188 — On Musical Temperament. 
the same theorem be applied as before, till no value of x can 
be obtained, and the temperaments for that scale will be the 
best adjusted possible. 3 
But as the scale which contains but 13 degrees, or 12 inter- 
vals, to the octave, is in much more general use than every 
other, we shall content ourselves with stating how the problem 
may be solved for scales containing any intermediate number 
of degrees, and proceed directly to the consideration of that 
which is so much the most practically important. 
Lema. 
No arrangement of the intervals in the common scale of 12 
degrees, which renders none of the Vths or 3ds sharp, 4 
none of the IlIds flat, can make any change in the aggre- 
gate temperaments of all the concords of the same name. 
We will conceive the 12 Vths of the Douzeave scale tobe 
arranged in succession, as CG, GD, DA. &c. embracing 7 
octaves. Let them at first be all equal: they will each be 
flattened 49. I say that no change in these Vths which pre- 
serves the two extreme octaves perfect, and renders none 
them sharp, can alter the sum of their temperaments. ie 
a, b, c, &c. be any quantities, positive or negative, by which 
the points C, G, D, &c. may be conceived to be raised above 
Vths. Then as the mean temperament Vth= V—49- the first 
Vth in the supposed arrangement will be V-49-4@ The dis 
tance from C to D will be, in like manner, 2.V—49+ 45 and 
consequently the Vth GD will be V--49+0--a. In ™ 
same manner the third Vth DE will be V--49¢-- at. 
Hence the temperament of CG—--49+<a, of Gp=--49t b 
--a, of DA=--49-+¢--b, &c. Adding the 12 temperamenl® 
together, we find their sum=--12% 49-+a+)+&e--4- 
&c. in which all the terms except the first destroy each 
other and leave, their sum=--12%49 which is the aggregate 
temperament of the twelve equal Vths in the scheme of equi! 
semitones. 
The same reasoning holds good if we bring these Vis 
within the compass of an octaye; since, if the octave be kept 
