434 Mr. R. H. M. Bosanquet on the Beats 



of beats per second. Then the mistimed octave is n : 2n±m; 

 the mistimed twelfth is n : on±.m; and so on. 



49. Beats of the mistimed octave, 



n : 2n±??2. 

 Number of beats =m. 



m variations of intensity of the lower note (n) are produced 

 by interference of notes n and n±m; and n±m is the first 

 combination-tone (difference-tone of form p — q), of the pri- 

 maries n and 2n±m. 



This rests chiefly on the observation that the beats, when 

 the octave harmonic is eliminated, consist entirely of variations 

 of intensity of the lower note. 



The existence of the first combination-tone in question 

 (p — q) is well known. It is easily demonstrated in the neigh- 

 bouring case of intervals not far removed from the fifth, when 

 the beats of the first two combination-tones are specially pro- 

 minent (secondary beats of Konig). 



50. Beats of the mistuned twelfth, 



n : 3n±m } 



Number of beats =m. 



m variations of intensity of the lower note (n) are produced 

 by interference of notes n and n±m. And n-±.m is the second 

 combination-tone (difference-tone of form 2p — q) of the pri- 

 maries n and 'dn±m. \ 



This rests also chiefly on the. observation that the beats, 

 when the third partials are eliminated, consist entirely of 

 variations of intensity of the lower note. 



The existence of the second combination-tone in question 

 (2p — q) is demonstrated in many cases by Konig. It is easily 

 heard in the case of intervals near the octave high in the 

 scale. It is also easily detected by the secondary beats which 

 it forms with the first combination-tone in the case of inter- 

 vals near the fifth — also less easily by the secondary beats 

 which it forms with the third combination-tone in intervals 

 near 2 : 5, at which point the second and third combination- 

 tones coincide. 



51. Beats of the mistuned fifteenth or. double octave, 



n :4n±m. 

 Number of beats = m. 



m variations of intensity of the lower note (n) are produced 

 by interference of notes n and n±m. And ?z±?n is the third 

 combination-tone (difference-tone of form 3p— q) of the pri- 

 maries n and 4n±m. 



This rests also chiefly on the observation that the beats, when 



