REStECTIXG SOUND AND LIGHT. 



551 



precisely confirmed, in a late dissertation of 

 M. Kneclit, published in the musical news- 

 paper of Leipsic. Perhaps, by a close atten- 

 tion to the harmonics entering into the con- 

 stitution of various sounds, more may be 

 done in their analysis than could otherwise 

 be expected. 



XVI. Of the Temperament of musical 

 Intervals. 



It would have been extremely conrenient 

 for practical musicians, and would have saved 

 many warm controversies among theoretical 

 ones, if three times the ratio of 4 to 5, or 

 four times that of 5 to 6, had been equal to 

 the ratio of 1 to 2. As it happens to be other- 

 wise, it has been much disputed in what in- 

 tervals the imperfection should be placed. 

 The Aristoxenians and Pythagoreans were 

 in some sense the beginners of the contro- 

 versy. Sauveur has given very comprehen- 

 sive tables of a great number of systems of 

 temperament ; and hisown now ranks among 

 the many that are rejected. Dr. Smith has 

 written a large and obscure volume, which, 

 for every purpose but for the use of an im- 

 practicable instrument, leaves the whole sub- 

 ject precisely where it found it. Kirnberger, 

 Marpurg, and other German writers, have 

 disputed with great bitterness, almost every 

 one for a particular method of tuning. It 

 is not with any confidence of success, that 

 one more attempt is made, which rests its 

 chief claim to preference, on the similarity 

 of its theory to the actual practice of the best 

 instrument makers. However we estimate 

 the degree of imperfection of two tempered 

 concords of the same nature, it will appear, 

 thatj the manner of dividing the tempera- 

 ment between them does not materially alter 

 its aggregate sum ; for instance, the imper- 



fection of a comma, in a major third, occa- 

 sions it to beat very nearly twice as fast as 

 that of half a comma. If indeed the im- 

 perfection were great, it might aflfect an in- 

 terval so materially as to destroy its charac- 

 ter; as, in some methods of temperament, a 

 minor third diminished by two commas ap- 

 proaches more nearly to the ratio 6 : 7, than 

 to 5 : 6 ; but, with this limitation, the sum 

 of harmony is nearly equal in all systems. 

 Hence, if every one of the twelve major and 

 minor thirds occurred equally often in the 

 compositions which are to be performed on 

 an instrument, it would be of no great con- 

 sequence, to the sum of the imperfections, 

 among which of the thirds they were di- 

 vided : and, even in this case, the' opinion 

 of the best practical authors is, that the dif- 

 ference of character produced by a differ- 

 ence of proportions in various keys, would 

 be of considerable advantage in the general 

 effect of modulation. But, when it is con- 

 sidered, that upon an average of all the mu- 

 sic ever composed, some particular keys oc- 

 cur 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 aggregate sum of all the im- 

 perfections, which occur in playing, must by 

 these means be diminished in the greatest 

 possible degree, and the diversity of charac- 

 ter at the same time preserved. Indeed, in 

 practice, this method, under different modi- 

 fications, has been almost universal ; for, 

 although many have pretended to an equal 

 temperament, yet the methods which they 

 have usually employed to attain it have been 

 evidently defective. It appears to me, that 

 every purpose may be answered, by making 

 the third C : E too sharp by a quarter of a 



