1864.] 



of Instruments with Fixed Tones* 



411 



Here e for III : e for Y = -1-94, or k~Ax=^ 1-94^, ^=ig^ -0009683, 

 log ?;=-1751830. 



This is the temperament calculated by Drobisch (Nachtrage, § 10) 

 from Delezenne's conclusion (Rec. Soc. Lille, 1826-27, pp. 9 and 10), 

 that the ear can detect an error of -284^ in the Ilird, and '146^ in the 

 Yth, which gives the comparative sensibility as -284 : •146 = 1*94. 



No. 24 (20). The errors of the Ilird and 3rd are as 2 : 5, but in opposite 

 directions. 



Here e for III : for e 3 = — 2 : 5, or bh — 20cG=2k—QsG, ^'=y\- k 

 = •0011561, log z;=-17493.52. See No. 27. 



No. 25 (46). The errors of the 3rd and Ilird are as 2 : 1, but in opposite 

 directions, or the errors of the Yth and 3rd are equal and opposite. 



Here e for 3 : e for III= — 2, or 2^— 6,r=^—4.r, or else x——k-\-dx\ 

 both give x=i /c= '0026975, log 1733938. 



Here the error of the Vth reaches the utmost limit of endurance. 



VII. Systems of least melodic errors. 



No. 26 (1). The interval errors of all the melodic intervals conjointly are 

 a minimum. 



Here the sum of the squares of the 23 interval errors in Table XII., or 

 32^'— 212/<:a; + 420^' = aminimum,^=2^f\i^=-0013616,logz?=-1747297. 



No. 27 (20). The melodic errors of the Ilnd, Ilird, 4th, Yth, YIth, and 

 Yllth conjointly, are a minimum. 



Here (e for II)^ + (e for III/ + (e for 4)H(e for Y)^ + (e for YI)^ + 

 (e for YII)^ or 4jr'+ (7^— 4a?)' + 2^' + (^-3^)' + (/(;-5^)2= a minimum, 

 tjc=:^jk, as in No. 24. 



This is Drobisch's *'most perfect possible" (moglich reinsfe) tempera- 

 ment (Poggendorff's Annalen, vol. xc. p. 353, as corrected in Nachtrage, 

 § 7). It is only the "most perfect possible " for the major scale. 



No. 28 (5). The melodic errors of the 3rd, Ilird, and 4th conjointly are 

 a minimum. 



Here (efor3)=^ + (e for III)^ + (e for 4)^ or (-^ + 3.r)' + (>^- 4^-^) 

 + a?'=a minimum, .r= 2^^=.0014525, log z;=-1746388. 



This is "Woolhouse's Equal Harmony (Essay on Musical Intervals, 

 p. 45). 



No. 29 (12). The melodic errors of the Ilird and Yth conjointly are a 

 minimum. 



Here (e for III)2+(e for Y)^ or x^ + (Jc—4xy= a minimum, x=^Jc, 

 as in No. 14. 



This is given by Drobisch (Nachtrage, § 8) as the simplest solution 

 of the problem." 



