Hindoo Division of the Octave. 



375 



of the octave into 22 equal intervals, do not deviate very widely from the 

 exact intervals, which are the foundation of the diatonic scale. 



Tor this purpose we shall only need to recall the values of the perfect 

 fifth and third in terms of equal temperament semitones of 12 to the 

 octave. A simple calculation will give us the values of the corresponding 

 intervals of the system. 



The perfect fifth is 7*01955 semitones, 



or 7± nearly. 

 The perfect third is 4— -13686 semitones, 

 or 4—^ nearly. 



To find the interval in semitones made by x units of the system of 22, 

 we have 



12 6^ 

 22 00 or 11 °°' 



Hence we obtain the following values : — 



System of 22. ' 



No. of Interval in Exact interval 



Intervals. units. semitones. in semitones. 



Major third 7 3-8182 3-8631 



Fifth 13 7-0909 7-0195 



Hence the fifth of the system of 22 is sharp by about -07, or g of a 

 comma very nearly. 



The major third is flat by *045, or 5 of a comma nearly. 



(Comma of -21506.) 



The system of 22 possesses, then, remarkable properties ; it has both 

 fifths and thirds considerably better than any other cyclical system having 

 so low a number of notes. The only objection, as far as the concords go, 

 to its practical employment for our own purposes lies in the fifths ; 

 these lie just beyond the limit of what is tolerable in the case of instru- 

 ments with continuous tones. (The mean tone system is regarded as 



the extreme limit ; this has fifths ^ of a comma flat.) For the purposes 



of the Hindoos, where no stress is laid on the harmony, the system is 

 already so perfect that improvement could hardly be expected. 



It is thus wrong to suppose that the system of 22 would need much 

 tempering to bring its concords into tune. These are probably quite as 

 accurate as rough and poorly toned instruments admit of. 



But although the consonance error of fifth and third is small, it is far 



