16 On Musical Temperament, 



the simpler to an 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 + 3d, whose ratio is /j, 

 when tempered ^^ of ^ comma, and the unison, whose ratio is 

 ^, when tempered 3 commas, are equally harmonious. But 

 all who have the least experience in tempered consonance* 

 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 ren- 

 der the VIII -j- Vth, and the VIII -|- 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 lllds, and consequently 

 the VIII 4" 6ths, are tempered ^ of a comma ; which, so far 

 from being offensive, will be positively agreeable to the ear. 

 This cannot be doubted by those who admit that the VIII -{- 

 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 

 VIII -{- 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, g^ ; instead of A, c, let him play A, cb, 

 &c. and compare the harmony of these with that of the VIII -j- 

 6ths, if he wants any farther evidence that Dr. Smith's mea- 

 sure of equal harmony is without foundation. 



It may be thought, that even the measure of equal harmony 

 laid down in the proposition, is more favourable to the com- 



