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



be used throughout this paper. See also the last columns in Tables XII. 

 and XIV. 



Sign. 



Interval. 



Example. 



Sign. 



Interval. 



Example. 



1st. 





c c 









Ilnd. 



Major Second .... 



cd 



2nd. 



Minor Second .... 



e f 



Illid. 



Major Third 



c e 



3rd. 



Minor Third 



eg 



IVth. 



Augmented Fourth 



f b 



4th. 



Perfect Fourth . . 



c f 



Vth. 



Perfect Fifth 



eg 



5th. 



Diminished Fifth.. 



bf^ 



Vlth. 



Major Sixth .... 



c a 



6th. 



Minor Sixth .... 



e c^ 



Vllth. 



Major Seventh . . 



cb 



7th. 



Minor Seventh . . 





Vlllve. 





c c^ 







IXth. 



Major Ninth 



cd^ 



9th. 



Minor Ninth 



e 



Xth. 



Major Tenth 



c e^ 



10th. 



Minor Tenth 



e g2 



In no system of temperament will it be possible to interfere with the 

 octave, the only unisonant concord. Hence 2 will remain unchanged. 

 Let the ratios of the tempered Ilird and Vth be T, v, which will replace 

 5 : 4 and 3 : 2 throughout the system of chords. Hence if we take four 

 successive perfect major triads in the form 4, 5,6 as OjE' G B d, dfi^^ay 

 t« c^tt t^^ and suppose them to be tempered so that the distinction be- 

 tween E and no longer exists, but that in each chord the pitch of the 

 second and third tones are T and v times that of the first tone respectively, 

 while the ratio of the octave remains unchanged, the ratio of each of the 

 above tones to C will be as under : — 



a G, B, d, a, c% 



1, T, V, Tv, v\ Tv\ v\ Tv\ v\ 



Hence, since e^ = 4E,yve have t;^ = 42^as the first condition of tempera- 

 ment, showing that we shall arrive at the same tone whether we take two 

 Vlllves and a tempered Ilird, or take four tempered Vths, as in Cc, cc^y 

 and Q G^ Gd, da^ ae^. In this case the above ratios reduce to 



a G, B, d, a, c% 



If we further call the interval of the mean tone m, the limma /, the 

 sharp % the flat t?, and the diesis the above ratios give 



d 



d 





c 



- 2(7"" 2' 





c 



2(7 2' 





B 



" B v^' 





c% 









~4(7 2"' 





d'^Q 



2«^X(2'-rZ)' 











Whence m=iJ/, 1=^-^, ' m^^dj^, m'f==2. 



