HARMONIC ANALYSIS. 97 



results will differ although the relation of the sinusoid to the resonators is 

 the same as before. This is different from the mathematical analysis, 

 where the results always remain the same whatever the period of the fun- 

 damental, provided the harmonic vibration has the same relation to it. 

 The relation of the two cases is shown by a comparison of the formulas 

 for the vibration a. sin jkt; the harmonic i has an ampUtude given by a^ 

 in (30); the resonator i responds with the amplitude given by c, in (35). 

 The two agree only when 



^7 sm-* jrc 



k 



The analogy is therefore not a valid one. In Chapters VIII and IX we 

 shall consider Helmholtz's theory of the vowels, which was based entirely 

 on this analogy. 



We have now to face the problem of deducing the pitch and ampli- 

 tude of the inharmonic sinusoid vibration from the results of a harmonic 

 analysis. 



In the case of a single vibration a. sin pt, the values a and p are to 

 be found when the coefficients Ci, a^, . . . hi, 62, ... in a harmonic 

 analysis with the fundamental k are known. For this calculation any 

 two values of a or of 6 will suffice. 



Thus from 



_ 2 akp. sinVj 

 ^'~ Ttip'-k') 



_ 2 akp. sin'Trf 



7r(p^-F) 



we obtain 



p^kyj 



Ci — tti 



from which a can be calculated by substitution. Since the values Oi, a^, 

 ... 61, 62, . . . were obtained from the ordinates yo, yi, yi, . . . which 

 are contaminated by the errors of measurement, the values of p from 

 different pairs of coefficients will not exactly agree; the method of least 

 squares requires that their average be taken. The matter is of no practical 

 importance because we never have to deal with a vibration composed 

 of a single element. 



