I al ee ee eT ae en 
SE ee ee ee eae ea a 
ee Sa ra a ee Nn ea Se Lee a EN eT Te ES Sh a ee ee ee ee 
On Musical Temperuncent. 23 
scheme of intervals for the natural scale, in which the harmo- 
ny 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 Ff G equal w either of the semi- 
tones found above, the intervals, reckoned from G as a key 
note, 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 the 
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 differ 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 ,,, the 
IlIds 4, and the 3ds } of acomma. The several differences 
are only;! ,4,and 3, 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. 
