HARMONIC ANALYSIS. 81 



A harmonic analysis of a wave (figure 82) from [a] in "Marshall" 

 gave the following result : 



Partial 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 



Amplitude 1.9 5.1 9.1 44.5 31.3 5.1 3.3 2.1 6.0 3.3 1.2 3.4 4.4 1.4 2.7 5.2 1.0 0.8 2.5 3.1 



The harmonic plot is given in figure 83. There are minima at the 8th, 

 11th, 14th, and 18th partials. The amplitudes for each of these is to be 

 divided proportionately between its two neighbors. For example, 2.1 is 

 divided into two parts in the ratio of 3.3 to 6.0, that is, into 0.7 and 1.4. 

 Similarly divided, the other numbers give 0.6, and 0.6; 0.9 and 0.5; 0.2 

 and 0.6 respectively. For the group around the first maximum we then 

 have 



(Ixl.9) + (2x5.1) + (3x9.1) + (4x44.5) + (5x31.3) + (6x5.1) + (7x3.3) + (8x0.7) , 

 1.9 + 5.1+9.1+44.5 + 31.3 + 5.1-1-3.3 + 0.7 "^'^ 



This means that the first component tone has the ordinal number 

 4.3 in the series of partials; its frequency is 4.3 times that of the funda- 

 mental, namely, 144.5 X 4.3=619.9. For the second component we find 



( 8 X 1.4) + (9 X 6.0) + (10 X 3.3) + (11 x 0.6) 



1.4 + 6.0+3.3 + 0.6 ~^-^- 



Its frequency is, therefore, 144.5 X 9.3= 1343.9. In the same way we find 

 that the group of partials between the minima at 1 1 and 14 has a weighted 

 mean of 11.5, giving a frequency of 1661.8, and the following group has 

 a mean of 17.6, giving a frequency of 2543.2. For the last group the 

 calculation becomes somewhat indefinite because the series is broken off 

 at a partial which can not be a minimum ; we can do no more than make a 

 rough approximation to the tone which must lie somewhere between the 

 19th and 20th partial, let us say at 



(19x2.5) + (20X3.1) 



2.5 + 3.1 



19.5, 



which gives the frequency 144.5 X 19.5 = 2817.8. 



It is important to approximate the amplitudes of the components. 

 This might perhaps be closely done by a complicated calculation, but as 

 a practical rule I suggest the following one. When the inharmonic does 

 not differ much from a harmonic the largest amplitude in the group is taken 

 for that of the inharmonic. When it differs much (for example, when it lies 

 about half-way between two harmonics), f of the amphtude of the largest 

 harmonic in the group is taken. The suggestion for this rule arises from 

 the fact that if an inharmonic is sUghtly changed so as to become a har- 

 monic, its amplitude will coincide with that of the harmonic, and from 

 the method of distribution of the harmonic results shown in figure 76. 

 In the present case each inharmonic lies nearer the middle point between 



