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LVII. On the Beats of Consonances of the Form h : 1. 

 By R. H. M. Bosanquet*. 



[Plates IV.-VII.] 



Ohns Law, and the Hypothesis of Resultant Displacements. 



1. r PHE doctrine known as Ohm's law states that the 

 JL simplest form of motion by which definite musical 

 pitch is defined to the ear is the pendulum-vibration. It may 

 be extended as follows: — In all cases in which Ohm's law 

 holds, the ear resolves any complex of two or more simulta- 

 neous pendulum-vibrations into the original pendulum-vibra- 

 tions of which they consist, and hears them as distinct and 

 independent sounds. 



I rest my belief in Ohm's law mainly on ordinary pheno- 

 mena, not on refinements or difficult observations ; and I 

 shall endeavour to make this my course throughout. 



2. So long as, in our mechanical arrangements, we approxi- 

 mate more and more nearly to the condition of things by 

 which we know that simple harmonic vibrations must be pro- 

 duced, we also approximate in the character of the resulting 

 sound to a pure and simple note of definite pitch, free from 

 harmonics and other accompaniments. 



So far as simple sounds are concerned, therefore, we re- 

 ceive Ohm's law as being at all events approximately true 

 generally, and in all probability absolutely true when sounds 

 of small intensity alone are considered. 



3; When two different sounds are heard together, we have 

 phenomena of which the following is a slight sketch. 



If the two sounds are very nearly of the same pitch, they 

 are not heard according to Ohm's law, i. e. separately and 

 independently, but in the form of resultant displacements. 

 The most important case is that in which the two sounds are 

 of nearly equal intensity. In this case one sound is heard, 

 intermediate in pitch between the two primaries, and oscilla- 

 ting in intensity between a certain maximum and nothing. 

 These oscillations are what are called the beats of imperfect 

 unisons. Now, as the two notes separate from one another in 

 pitch, the character of the phenomenon changes; and at a 

 certain point the two notes begin to be heard separately and 

 independently, beside the beats which accompany them. It 

 is this phenomenon that is accounted for by Helmholtz's 

 theory of the existence of vibrating bodies, in the ear, having 

 sympathy of a certain definite degree with the various notes. 



4, Helmholtz ascertained the approximate degree of this 

 * Communicated by the Physical Society, 



