On Musical Temperament. i7 
‘plex consonances than the conclusion of experience will war- 
rant. But when it is asserted by practical musicians, that the 
octave will bear less tempering than the Vth, the Vth less than 
the IIId, &c., they doubtless intend to estimate the tempera- 
ment by the rate of beating, and to imply, that when different 
consonances to the same base are made to beat equally fast, 
the simpler are more offensive than the more complex conso- 
nances. This is entirely consistent with the proposition; for 
when equally tempered, the more complex consonances will 
beat more rapidly than the more simple; if on the same base, 
very nearly in the ratio of their major terms. (Smith’s Har. 
Prop. XI. Cor: 4.) If, for example, an octave, a Vth and a 
{IId on the same base were made to beat with a rapidity which 
is as the numbers 2, 3, and 5, no unprejudiced ear would prob- 
ably pronounce the octave less harmonious in its kind than the 
To those, on the other hand, who may incline to a measure 
of equal harmony between that laid down in the proposition 
and that of Dr. Smith, on account of the rapidity of the beats 
of the moré complex consonahces, it may be sufficient to re- 
ply, that if the beats of a more complex cohsonance are moré 
rapid than those of a simpler one, when both are equally tem- 
pered, those of the latter, czeteris paribus, are more distinct. 
It is the distinctness of the undulations, in tempered conso- 
hances, which is one of the principal causes of offence to the 
ear. 
Scholium 9; 
It will be proper to explain, in this place, the notation of mu« 
sical intervals, which will be adopted in the following pages. 
Tt is well known that musical intervals are as the logarithms of 
their corresponding ratios, If, therefore, the octave be re- 
presented by .30103, the log. of 2, the value of the Vth will 
be expressed by .17509; that of the major tone by .05115 ; 
that of the comma by .90540, &c. But in order to avoid the 
prefixed cyphers, in calculations where so small intervals as 
the temperaments of different concords are concerned, we will 
Van. I...No. 1. § 
