416 



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



No. 58 (41). There is no Ilird wolf. 



Here — ^ + 8a?=0, a;=-0000613, log « = -1760300. This is almost ex- 

 actly No. 32 (42). 



No. 59 (42). There is no 3rd wolf. 



Here 5—9^=0, a?=-0000545, log t?=- 1760368. 



F. Systems of Unequal Temperament. 



In a defective equal temperament the same just concordance is represented 

 by two different discordances. As performers limited themselves to twelve 

 tones to the octave, those who found the Hemitonic temperament No. 50 

 (35) too rough, accepted this variety of representatives of the same con- 

 cordance as the basis of a temperament, hoping to have better Illrds in 

 the usual chords, without the wolves of the defective temperament. Others 

 conceived that an advantage would be gained by altering the character of 

 the different keys. Thus arose unequal temperament, properly so called, 

 which must be carefully distinguished from any defective equal temperament 

 with which it is popularly confused. 



Arrange the twelve unequally tempered chords as follows, where the 

 identical numbers indicate identical chords with different names : — 



\. C E G. 7. cp 7. G\; ^. 



2. G B d. 8. C% G^, 8. Db F A), 



3. D F% A, 9. G% dp 9. J} c eb. 



4. A c% e. 10. J)^ Fx Ap 10. G 



5. E G^B. 11. A^ cx ep 11. d /. 



6. B d%fp 12. E% Gx Bp 12. F A c. 



Let Tn, 4, Vn be the ratios of the Ilird, 3rd, and Vth in the ?ith chord, 

 so that, for example, in the 6th chord d^=T^ . B, f% = t^. dp f^^v^ . B. 

 Then it is evident from the above scheme that there exist 12 pairs of 

 equations between these 36 ratios, of the form 



Tn . 4=«^n and AT,,=V^ . . V„+2 • 'Vn+Z 



(where, when the subscript numbers exceed 12, they must be diminished 

 by 12), and one condition. 



Put log r„=log log 4=log f-s",,, log v^=\og |— a?^ then the 



above equations become 



2'n = ^n + 2/n, 



^»7^ + a7, + ^3 + ^4 + ^o+^'^^6 + ^7 + ^8 + ^9 + ^'l0 + ^U + ^12 = *005885I, 



which represent 25 equations, where the second set of 1 2 may be replaced 

 by the following, which are readily deduced from them and the last con- 

 dition : — 



