1864.] of Instruments with FioGed Tones, 409 



No. 6 (21). The Ilird and 4th beat equally, and in the same direction. 



Here /3 for III=/3 for 4, or 5^—20.^=4^, .^=^^=-0011239, 

 log t>=- 1749674. 



No. 7 (18). The 6th and 4th beat equally, and in opposite directions. 



Here/3 for 6+/3.for 4=0, or (-8^+32^) + 4^=0, a7=|^=-0011989, 

 log «;= 1748924. 



No. 8 (16). The 3rd and Vth beat equally, and in the same direction. 



Here /3 for 3=/3 for Y, or -6^+ 18^= -3a;, a7=f .r=-0015414, 

 log v=-l 745199. See No. 20. 



No. 9 (13). The Vlth and Yth beat equally, and in opposite directions. 



Here /3 for VI+^S for V=0, or (pk-l^cc) - 3^=0, ^= i^8^=-0014986, 

 log v=-1745927. 



This coincides with Dr. Smith's system of equal harmony, as contained 

 in the Table facing p. 224 of his * Harmonics,' 2nd ed. 



No. 10 (9). The 3rd and 4th beat equally, and in opposite directions. 



Here /3 for 3 +/3 for 4=0, or (-6^+ 18.r) + 4.i7=0, .^^=^^=•0014713, 

 log tj=-l 746200. 



No. 11.(2)- The YIth and 4th beat equally, and in the same direction. 



Here /3 for YI=/3 for 4, or ^k—\bx=4x, ^=y53^= -00141 97, 

 log ^=-1746716. 



III. Systems in which the harmony of two concords is in a given ratio. 



No. 12 (24). The beats of the Ilird and Yth are as 5 : 3, but in opposite 

 directions. 



Here /3 for III : /3 for Y= - 5: 3, or 15y^-60.r= \5.v, ^=i-^= '00 10790, 

 log «j=-1750123. 



M. Romieu gives this temperament under the title of " syst^me tempere 

 de 1 comma," Mem. de I'Acad. 1758. See No. 18. 



No. 13 (2). The beats of the 3rd and Yth are as 2 : 1, and in the same 

 direction. 



Here /3 for 3 : /3 for Y=2, or -6^+ 18^= — 6^, a?=p, as in No. 2. 



No. 14 (12). The beats of the 3rd and Yth are as 5 : 2, and in the same 

 direction. 



Here/3for3:/3forY=5:2,or-12^+36^=-15^,^=yV^=-0012694, 

 logtj=-1748219. See No. 29. 



lY. Systems of least harmonAc errors. 



No. 15 (7). The harmonic errors of all the harmonic intervals conjointly 

 are a minimum. 



This is determined by putting the sum of the squares of the beat meters. 



