1868. | On Musical Scales. 343 
in the ensemble of all the twelve; bearing in mind, however, that 
the Fifth is a far more important harmonic than the Third.* 
With this object, setting out, we will suppose, with the key of C 
on the pianoforte as our fundamental scale, let us denote by C, D, 
E, F, G, A, B, C’, the respective numbers above assigned as express- 
ing the intervals from C of the natural notes so named, taking, in 
the first instance, 170 for the value of Re; or, lets C=0; D=170; 
B= 322; F=415; G=585; A=737; B=907; C=1000: and 
let «, B, y, 5, ¢ be taken to represent the numbers, at present un- 
known, to be assigned to the intermediate notes Cf=Dp, Df= 
Ep, Fg=Gp, GE=Ap, and AZ=Bp; so that 
CaDBEFyGShAcCBOC 
shall form the complete chromatic scale within the compass of an 
octave. Now, if we assume, for our fundamental or key note, each, 
in succession, of the twelve elements C,a,D, . . . . B, of the 
scale, the derivative scale will be formed by taking that note for 
our Do, the third from it in succession (counting the fundamental 
note as the first) for Ae, the fifth im order for M%, the sixth for Fa, 
the eighth for Sol, and so on: so that our aim must be to make as 
many of such derivative intervals (Mi—Do) as possible, perfect 
Thirds, as many from Do to Fa perfect Fourths, and from Do to 
Sol perfect Fifths as possible. These last conditions are identical ; 
a Fourth ascending being equivalent to a Fifth descending. And 
it is evident that by proceeding thus through the whole scale, we 
shall get as many Thirds and Fifths as it is possible to make by 
combining its notes two and two. Our object then is, in effect, to 
assign such numerical values to our unknown symbols, z, B, ¥, 5, 
e, as shall satisfy as many as possible of the following equations. 
Iie 
E-C = 322, F—a= 322, v pea G— B= 322, 5-H= 322, A—F= 322, 
€— y= 322, B—G= 322, 5—C (=1000-322) = 678, A—a = 678, e—D= 678, 
B-—B=678. 
And (IL) 
F— C=415, y—a=415, G-D=415, 8-6 = 415, A—E = 415, e -F= 415, 
B — y= 415, G—C (=1000 — 415)=585, 5 — a=585, A—D=585, «—8 = 585, 
B-—E=585. 
Of these 24 equations, eight, viz.: H—C = 322, A-F = 822, 
B-G = 322, F-C=415, Gd-D=415, A-E=415, G-C 
= 585, and B—EK = 585, are satisfied by the numerical values 
already assigned to these several letters; but the value of D so 
assigned (170) renders the equation A—D = 585 self-contradic- 
tory, and this equation must be left unsatisfied, giving, as we have 
already observed, an imperfect Fifth. 
* I believe this is the general opinion among musicians; and it is justified by 
the important part it plays as the dominant of the scale, and as giving fullness 
and roundness to the harmony of the common chord, 
VOL. V. 2.8 
