1864.] 



of Instruments with Fixed Tones, 



415 



— •0058851 + 12.r= ~ ^— 5 -fl2jr, where 5=-0004901*. Such interval 

 errors are termed wolves, from their howling discordance. In Table XIII. 

 will be found an enumeration of all the wolves, with a notation for them, and 

 an expression of their interval errors and beat meters in terms of k, s, and x. 



No. 51 (33). System of least wolf melodic errors. 

 The sum of the squares of the wolf interval errors, or 

 + 2i^s + 65' - 4 ( 1 U + 285) X + 266x\ 

 is a minimum. Hence 22^ + 565—266^, or a?='0005495, log t'=' 17554 18. 



No. 52 (39). System of least wolf harmonic errors. 

 The sum of the squares of the wolf beat meters, or 



25F -f- 1 75s' + 50/^5- (550^ + 30 72*) x + lS662x\ 

 is a minimum. Hence 275^+ 1536s= 13662.r, or ^='0001638, log v 

 = •1759275, as in No. 36 (39). 



No. 53 (35). The wolf interval errors are equal to the usual int<5rval errors, 

 that is, there are no wolves, or there are none but wolves. 



In this case log 3=0, or, since ^=ff\> : f%—2'' : 7 log 2=12 log v. 

 Hence this system is the cycle of 12, No. 50. When I is greater than 1, 



is sharper than f% and log v is less than -j^ log 2, or '1756008. But 

 if I is less than \, is flatter than J% and log v is less than -^-^ log 2, or 

 •1756008. The latter case is, according to Drobisch, indispensable for 

 musical theory and violin practice (Musik. Tonbestim. Einleit.). Since 

 this temperament thus forms the boundary of the two other classes, distin- 

 guished by g\) being flatter or sharper tban f^, Drobisch terms it the 

 "mean" temperament (ibid. § 51). It is this property of making =f^ 

 which renders this temperament so popular, as the ear is never distressed 

 by the occurrence of intervals different from those expected, and the whole 

 number of tones is reduced to 12. 



No. 54 (31). The wolf interval error of the Illrd is to its usual interval 

 error as 14 : 5. 



This gives — s+8j? : h—Ax—U : 5, or 96.r= 14A + 55, -00081 23, 

 log v=* 1752790. This is Marsh's system of temperament ; see Phil. Mag. 

 vol. xxxvi. p. 437, and p. 39 seqq. Schol. 8. 



No. 55 (36). The wolf errors of the Ilird and Vth conjointly are a minimum. 



Here {—s-\-Sxy-^{—k—s-\-\\xy is a minimum, whence 11^+ 19* 

 = 185a^, a^=-0003712, log 175 7201. 



No. 56 (37). The wolf errors of the Vth and Ilird are equal and opposite. 

 Here-A;— «+ll^=5-8a?, 19a?=/c+25, ^=-0003356, log2;='1757557. 



No. 57 (34). There is no Vth wolf. 



Here —/c— 5 + 11^=0, a?=-0005351, log ??t=- 1755562. 



* It appears from Proceedings, vol. xiii. p. 95, that 5 must be nearly the loga- 

 rithm of the schisma or log ^. Actual calculation shows that s and log 51 agree 

 to 14 places of decimals. 



