House & Garden 
v5:<3= 
4 * 5 -= 
< 3'.4 = 
2.3 = 
MINOR) THIRD 
MAJOR 3RD ' 
FOURTtF 
5TH' 
VZ OCTAVE; 
IO'.it 
3 :14 
8 :13 
7 :12 
G : 11 
5 ": 10 
4 : 3 
a: 8 
2: 7 
1 : G 
GRAPHICAL EXPRESSION 
OF MUSICAL INTERVALS. 
degrees, because the numbers 4 and 7 
represent (very nearly) the ratio between one- 
half the base and the altitude of an equi¬ 
lateral triangle. According to Gwilt, the 
Gothic chapels 
at Windsor and 
Oxford are 
divided longi¬ 
tudinally by 
four, and trans¬ 
versely by seven 
equal parts. 
A distinguish¬ 
ing characteristic 
of the series of 
ratios which 
represents the 
consonant inter¬ 
vals within the compass of an octave is 
that it advances by the addition of 1 to 
both terms, 1 : 2, 2 :3, 3 :4, 4 : 5, and <; : 6. 
Such a series always approaches unity, just 
as, represented graphically by means of 
parallelograms, it tends toward a square. 
According to W. Watkiss Lloyd,—in an 
article published in The American Architect 
of March 31st, 1888—the scale of ratios 
which determined all 
the important pro¬ 
port i o n s of the 
Parthenon is of this 
order, advancing by 
consecutive differ¬ 
ences of 5, as shown 
in figure four. Mr. 
Lloyd goes on to say : 
“The oblong plan of 
the temple has exactly 
the proportion of 
breadth to length of 
4: 9,-131.341 front, 
228.141 on flank, 
(error, . 012). The 
same proportion is 
repeated in the well 
marked definition of 
breadth of top step 
—the hundred Attic 
feet—and the height 
from this step to the 
top of the horizontal cornice.” (It will be 
noted that this, the most important ratio, is 
not the simplest.) 
2:3 
1:2. 
SCALE SHOWING PRINCIPAL PRO¬ 
PORTIONS OF THE PARTHENON— 
FIGURE FOUR 
art. 
Ls 
aws 
the 
re are. 
It would be a profitless task to attempt to 
formulate exact rules of architectural pro¬ 
portion, based upon the laws of musical 
harmony. The two arts are too different 
from each other 
for that; and 
moreover the 
last appeal must 
always be to the 
eye, and not to 
a mathematical 
formula, just as 
in music the last 
appeal is to the 
ear. Nothing is 
truer than that 
“the concept is 
unfruitful i n 
but they discover 
themselves to the artist as he proceeds, and 
are for the most part incommunicable. No 
masterpiece was ever fashioned by means of 
predetermined formulas of beauty, though 
from every masterpiece such formulas may 
be deduced. And these are useful and 
valuable, not as a substitute for inspiration, 
but as a guide : not as wings, but as a tail. 
In the present 
instance, perhaps all 
that it is necessary 
for the architectural 
designer to consider is 
that important ratios 
of height and width 
should be composed 
of quantitively small 
numbers ; and that if 
possible, they should 
obey some simple 
law of numerical pro¬ 
gression. From this 
basic simplicity com¬ 
plexity will follow, 
but it will be an 
ordered and harmoni¬ 
ous complexity, like 
that of a tree, or of a 
symphony. 
In the same way 
that a musical com¬ 
position implies the division of time into 
equal and regular beats, so a work of archi¬ 
tecture should have for its basis some unit of 
59 
