Sounding of two Notes. 429 



d lY 



e IV 



8: 9 

 4: 5 



d 



— ra is distinctly perceptible. 



— m is loud. 



f lv 



5632 v. s. 



3: 4 

 8:11 



r 



9" 



— m is equally loud. 

 d" m and mf loud. 



9 lv 



2: 3 



d" 



d" m — m f quite loud. 



6656 v. s. 



8:13 



e m 



g" m and mf both audible. 



oF 



3: 5 



f" 



/ // m and m' audible. 



7168 v. s. 



4: 7 



g m 



c" m audible, mf louder than m. 



7936 v. s. 

 8064 v. s. 



8:15 



16:31 

 32:63 





d m quite inaudible, mf distinctly 



perceptible. 

 d mf only distinctly audible. 

 c mf perceptible. 



If the entire series of observations here set forth with their 

 results be reviewed, it will be found that, taken as a whole, 

 they show as follows : — 



(1) The lower beats ra, as well as the upper beats m! — n—m 

 of an interval n :hn + m (h=l, 2, 3. . .), when the number of 

 beats and the intensity of the primary notes are sufficient, 

 change into beat-notes ; for example the notes of the ratio 

 8 : 15, C : b, allow ra' = 8 beats to be heard, and the notes of 



the beat-note mf = c, the notes c IV : b lx 

 Further, with the notes of the ratio 4 : 15 

 (n : Sn + m), C : 6, a distinct roll of 16 upper beats is heard, 

 and with the notes of the same ratio, c'" : d 1Y , the upper beat- 

 note m / = c / . 



(2) The beat-notes in the high octaves, and the singly au- 

 dible beats in the low ones, are always equal to the two dif- 

 ferences of the double vibrations of the higher primary notes, 

 and of the two upper and lower notes of the harmonic series 

 lying next above and below the deeper primary note, and not, 

 as has been hitherto assumed, simply equal to the difference of 

 the double vibrations of the two primary notes. For example, 

 the notes of the ratio 4:9, d" : d 1Y allow the beat-note 

 m=l = c', and no trace of the note 9— 4 = 5 = e /// , to be heard; 

 d" : e 1Y (2:5) gives m=l = c // , and no trace of g" ! . The 

 ratio n : 2n + m, 4 : 11, formed by the notes 2048 (c f,f ) and 

 5632 v. s., allows further the beat-notes m=3=g f/ and 

 m f = l = d to be perceived, and no trace of the note 

 7 = 3584 v.s. 



(3) Of the beat-notes of the higher octaves m and mf, as 

 well as of the singly audible beats m and mf of the lower ones, 



m alone is audible when m is much less than -, mf when 



