Theorems on Musical Temperament. 435 



Smith has shown, in his truly excellent " Harmonics," 

 Prop. III. and XVill, that the sum of 5 of the mean Tones 

 (1) and two of the major Limmas (L) are equal to the 

 Octave, in all such systems. Tims in the system of M. 

 Henflmg (Schol. X. p. 5:') wherein the Octave is divided 

 into 50 equal parts, six different sets of T and L may be 

 tound to answer the above condition, viz. 



T L 



3X10 + 2X = 50 



5x 8 + 2X 5 = 50 



5 X C -f 2 X 10 = 50 



5x 4 + 2x15 = 50 



5 X 2 + 2 X 20 = 50 



5 X + 2 X 25 = 50 

 Which seta of answers may, in general, be obtained bv 

 this rule, viz. 



From the given number of equal parts in the Octave (a), 

 deduct successively the even numbers in the series 0, 2, 4, 

 6, 8, &c. until a remainder is found, divisible by 5, or which 

 ends with or 5, and let such eveyi subtrahend be called h-. 



then will ~ be the greatest value of T, and t. the least 



or corresponding value of L; and all the other corresponding 

 values of T will decrease from this by 2, and those of L 

 increase by 5, in succession, as in the example above. It 

 will however be unnecessary, to carrv this process on any- 

 further, than till L becomes equal to T ; since in all prac- 

 tical systen)s, the value of L cannot differ very greatly from 

 the half of T, and thus the second line in the above ex- 

 ample, is the only practical system that results from a divi- 

 sion of the Octave into 50 equal parts 5 and so of any other 

 value of a. 



In practice therefore, the value of I (or 2L) will be re- 

 stricted to some of those even subtrahends that produce 

 practical systems, and which may be determined, in my new 

 notation, by 



Theorem 7. — r = — x 61-421264S — 9'23622212r, 



the flat temperament of the Fifths, in the system having a 

 equal parts in the Octave. 



Or, in reciprocals of common logarithms, by 



Theor. 8. — r = — x •0301029,99566 — -0045267,3834. 



a 



For example, in Mercator's System, mentioned in Schol. 1, 



a = 53 and b = 8, and we have 8 x 61-421264 -r- 53 



-- 9-236222 = —•034911212 the flat Temperament of the 



K t 2 Fifth : 



