observed in writing for just intona- 

 tion; and I feel sure that until just intonation is studied, and 

 the fact that it must be specially written for is recognized, no 

 real benefit can be got from its employment. 



A question of primary importance, which has perhaps scarcely 

 received sufficient attention, is, Is there any natural standard for 

 excellence in melodic sequences ? and if so, what is it ? 



Mr. Ellis (Proceedings of the Royal Society, 1864) and, I 

 believe, Mr. Sedley Taylor, accept Helmholtz's authority for the 

 position that the sequences of the diatonic scale are the best, 

 and that in proportion as the sequences of any other scale differ 



