624 PHILOSOPHICAL TRANSACTIONS. [ANNO 1800. 



of harmony is nearly equal in all systems. Hence, if every one of the 12 major 

 and minor 3ds occurred equally often in the compositions which are to be performed 

 on an instrument, it would be of no great consequence, to the sum of the imper- 

 fections, among which of the 3ds they were divided: and even in this case the opi- 

 nion of the best practical authors is, that the difference of character produced by 

 a difference of proportions in various keys, would be of considerable advantage in 

 the general effect of modulation. But when it is considered, that on an average 

 of all the music ever composed, some particular keys occur at least twice as often 

 as others, there seems to be a very strong additional reason for making the harmony 

 the most perfect in those keys which are the most frequently used; since the aggre- 

 gate sum of all the imperfections which occur in playing, must by this means be 

 diminished in the greatest possible degree, and the diversity of character at the 

 same time preserved. Indeed in practice this method, under different modifica- 

 tions, has been almost universal; for though many have pretended to an equal tem- 

 perament, yet the methods which they have employed to attain it have been evi- 

 dently defective. It appears to me, that every purpose may be answered, by 

 making c to e too sharp by a quarter of a comma, which will not offend the nicest 

 ear; e to g*, and a* to c, equal; p* to a* too sharp by a comma; and the major 

 3ds of all the intermediate keys more or less perfect, as they approach more or less 

 to c in the order of modulation. The 5ths are perfect enough in every system. 

 The results of this method are shown in table 12. In practice, nearly the same 

 effect may be very simply produced, by tuning from c to f, b*, e', g*, c*, f* 6 



perfect 4ths; and c, g, d, a, e, b, f*, 6 equally imperfect 5ths, pi. 10, fig. 52. 

 If the unavoidable imperfections of the 4ths be such as to incline them to sharp- 

 ness, the temperament will approach more nearly to equality, which is preferable 

 to an inaccuracy on the other side. An easy method of comparing different systems 

 of temperament is exhibited in fig. 53, which may easily be extended to all the 

 systems that have ever been invented. 



Table 12 A shows the division of a 



monochord corresponding 

 to each note, in the system 

 proposed, b, the logarithm 

 of the temperament of each 

 of the major 3ds. c, of 

 the minor 3ds. n, of the 

 5ths; c and d being both 

 negative. 



Thus I have endeavoured 

 to advance a few steps only 

 in the investigation of some 

 very obscure but interesting subjects. As far as I know most of these observations 

 are new ; but if they should be found to have been already made by any other per- 



1 *», G#, C*, F* - 



2 F, B 6 , E, B 



3 C, G, D, A 



.0000000 



.0004597 



.001156*2 



