24 On Musical Temperament. 



Scholium 3. 



Although the foregoing would be the best division of the 

 musical scale, if our sole object were to render the harmony 

 of its concords as nearly equal as possible, yet the two other 

 considerations, stated at the beginning of the essay, must by 

 no means be neglected, as has been done by Dr. Smith. It 

 seems to be universally admitted, that the sum of the tempera- 

 ments may be increased to a certain extent, in order to equalize 

 the harmony of the concords ; otherwise the natural scale of 

 major and minor tones, which makes the sum of the tempera- 

 ments of the Vths, Illds, and 3ds but 2 commas, ought to be 

 left unaltered. Yet how far this principle ought to be carried, 

 may be a matter of doubt. If we make the Illds perfect, and 

 flatten the Vths and 3ds each i c, according to the old system 

 of mean tones, we shall have the smallest aggregate of tempe- 

 raments which admits of the different concords of the same 

 name being rendered equally imperfect ; but this amounts to 

 2i commas. Thus far, however, it seems evidently proper to 

 proceed. If we go still farther, and endeavour to equalize 

 the harmony of the concords of different names, it may be ques- 

 tioned whether nearly as much is not lost as gained ; for the 

 aggregate temperaments are increased, in Dr. Smith's scale, 

 to 2| c, and in that of the above proposition to 2| c. The 

 system of mean tones, although more unequal in its harmony 

 when but two notes are struck at once, yet when the chords 

 are played full, as they generally are on the organ, never 

 oflfends the ear by a transition from a better to a worse har- 

 mony. For every triad is equally harmonious ; being com- 

 posed of a perfect Hid, and a Vth and 3d, tempered each i c, 

 or of their complements to, or compounds with octaves, which, 

 in their kinds, are equally harmonious. 



Again, if different chords, in practice, vary in the frequency 

 of their occurrence, this will be a sufficient reason for deviating 

 from the system of equal temperament. Suppose, for exam- 

 ple, that a given sum of temperament is to be divided betweea 

 two Vths, one 5f which occurs in playing ten times as often as 



