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



resultant "of two pendulum-curves do not, as a rule, exhibit 

 any appearances corresponding to pendulum-curves having 

 the period of the Smith's heat, except in a very small number 

 of cases, the conditions for which can only be fulfilled by acci- 

 dent ; (2) that the increases and diminutions of the maximum 

 displacement which form what we have called the bows of the 

 harmonic curves, correspond in duration with the Smith's 

 beat, but not in the period of the harmonic curves of which 

 they form part. 



98. We infer from the previous parts of this paper : — 

 (3) that the variations of maximum displacement which are 

 represented in these figures by the bows of the harmonic 

 curves, give rise by transformation to pendulum-vibrations 

 having the same frequency as those variations — these being 

 the notes which Konig calls beat-notes, and Helmholtz differ- 

 ence-tones of various orders ; and (4) that the actual beats 

 of mistune.d consonances of the form h : 1, as heard by the ear, 

 are given rise to by the interference of these beat-notes or 

 difference-tones with the lower note of the combination. 



99. The upper numbers prefixed to the Plates of curves are 

 the ratios of the wave-lengths ; the lower ones the ratios of the 

 amplitudes. 



Typical curve of case I. . f J- 



9: 8 



Typical curve of case II. . |-J- 



Typical curve of case III. § -J-, or rather J x f £ 



9:2 10 : 1 



Postscript. — The curves shown in the Plates are all illustrations 

 of the subject of this paper, with the exception of three sets ; 

 namely, the combinations of vibrations whose wave-lengths are 

 nearly as 4 : 5, as 2:3, and as 2:5. These have been given for 

 the sake of completeness in the collection of curves, and that 

 readers may have the opportunity of seeing the nature of the dif- 

 ference between such curves as these, which may be said to belong 

 to mistimed consonances of the form h : h, and our normal forms 

 which belong to mistimed consonances of the form h : 1. 



