sufficiently complex to stimulate the eye to resolve it, and yet so
simple that it can be analyzed at a glance; as a center with two equal
sides, it is possessed of symmetry, and as the sum of an odd and even
number (3 and 4) it has vitality and variety. All these properties a
work of architecture can variously reveal (Illustration 77). Fifteen,
also, is a number of great perfection. It is possible to arrange the
first 9 numbers in the form of a "magic" square so that the sum of
each line, read vertically, horizontally or diagonally, will be 15.
Thus:
4 9 2 = 15
3 5 7 = 15
8 1 6 = 15
-- -- --
15 15 15
Its beauty is portrayed geometrically in the accompanying figure which
expresses it, being 15 triangles in three groups of 5 (Illustration
86). Few arrangements of openings in a façade better satisfy the eye
than three superimposed groups of five (Illustrations 76-80). May not
one source of this satisfaction dwell in the intrinsic beauty of the
number 15?
In conclusion, it is perhaps well that the reader be again reminded
that these are the by-ways, and not the highways of architecture: that
the highest beauty comes always, not from beautiful numbers, nor
from likenesses to Nature's eternal patterns of the world, but from
utility, fitness, economy, and the perfect adaptation of means to
ends. But along with this truth there goes another: that in every
excellent work of architecture, in addition to its obvious and
individual beauty, there dwells an esoteric and universal beauty,
following as it does the archetypal pattern laid down by the Great
Architect for the building of that temple which is the world wherein
we dwell.
VII
FROZEN MUSIC
In the series of essays of which this is the final one, the author has
undertaken to enforce the truth that evolution on any plane and on
any scale proceeds according to certain laws which are in reality only
ramifications of one ubiquitous and ever operative law; that this law
registers itself in the thing evolved, leaving stamped thereon as it
were fossil footprints by means of which it may be known. In the arts
the creative spirit of man is at its freest and finest, and nowhere
among the arts is it so free and so fine as in music. In music
accordingly the universal law of becoming finds instant, direct
and perfect self-expression; music voices the inner nature of the
_will-to-live_ in all its moods and moments; in it form, content,
means and end are perfectly fused. It is this fact which gives
validity to the before quoted saying that all of the arts "aspire
toward the condition of music." All aspire to express the law, but
music, being least encumbered by the leaden burden of materiality,
expresses it most easily and adequately. This being so there is
nothing unreasonable in attempting to apply the known facts of musical
harmony and rhythm to any other art, and since these essays concern
themselves primarily with architecture, the final aspect in which
that art will be presented here is as "frozen music"--ponderable form
governed by musical law.
Music depends primarily upon the equal and regular division of time
into beats, and of these beats into measures. Over this soundless and
invisible warp is woven an infinitely various melodic pattern, made
up of tones of different pitch and duration arithmetically related
and combined according to the laws of harmony. Architecture,
correspondingly, implies the rhythmical division of space, and
obedience to laws numerical and geometrical. A certain identity
therefore exists between simple harmony in music, and simple
proportion in architecture. By translating the consonant
tone-intervals into number, the common denominator, as it were, of
both arts, it is possible to give these intervals a spatial, and
hence an architectural, expression. Such expression, considered as
proportion only and divorced from ornament, will prove pleasing to
the eye in the same way that its correlative is pleasing to the ear,
because in either case it is not alone the special organ of sense
which is gratified, but the inner Self, in which all senses are one.
Containing within itself the mystery of number, it thrills responsive
to every audible or visible presentment of that mystery.
[Illustration 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 architecture 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 façade 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 practically
in the proportion of 8, 7, 6, 5, 4, 3, allowing in each case for the
amount hidden by the projection of the cornice below; each order being
accurate as regards column, entablature, etc. It is of interest to
compare this with Ruskin's idea in his _Seven Lamps_, where he takes
the case of a plant called Alisma Plantago, in which the various
branches diminish in the proportion of 7, 6, 5, 4, 3, respectively,
and so carry out the same idea; on which Ruskin observes that
diminution in a building should be after the manner of Nature.
[Illustration 90: ARCADE OF THE CANCELLERIA]
It would be a profitless task to formulate exact rules of
architectural proportion 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. Laws there
are, but they discover themselves to the artist as he proceeds, and
are for the most part incommunicable. Rules and formulæ are useful and
valuable not as a substitute for inspiration, but as a guide: not as
wings, but as a tail. In this connection perhaps all that is necessary
for the architectural designer to bear in mind is that important
ratios of length and breadth, height and width, to be "musical" should
be expressed by quantitively small numbers, and that if possible they
should obey some simple law of numerical progression. From this basic
simplicity complexity will follow, but it will be an ordered and
harmonious complexity, like that of a tree, or of a symphony.
[Illustration 91: THE PALAZZO VERZI AT VERONA (LOWER PORTION ONLY).
A COMPOSITION FOUNDED ON THE EQUAL AND REGULAR DIVISION OF SPACE, AS
MUSIC IS FOUNDED ON THE EQUAL AND REGULAR DIVISION OF TIME.]
[Illustration 92: ARCHITECTURE AS RHYTHM. A DIVISION OF SPACE
CORRESPONDING TO 3/4 AND 4/4 TIME.]
In the same way that a musical composition implies the division of
time into equal and regular beats, so a work of architecture should
have for its basis some unit of space. This unit should be nowhere too
obvious and may be varied within certain limits, just as musical time
is retarded or accelerated. The underlying rhythm and symmetry will
thus give value and distinction to such variation. Vasari tells how
Brunelleschi. Bramante and Leonardo da Vinci used to work on paper
ruled in squares, describing it as a "truly ingenious thing, and of
great utility in the work of design." By this means they developed
proportions according to a definite scheme. They set to work with a
division of space analogous to the musician's division of time.
The examples given herewith indicate how close a parallel may exist
between music and architecture in this matter of rhythm (Illustrations
91-93).
[Illustration 93]
It is a demonstrable fact that musical sounds weave invisible patterns
in the air. Architecture, correspondingly, in one of its aspects,
is geometric pattern made fixed and enduring. What could be more
essentially musical for example than the sea arcade of the Venetian
Ducal Palace? The sand forms traced by sound-waves on a musically
vibrating steel plate might easily suggest architectural ornament did
not the differences of scale and of material tend to confuse the mind.
The architect should occupy himself with identities, not differences.
If he will but bear in mind that architecture is pattern in space,
just as music is pattern in time, he will come to perceive the
essential identity between, say, a Greek rosette and a Gothic
rose-window; an arcade and an egg and dart moulding (Illustration 94).
All architectural forms and arrangements which give enduring pleasure
are in their essence musical. Every well composed façade makes harmony
in three dimensions; every good roof-line sings a melody against the
sky.
[Illustration 94: ARCHITECTURE AS PATTERN]
CONCLUSION
In taking leave of the reader at the end of this excursion together
among the by-ways of a beautiful art, the author must needs add a
final word or two touching upon the purpose and scope of these essays.
Architecture (like everything else) has two aspects: it may be viewed
from the standpoint of utility, that is, as construction; or from the
standpoint of expressiveness, that is, as decoration. No attempt has
been made here to deal with its first aspect, and of the second
(which is again twofold), only the universal, not the particular
expressiveness has been sought. The literature of architecture is rich
in works dealing with the utilitarian and constructive side of the
art: indeed, it may be said that to this side that literature is
almost exclusively devoted. This being so, it has seemed worth while
to attempt to show the reverse of the medal, even though it be "tails"
instead of "heads."
It will be noted that the inductive method has not, in these pages,
been honored by a due observance. It would have been easy to
have treated the subject inductively, amassing facts and drawing
conclusions, but to have done so the author would have been false
to the very principle about which the work came into being. With the
acceptance of the Ancient Wisdom, the inductive method becomes
no longer necessary. Facts are not useful in order to establish a
hypothesis, they are used rather to elucidate a known and accepted
truth. When theosophical ideas shall have permeated the thought of
mankind, this work, if it survives at all, will be chiefly--perhaps
solely--remarkable by reason of the fact that it was among the
first in which the attempt was made again to unify science, art and
religion, as they were unified in those ancient times and among those
ancient peoples when the Wisdom swayed the hearts and minds of men.