[Illustration: Fig. 7]
This is an example of Direction-Harmony. All the points or positions
lie in one and the same direction from the premise-point “A.” The
distances from “A” vary. There is no Harmony of Intervals.
Directions being defined by angles of divergence, we may have a Harmony
of Directions in the repetition of similar angles of divergence: in
other words, when a certain change of direction is repeated.
[Illustration: Fig. 8]
In this case the angles of divergence are equal. There is a
Harmony, not only in the repetition of a certain angle, but in the
correspondence of the intervals.
[Illustration: Fig. 9]
This is an example of Harmony produced by the repetition of a certain
alternation of directions.
[Illustration: Fig. 10]
In this case we have Harmony in the repetition of a certain relation
of directions (angles of divergence). In these cases, Fig. 9 and Fig.
10, there is Harmony also, in the repetition of a certain relation of
intervals.
23. Two or more positions may lie at the same distance from a given
point taken as a premise-point. In that case the positions, having a
certain distance in common, are, to that extent, in Harmony.
[Illustration: Fig. 11]
This is an example of Distance-Harmony. All the points are equally
distant from the premise-point “A.” The directions vary.
We may have Distance-Harmony, also, in the repetition of a certain
relation of distances.
[Illustration: Fig. 12]
This is an illustration of what I have just described. The Harmony is
of a certain relation of distances repeated.
24. Intervals, that is to say intermediate spaces, are in Harmony when
they have the same measure.
[Illustration: Fig. 13]
In this case we have, not only a Harmony of Direction, as in Fig. 7,
but also a Harmony of Intervals.
[Illustration: Fig. 14]
In this case the points are in a group and we have, as in Fig. 11, a
Harmony of Distances from the premise-point “A.” We have also a Harmony
of Intervals, the distances between adjacent points being equal. We
have a Harmony of Intervals, not only when the intervals are equal, but
when a certain relation of intervals is repeated.
[Illustration: Fig. 15]
The repetition of the ratio one to three in these intervals is
distinctly appreciable. In the repetition we have Harmony, though we
have no Harmony in the terms of the ratio itself, that is to say, no
Harmony that is appreciable in the sense of vision. The fact that one
and three are both multiples of one means that one and three have
something in common, but inasmuch as the common divisor, one, cannot be
visually appreciated, as such (I feel sure that it cannot), it has no
interest or value in Pure Design.
[Illustration: Fig. 16]
The relation of intervals is, in this case, the relation of
three-one-five. We have Harmony in the repetition of this relation of
intervals though there is no Harmony in the relation itself, which is
repeated.
[Illustration: Fig. 17]
In this case, also, we have Interval-Harmony, but as the intervals in
the vertical and horizontal directions are shorter than the intervals
in the diagonal directions, the Harmony is that of a relation of
intervals repeated.
25. In moving from point to point in any series of points, it will be
found easier to follow the series when the intervals are short than
when they are long. In Fig. 17 it is easier to follow the vertical or
horizontal series than it is to follow a diagonal series, because in
the vertical and horizontal directions the intervals are shorter.
[Illustration: Fig. 18]
In this case it is easier to move up or down on the vertical than in
any other directions, because the short intervals lie on the vertical.
The horizontal intervals are longer, the diagonal intervals longer
still.
[Illustration: Fig. 19]
In this case the series which lies on the diagonal up-left-down-right
is the more easily followed. It is possible in this way, by means of
shorter intervals, to keep the eye on certain lines. The applications
of this principle are very interesting.
26. In each position, as indicated by a point in these arrangements,
may be placed a composition of dots, lines, outlines, or areas. The
dots indicate positions in which any of the possibilities of design
may be developed. They are points from which all things may emerge and
become visible.