Temperaments of different musical Systems. 49 



the sharp wolf. In theorem 3, we.get the sharp tempera- 



3 5 



ment of the major third = ft- £ + — m, an d its wolf 



1 "* 



= 12-2 + fi m. And intheorem6, the sharp tempera- 

 ment of the major sixth = 6-- 2 +~m. and its 



ir 20 21 



wolf IS^*.+ To -m. 



This is Mr. J. Marsh's approved method of tuning 

 a douzeave. Theory of Harmonics, page 13. 



The system nearest to this of any which I remem- 

 ber to have met with, is that of a division of the oc- 

 tave into 67 equal parts, (see M. Sauveur's table above 

 referred to), where — 1 '764552 is the flat temperament 

 of the fifth , which here is — 1*65742; and the same 

 differs considerably from the other system recom- 

 mended by Mr. Marsh page 18, which, perhaps after 

 the example of Dr. Smith, he has borrowed from 

 M. Henfling without acknowledgement. See my 10th 

 scholium. 



By Dr. Smith's Harmonics, ed edit. p. 84, prop.XI> 

 latter part of cor. 3, f when the bases and beats (of 

 two tempered consonances) are the same, the tempera- 

 ments have ultimately* the inverse ratio of the major 

 terms" of the perfect ratios of these consonances. 

 Whence 



Scholium 9. If a douzcave be required, wherein the Jiftk 



(■§-) and the major third [$) to the same base shall beat 



equally quick, the former flat and the latter sharp ; we 



•— t lis— At 



have from theorems I and 3, as 5 : 3 : : : 



s s 9 



whence 55s— 20r= — 3r, or 55s=23r, and — = -- — 



' s 23 



_/ u— At . 

 also, 5:3:: : , whence 5a— 20/= —3/, 



3 u u ' * 



* The ultimate ratios are in tkese cases, very near to the exact ratios : 



thus in scholium 9, — -c, or 2-2930132, results from the ultimate ratios ; 



the true temperament being '2-S93G932, as derived from the length of string 

 I of the Vth, in the equation 4/* — P — f ; the difference being lels than 



th of a X, an interval altogether imperceptible in practice. 



Vol. 36. No. 147. July 1S10. D 01 



