THE FOURlEklAN ANALYSIS OF WA VF-PdRMS. 1223 



This analysis displays only one considerable reinforcement, but it extends 

 in varying strength over the sixth, seventh, eighth, and ninth partials. The 

 resonance chamber in this case is the inter-velar or pharyngeal cavity. Call 

 this the a-resonance. How can we value the proper pitch of that resonance 1 

 Hermann has employed what he terms a centre-of-gravity calculation ; that 

 is to say, he proceeds as if he were finding the centre of gravity of four 



10 



4-2 



8-5 



3-2 



139 



447 



50-2 



13-6 



14-6 



Fig. 436. — Diagram showing the amplitude in the partials of the German 

 long a, sung and analysed by Hermann. 



heavy points, situated, as above, on the axis of x, and each weighing as 

 many units as there are units in the respective amplitudes. Having thus 

 discovered the "mean partial," he multiplies the fundamental by it to get 

 the value of the resonance itself. Both operations are combined in the 

 equation — 



n 



_ lyp'a' +p"a"+p'"a'" + etc. 

 a +a +a +etc. 



where N and n are again the pitch-numbers of the note and of the desired 

 resonance, respectively ; p, p", p" are successive integers, the numbers of 

 the reinforced partials ; and a', a", a" are the amplitudes found for these 

 partials respectively. For the resonance displayed in the above Fig. 436, the 

 process is as follows : — 



n 



98 x 



(6x13-9) + (7x44 -7 ) + (8x50- 2) + (9x13 -6) 

 (13-9 + 44 : 7 + 50-2 + 13-6) 



737 vibs. 



Pipping 1 objects to the assumption that the partials, as in the above 

 figure, are equidistant from one another. It is true that they are equidistant, 

 when considered as an arithmetical progression. Thus, first partial, 98 vibs. ; 

 second partial, 196 vibs. ; third partial, 294 vibs., etc. But musical distances 

 are not measured in arithmetical, but in geometrical progression, and the equi- 

 distant partials are not the first, second, third, fourth, etc., but the first, 

 second, fourth, eighth, etc. The abscissae in the. figure ought not, therefore, 

 to be measured off in proportion to the numbers 1, 2, 3, 4, etc. ; but to log 2 1, 

 log 2 2, log 2 3, log 2 4, etc., or simply log 1, log 2, log 3, log 4, etc., seeing the 



1 "Ueber die Tonfarbe der Vocale," Ztachr. f. Biol., Bd. xxvii. S. 1 (being detailed 

 analysis of twenty-four vowel phonograms); also, " Zur Lehre von den Vocalllangen, " 

 ibid., Bd. xxxi. S. 524-583, containing controversy with Hermann as to real meaning "I 

 Fourierian analysis; also, " Ueber die Theorie der Vocale," Acta soc. scient. Fennicse, Bd. 

 xx. part 11, replies to Lloyd, and gives more analyses. 



