On Musical Tcmperammt. 23 



acheme of intervals for the natural scale, in which the harmony 

 of all the different consonances is rendered as nearly equal as 

 possible. 



Cor. 1. In the same manner it may be shown that these 

 temperaments are the best, among those which approach as 

 nearly as possible to equal harmony, for the artificial scale ; 

 provided that it is furnished with distinct sounds for all the 

 sharps and flats in common use. By inserting a sound between 

 F and G, making the interval F^^ G equal to either of the semi- 

 tones found above, the intervals, reckoned from G as a key 

 Hote, will be exactly the same in respect to their temperaments, 

 as the corresponding ones reckoned from C. The same thing 

 holds, whatever be the number of flats and sharps. It is sup- 

 posed, however, that the flat of a note is never used for th^e 

 sharp of that next below, or the contrary ; and hence this 

 scheme of temperament would only be adapted to an instru- 

 ment, furnished with all the degrees of the enharmonic scale ^ 

 or, at least, with as many as are in common use. 



Cor. 2. This scale will diiTer but little in practice from the 

 one deduced, with so much labour, by Dr. Smith, from his 

 criterion of equal harmony ; which flattens the Vths j^-^, the 

 Illds i, and the 3ds ^ of a comma. The several differences 

 are only ^^, Z^^, and J^ of a comma. Hence, as his measure 

 of equal harmony differs so widely from that of Proposition I. 

 we may infer that the consideration of equalizing the harmony 

 of the concords of different names can have very little practi- 

 cal influence on the temperaments of the scale. Should it, 

 therefore, be maintained that the criterion laid down in Prop. 

 I. is not mathematically accurate ; yet, as it must be allowed, 

 in the most unfavourable view, to correspond far better with 

 the decisions of experience than that of Doctor Smith, the 

 chance is, that, at the lowest estimate, the temperaments de- 

 duced from it approach much more nearly to correctness. 

 Hence it is manifest that equal temperament may be made, 

 without any sensible error in practice, the criterion of equal 

 harmony. 



