102 Mr. R. H. M. Bosanquet on the Mathematical 



sequently it may be expected, from the symmetry, that minute 

 exactness of the position is not of special importance. Putting, 



then, £ = 7— b = -, (7) reduces to 



7r(A 2 -X 2 )\ tan^=XA 2 , .... (8) 



which gives an infinite number of values of X when A, X are 

 assigned. 



I now assume A = / (the reed the octave of the string), and 



X = — , as a pair of values such as may easily occur, and con- 

 venient for calculation, for the sake of seeing the general nature 

 of the results to be expected. 



The equation (8) can then be put in the form 



I 



The numbers placed under the head -, in the Table which 



X 



follows, are approximate values of the first five roots of the above 

 equation. Proceeding further, we should find a root lying be- 

 tween every consecutive pair of integers. 



The second column contains the values of the ratios - reduced 



X 



to equal-temperament semitones ; it gives the pitch of the note 

 sounded with reference to the octave of the string. 



The third column gives the pitch of the note sounded with 

 reference to the lowest note of the combination, both in equal- 

 temperament semitones and by description. 





Pitch, in equal. 



Pitch, referred to lowest note of combination. 



1 



temperament 

 semitones, 









\ 



referred to -' 



Equal-tem- 







perament 



Description. 







semitones. 





•5868 



-9-230 







1-441 



+ 6326 



15-556 



Flat major tenth. 



2-357 



16132 



25 362 



Sharp minor sixteenth. 



3295 



20-646 



29-876 



Flat two octaves and tritone. 



4-25 



25 052 



34-282 



Sharp two octaves and minor seventh. 



Although it has not been possible to get a complete deter- 

 mination of the elements of any experiment, yet the following 



