14 On Musical Temperament, 



If, thirdly, we omit two-thirds of the pulses of the lower 

 unison, retaining the octave ac of the last case, we shall have 

 AD, ac, the times of vibration of imperfect Vths, to which, 

 and to all other concords, the same reasoning may be applied 

 as above. It may be briefly exhibited thus ; since the inter '^ 

 mission of the coincidences CV, E'e' of the perfect anisons, 

 an octave below A'B', does not render the Vth A'D'G a'c'e'g less 

 perfect than the unison A'c' a'c', each being perfect in its kind ; 

 so neither does the intermission of the corresponding disloca- 

 tions Cc, Ee, of the tempered unisons, in the imperfect Vth, 

 ADG, aceg, render it less harmonious in its kind than the tem- 

 pered unison AB, ab, from which it is derived in exactly the 

 same manner that the perfect Vth is derived from the perfect 

 unison. 



The consonances thus derived, as has been shown by Dr. 

 Smith, will have the same periods, and consequently the same 

 beats, with the imperfect unisons. It is obvious, likewise, that 

 they will all be equally tempered. Let m AB, and n ab, be a 

 general expression for the times of vibration of any such con- 

 sonance. The tempering ratio of an imperfect consonance is 

 always found by dividing the ratio of the vibrations of the im- 

 perfect by that of the corresponding perfect consonance. But 



. — -. = —T- : which is evidently the tempering ratio of 



n ab n ab 



the imperfect unisons. 



Hence, so far as any reasoning, founded on the abstract na- 

 ture of coexisting pulses can be relied on, (for, in a case of 

 this kind, rigid demonstration can scarcely be expected,) we 

 are led to conclude that the harmoniousness of diflerent con- 

 sonances is proportionally diminished when they are equally 

 tempered. 



The remaining part of the proposition, viz. that conso- 

 nances differently tempered have their harmoniousness dimi- 

 nished, or their harshness increased, in the direct ratio of their 

 temperaments, will be evident, when we consider that the 

 temperament of any consonance is the sole cause of its harsh- 

 ness, and that the effect ought to be proportioned to its ade- 



