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 3; since the inter- 
mission of the coincidences C‘c’, E’e’ of the perfect unisons, 
an octave below A’B’ does not render the Vth A'D’G a'c’e’g' 
less perfect than the unison A’<’ a’c’, each being perfect in its 
kind; so neither does the intermission of the corresponding dis- 
locations 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 nab, 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 
cra Sa which is evidently the tempering ratio of 
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 different 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 propértioned to its ade- 
