On Musical Temperament, i ) 



although in a loose and conjectural manner, to make the pro- 

 minent chords of the simplest keys the nearest to perfection, 

 whilst a greater temperament is thrown upon those which oc- 

 cur only in the more complex keys. Thus Dr. Young, in the 

 Philos. Trans, for 1800, recommends a scheme which increa- 

 ses the temperament of the Illds, on the key note of the 

 successive keys, as we modulate by fifths from C, nearly in 

 arithmetical progression. Earl Stanhope assigns as a reason 

 for the small temperament which is given to several of the 

 Illds in his system, that they are on the tonic of the simpler 

 keys. The irregularities in Mr. Hawkes's scheme may be 

 traced to the same cause. And, with the instrument-makers, 

 it is a favourite maxim to lay the wolf, as they term it, where 

 it will be most seldom heard. 



But if the above consideration deserves any weight at all, it 

 deserves to be accurately investigated. Not only ought the 

 relative frequency of different chords to be ascertained with 

 the greatest accuracy, of which the nature of the subject is sus- 

 ceptible, but the degree of weight which this consideration 

 ought to have, when compared with the two others above-men- 

 tioned, should be determined : for it is plain that neither of 

 them ought to be ever left out of view. 



Accordingly, the principal design of the following propositions 

 will be to investigate the actual frequency of occurrence of 

 different chords in practice ; and from this and the two other 

 above-mentioned considerations united, to deduce the best 

 system of temperament for a scale, containing any given num- 

 ber of sounds to the octave, and particularly for the common 

 Douzeave, or scale of twelve degrees. 



Proposition I. 



All consonances may be regarded, without any sensible error 

 in practice, as equally harmonious in their kinds, when 

 equally tempered ; and when unequally tempered, within 

 certain limits, as having their harmoniousness diminished in 

 the direct ratio of their temperaments. 

 As different consonances, when perfect, are not pleasing to 



the ear in an equal degree, some approaching nearer to the 



