On Musical Temperament, 17 



plex consonances than the conclusions of experience will war- 

 rant. But when it is asserted by practical musicians, that the 

 octave will bear less tempering than the Vth, the V^th less than 

 the Illd, Sic, they doubtless intend to estimate the tempera- 

 ment by the rate of beating, and to imply, that when diti'erent 

 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 

 Hid on the same base were made to beat with a rapidity which 

 is as the numbers 2, 3, and 5, no unprejudiced ear would pro- 

 bably pronounce the octave less harmonious in its kind than 

 the Hid. 



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 more complex consonances, it may be sufficient to reply, 

 that if the beats of a more complex consonance are more 

 rapid than those of a simpler one, when both are equally tem- 

 pered, those of the latter, caeteris paribus, are more distinct. 

 It is the distinctness of the undulations, in tempered conso- 

 nances, which is one of the principal causes of offence to the 

 ear. 



Scholium 2. 



It will be proper to explain, in this place, the notation of mu- 

 sical intervals, which will be adopted in the following pages. 

 It 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 .05116 ; 

 that of the comma by .00540, &c. But in order to avoid the 

 prefixed ciphers, in calculations where so small intervals as 

 the temperaments of different concords are concerned, we will 



Vox. I.. .No. 1. 3 



