“ Frozen Music 
y y 
original interval as given above. 2:3, the fifth, 
gives 3:4, the fourth; 4:5 the major third, 
gives 5:8, the minor sixth; 5:6, the minor 
third gives 6:10, or 3 : 5, the major sixth. 
THE- RELATION BETWEEN THE- 
SUBMINOR, SEVENTH (4'.7) AND 
THE. EQUILATERAL TRIANGLE.^ 
FIGURE TWO 
Of these various consonant intervals the 
octave, fifth, and major third are the most 
important because the most perfect. It 
will be noted that all of the intervals above 
given are expressed by means of the num¬ 
bers i, 2, 3, 4, 5, and 6, except the minor 
sixth ; and this, of all consonant intervals, is 
the most imperfect. The subminor seventh, 
whose ratio is 4:7, 
though included 
a m ong the dis¬ 
sonances forms, ac¬ 
cording to Helmoltz, 
a more perfect 
consonance with the 
tonic than the minor 
sixth. 
A natural deduc¬ 
tion from these tacts 
is that relations of 
architectural length 
and breadth, height 
and width, to be 
“ musical ” should be 
capable of being 
expressed by ratios 
of quantitively small 
numbers. Although, 
generally speaking, 
the simpler the ratio 
the more perfect 
the consonance, yet 
the intervals of the 
fifth a n d m a j o r 
third (2:3, and 4:5) 
are more pleasing than the octave (1 : 2), 
which is too obviously a repetition of the 
original note. From this it is reasonable to 
assume (and the assumption is borne out by 
experience) that proportions, the numerical 
ratios of which the eye resolves too readily,, 
become at last wearisome. The relation 
should be felt rather than fathomed. There 
should be a perception of identity, and also of 
difference. As in music, where dissonances 
are introduced to give value to consonances 
which follow them, so in architecture simple 
ratios should be employed in connection with 
those more complex. 
H armonics are those tones which sound 
with and reenforce any musical note when 
struck. The distinguishable harmonics of 
the tonic are given in figure one. They 
yield the ratios, 1: 2, 2:3, 4 : 5, and 4 : 7. 
The note and its harmonics form a natural 
chord. They may be compared to the 
widening circles which appear in still water 
when a stone is dropped into it ; for when a 
musical sound disturbs that pool of silence 
which we call the air, it ripples into over¬ 
tones which, becoming fainter and fainter, die 
away into the original 
silence. It would 
seem that the com¬ 
binations of numbers 
which express these 
overtones, if trans¬ 
lated into terms of 
space, should yield 
proportions agreeable 
to the eye. Figure 
three illustrates a 
simple application of 
these ratios to archi¬ 
tecture. The sub¬ 
minor seventh (4: 7), 
used in this way, in 
connection with the 
simpler intervals of 
the octave (1 : 2), and 
the fifth (2:3), is 
particularly pleasing, 
because it is neither 
too obvious nor too 
subtle. This interval 
is important from the 
fact that it expresses 
the angle of sixty 
ARCHITECTURE AS HARMONY too scale) 
VARIOUS RENAIASANGE WINDOWS 
FIGURE THREE 
