ration 87]

If a vibrating string yielding a certain musical note be stopped in
its center, that is, divided by half, it will then sound the octave
of that note. The numerical ratio which expresses the interval of
the octave is therefore 1:2. If one-third instead of one-half of the
string be stopped, and the remaining two-thirds struck, it will yield
the musical fifth of the original note, which thus corresponds to the
ratio 2:3. The length represented by 3:4 yields the fourth; 4:5 the
major third; and 5:6 the minor third. These comprise the principal
consonant intervals within the range of one octave. The ratios of
inverted intervals, so called, are found by doubling the smaller
number of the 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.

[Illustration 88: ARCHITECTURE AS HARMONY]

Of these various consonant intervals the octave, fifth, and major
third are the most important, in the sense of being the most perfect,
and they are expressed by numbers of the smallest quantity, an odd
number and an even. It will be noted that all the intervals above
given are expressed by the numbers 1, 2, 3, 4, 5 and 6, except the
minor sixth (5:8), and this is the most imperfect of all consonant
intervals. The sub-minor seventh, expressed by the ratio 4:7 though
included among the dissonances, forms, according to Helmholtz, a more
perfect consonance with the tonic than does the minor sixth.

A natural deduction from these facts 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, preferably an odd number and an even. Although generally
speaking the simpler the numerical ratio the more perfect the
consonance, yet the intervals of the fifth and major third (2:3 and
4:5), are considered to be 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 archite