410 Mr. A. J. Ellis on the Temperament [June 16, 



or (by Table XII.) 150F— 107S^a7+1998a?^=a minimum, which gives 

 ^-T-hh /^ = -0014.554, log r=-l 746359. 

 If we had used the sum of the squares of the beat factors^ we should 

 have obtained an equation of 1 6 dimensions in which gives logu=* 1746387. 

 The difference between the two values of log is not appreciable to the ear. 



No. 16 (14). The harmonic errors of the 3rd, Ilird, and Yth conjointly 

 are a minimum. 

 Here (/3 for 3)H(/3 for III)H(/3 for Y)^ or 

 (6^— 18a?)^+(5/i;— 20a?)^ + 9a:^=a minimum, which gives 

 ^=2j|.^=.ooi5309, log « = • 1744404. 



No. 17 (6). The harmonic errors of the Yth and IlIrd conjointly are a 

 minimum. 



Here (/3 for Y)2+(/3 for III)% or 9^^'+ (5A-— 20.r)^=a minimum, 

 ^=i^-0 ^=-0013190, log t^=-1747723. 



B. Melodic Systems of Equal Temperament. 



Y. Systems of equal or equal and opposite interval errors. 



No. 18 (24). The interval errors of the Ilird and Yth are equal and 

 opposite. 



Here e for III+ e for Y=0, or h--Ax=iX, x=\;Jcy as in No. 12. 



No. 19 (2). The interval errors of the 3rd and Yth are equal. 

 Here e for 3 =e for Y, or —k-\-3oc=—x, x—^k, as in No. 2, 



No. 20 (16). The interval errors of the Ilird and 3rd are equal. 

 Here e for III=e for 3, or ^— 4^=— ^ + 3^, x=^k, as in No. 8. 



YI. Systems in which the interval errors of two intervals are in a 

 given ratio. 



No. 21 (17). The errors of the Ilird and Yth are as 5 : 3, but in opposite 

 directions. 



Here e for III : e forY =—5:3, or 3Z;— 12^ = 5^^, ^^=y3-;5;=-00 15750, 

 logir=-1745163. 



This is the theoretical determination of M. Romieu's anacratic tempera- 

 ment (Mem. de I'iVcad. 1758, p. 510), to which, however, lie has in 

 practice preferred No. 22. 



No. 22 (29). The errors of the Ilird and Yth are as 2 : 1, but in opposite 

 directions. 



Here e for III : e for Y= — 2, or ^—4^=2^, ir=^/<;= '0008975, 

 log 2; = -175 1938. 



This is M. Romieu's anacratic temperament. See No. 21. 



No. 23 (26). The errors of the Ilird and Yth are as 1-94: 1, and in 

 opposite directions, 



