6 On Musical Temperament. 
the simpler to au absolute level, in point of agreeableness, with 
the more complex ; which, as has been shown, is not the ob- 
ject to be aimed at in adjusting their comparative tempera- 
ments. But, in truth, his measure is far more favourable to 
the complex consonances than equal harmony, even in this 
sense, would require ; and in a great number of instances, 
leads to the grossest absurdities. Two consonances, accord- 
ing to him, are equally harmonious, when their temperaments 
are inversely as the products of the least numbers expressing 
their perfect ratio. If so, the VIII 4+ 3d, whose ratio is 4» 
when tempered ., of a comma, and the unison, whose ratio is 
+, when tempered 3 commas, are equally harmonious. But 
all who have the least experience in tempered consonances 
will pronounce, at once, that the former could scarcely be dis- 
tinguished by the nicest ear from the corresponding perfect 
concord, while the latter would be a most offensive discord. 
One instance more shall suffice. The temperaments to render 
the VIII + Vth, and the VEIL + 6th equally harmonious, 
are laid down in his tables to be as 80:3. We will now sup- 
pose an instrument perfectly tuned in Dr. Smith’s manner, and 
furnished with all the additional sounds which constitute his 
changeable scale. In this system, the IIfds, and consequently — 
the VIII + 6ths, are tempered } of a comma; which, so far from — 
from being offensive, will be positively agreeable to the ear. — 
This cannot be doubted by those who admit that the VITL + 
6ths, in the common imperfect scales, when tempered at a me- — 
dium nearly seven times as much, make tolerable harmony. 
Yet, according to the theory which we are opposing, the — 
VILE + Vth will be equally harmonious when tempered nearly 
a minor semitone. Now let any one, even with the common ; 
instruments, whenever an VIII + Vth occurs, strike the semi- 
tone next above or below: for example, instead of playing — 
C, g, let him play C, gif ; instead of A, e, let him play A, € 
&c. and compare the harmony of these with that of the VIII + 
6ths, if he wants any farther evidence that Dr. Smith’s mea- 
‘sure of equal harmony is without foundation. 
: 
4 
It may be thought, that even the measure of equal harmony — 
laid down in the proposition, is more favourable to the com- ; 
