cture simple
ratios should be employed in connection with those more complex.

[Illustration 89]

Harmonics are those tones which sound with, and reinforce any musical
note when it is sounded. The distinguishable harmonics of the tonic
yield the ratios 1:2, 2:3, 3:4, 4:5, and 4:7. A 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 the quietude of that pool of silence which
we call the air, it ripples into overtones, which becoming fainter
and fainter, die away into silence. It would seem reasonable to assume
that the combination of numbers which express these overtones, if
translated into terms of space, would yield proportions agreeable
to the eye, and such is the fact, as the accompanying examples
sufficiently indicate (Illustrations 87-90).

The interval of 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 ratio of 4:7 is important for the reason
that it expresses the angle of sixty degrees, that is, the numbers 4
and 7 represent (very nearly) the ratio between one-half the base and
the altitude of an equilateral triangle: also because they form part
of the numerical series 1, 4, 7, 10, etc. Both are "mystic" numbers,
and in Gothic architecture particularly, proportions were frequently
determined by numbers to which a mystic meaning was attached.
According to Gwilt, the Gothic chapels of Windsor and Oxford are
divided longitudinally by four, and transversely by seven equal parts.
The arcade above the roses in the facade of the cathedral of Tours
shows seven principal units across the front of the nave, and four in
each of the towers.

A distinguishing characteristic of the series of ratios which
represent the consonant intervals 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 5:6. Such a series always approaches unity, just as,
represented graphically by means of parallelograms, it tends toward a
square. Alberti in his book presents a design for a tower showing his
idea for its general proportions. It consists of six stories, in a
sequence of orders. The lowest story is a perfect cube and each of the
other stories is 11-12ths of the story below, diminishing p