THE ARITHMETIC OF BEAUTY

Although architecture is based primarily upon geometry, it is possible to express all spatial relations numerically: for arithmetic, not geometry, is the universal science of quant.i.ty. The relation of ma.s.ses one to another--of voids to solids, and of heights and lengths to widths--forms ratios; and when such ratios are simple and harmonious, architecture may be said, in Walter Pater's famous phrase, to "aspire towards the condition of music." The trained eye, and not an arithmetical formula, determines what is, and what is not, beautiful proportion. Nevertheless the fact that the eye instinctively rejects certain proportions as unpleasing, and accepts others as satisfactory, is an indication of the existence of laws of s.p.a.ce, based upon number, not unlike those which govern musical harmony. The secret of the deep reasonableness of such selection by the senses lies hidden in the very nature of number itself, for number is the invisible thread on which the worlds are strung--the universe abstractly symbolized.

Number is the within of all things--the "first form of Brahman." It is the measure of time and s.p.a.ce; it lurks in the heart-beat and is blazoned upon the starred canopy of night. Substance, in a state of vibration, in other words conditioned by number, ceaselessly undergoes the myriad trans.m.u.tations which produce phenomenal life. Elements separate and combine chemically according to numerical ratios: "Moon, plant, gas, crystal, are concrete geometry and number." By the Pythagoreans and by the ancient Egyptians s.e.x was attributed to numbers, odd numbers being conceived of as masculine or generating, and even numbers as feminine or parturitive, on account of their infinite divisibility. Harmonious combinations were those involving the marriage of a masculine and a feminine--an odd and an even--number.

[Ill.u.s.tration 72: A GRAPHIC SYSTEM OF NOTATION]

Numbers progress from unity to infinity, and return again to unity as the soul, defined by Pythagoras as a self-moving number, goes forth from, and returns to G.o.d. These two acts, one of projection and the other of recall; these two forces, centrifugal and centripetal, are symbolized in the operations of addition and subtraction. Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. In order to think correctly it is necessary to consider the whole of numeration, computation, and all mathematical processes whatsoever as _the division of the unit_ into its component parts and the establishment of relations between these parts.

[Ill.u.s.tration 73]

[Ill.u.s.tration 74]

The progression and retrogression of numbers in groups expressed by the multiplication table gives rise to what may be termed "numerical conjunctions." These are a.n.a.logous to astronomical conjunctions: the planets, revolving around the sun at different rates of speed, and in widely separated orbits, at certain times come into line with one another and with the sun. They are then said to be in conjunction.

Similarly, numbers, advancing toward infinity singly and in groups (expressed by the multiplication table), at certain stages of their progression come into relation with one another. For example, an important conjunction occurs in 12, for of a series of twos it is the sixth, of threes the fourth, of fours the third, and of sixes the second. It stands to 8 in the ratio of 3:2, and to 9, of 4:3. It is related to 7 through being the product of 3 and 4, of which numbers 7 is the sum. The numbers 11 and 13 are not conjunctive; 14 is so in the series of twos, and sevens; 15 is so in the series of fives and threes. The next conjunction after 12, of 3 and 4 and their first multiples, is in 24, and the next following is in 36, which numbers are respectively the two and three of a series of twelves, each end being but a new beginning.

[Ill.u.s.tration 75]

It will be seen that this discovery of numerical conjunctions consists merely of resolving numbers into their prime factors, and that a conjunctive number is a common multiple; but by naming it so, to dismiss the entire subject as known and exhausted, is to miss a sense of the wonder, beauty and rhythm of it all: a mental impression a.n.a.logous to that made upon the eye by the swift-glancing b.a.l.l.s of a juggler, the evolutions of drilling troops, or the intricate figures of a dance; for these things are number concrete and animate in time and s.p.a.ce.

[Ill.u.s.tration 76]

The truths of number are of all truths the most interior, abstract and difficult of apprehension, and since knowledge becomes clear and definite to the extent that it can be made to enter the mind through the channels of physical sense, it is well to accustom oneself to conceiving of number graphically, by means of geometrical symbols (Ill.u.s.tration 72), rather than in terms of the familiar arabic notation which though admirable for purposes of computation, is of too condensed and arbitrary a character to reveal the properties of individual numbers. To state, for example, that 4 is the first square, and 8 the first cube, conveys but a vague idea to most persons, but if 4 be represented as a square enclosing four smaller squares, and 8 as a cube containing eight smaller cubes, the idea is apprehended immediately and without effort. The number 3 is of course the triangle; the irregular and vital beauty of the number 5 appears clearly in the heptalpha, or five-pointed star; the faultless symmetry of 6, its relation to 3 and 2, and its regular division of the circle, are portrayed in the familiar hexagram known as the Shield of David.

Seven, when represented as a compact group of circles reveals itself as a number of singular beauty and perfection, worthy of the important place accorded to it in all mystical philosophy (Ill.u.s.tration 73). It is a curious fact that when asked to think of any number less than 10, most persons will choose 7.

[Ill.u.s.tration 77]

Every form of art, though primarily a vehicle for the expression and transmission of particular ideas and emotions, has subsidiary offices, just as a musical tone has harmonics which render it more sweet.

Painting reveals the nature of color; music, of sound--in wood, in bra.s.s, and in stretched strings; architecture shows forth the qualities of light, and the strength and beauty of materials. All of the arts, and particularly music and architecture, portray in different manners and degrees the truths of number. Architecture does this in two ways: esoterically as it were in the form of harmonic proportions; and exoterically in the form of symbols which represent numbers and groups of numbers. The fact that a series of threes and a series of fours mutually conjoin in 12, finds an architectural expression in the Tuscan, the Doric, and the Ionic orders according to Vignole, for in them all the stylobate is four parts, the entablature 3, and the intermediate column 12 (Ill.u.s.tration 74). The affinity between 4 and 7, revealed in the fact that they express (very nearly) the ratio between the base and the alt.i.tude of the right-angled triangle which forms half of an equilateral, and the musical interval of the diminished seventh, is architecturally suggested in the Palazzo Giraud, which is four stories in height with seven openings in each story (Ill.u.s.tration 75).

[Ill.u.s.tration 78]

[Ill.u.s.tration 79]

[Ill.u.s.tration 80]

[Ill.u.s.tration 81]

Every building is a symbol of some number or group of numbers, and other things being equal the more perfect the numbers involved the more beautiful will be the building (Ill.u.s.trations 76-82). The numbers 5 and 7--those which occur oftenest--are the most satisfactory because being of small quant.i.ty, they are easily grasped by the eye, and being odd, they yield a center or axis, so necessary in every architectural composition. Next in value are the lowest multiples of these numbers and the least common multiples of any two of them, because the eye, with a little a.s.sistance, is able to resolve them into their const.i.tuent factors. It is part of the art of architecture to render such a.s.sistance, for the eye counts always, consciously or unconsciously, and when it is confronted with a number of units greater than it can readily resolve, it is refreshed and rested if these units are so grouped and arranged that they reveal themselves as factors of some higher quant.i.ty.

[Ill.u.s.tration 82]

[Ill.u.s.tration 83: A NUMERICAL a.n.a.lYSIS OF GOTHIC TRACERY]

There is a raison d'etre for string courses other than to mark the position of a floor on the interior of a building, and for quoins and pilasters other than to indicate the presence of a transverse wall.

These sometimes serve the useful purpose of so subdividing a facade that the eye estimates the number of its openings without conscious effort and consequent fatigue (Ill.u.s.tration 82). The tracery of Gothic windows forms perhaps the highest and finest architectural expression of number (Ill.u.s.tration 83). Just as thirst makes water more sweet, so does Gothic tracery confuse the eye with its complexity only the more greatly to gratify the sight by revealing the inherent simplicity in which this complexity has its root. Sometimes, as in the case of the Venetian Ducal Palace, the numbers involved are too great for counting, but other and different arithmetical truths are portrayed; for example, the multiplication of the first arcade by 2 in the second, and this by 3 in the cusped arches, and by 4 in the quatrefoils immediately above.

[Ill.u.s.tration 84: NUMERATION IN GROUPS EXPRESSED ARCHITECTURALLY]

[Ill.u.s.tration 85: ARCHITECTURAL ORNAMENT CONSIDERED AS THE OBJECTIFICATION OF NUMBER. MULTIPLICATION IN GROUPS OF FIVE; TWO; THREE; ALTERNATION OF THREE AND SEVEN]

[Ill.u.s.tration 86]

Seven is proverbially the perfect number. It is of a quant.i.ty sufficiently complex to stimulate the eye to resolve it, and yet so simple that it can be a.n.a.lyzed 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 (Ill.u.s.tration 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 (Ill.u.s.tration 86). Few arrangements of openings in a facade better satisfy the eye than three superimposed groups of five (Ill.u.s.trations 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 enc.u.mbered 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 s.p.a.ce, and obedience to laws numerical and geometrical. A certain ident.i.ty 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.

[Ill.u.s.tration 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 princ.i.p.al 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.

[Ill.u.s.tration 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 quant.i.ty, 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 quant.i.tively 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 repet.i.tion of the original note. From this it is reasonable to a.s.sume (and the a.s.sumption 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 ident.i.ty, 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.

[Ill.u.s.tration 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 a.s.sume that the combination of numbers which express these overtones, if translated into terms of s.p.a.ce, would yield proportions agreeable to the eye, and such is the fact, as the accompanying examples sufficiently indicate (Ill.u.s.trations 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 alt.i.tude 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 princ.i.p.al 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 compa.s.s 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.

[Ill.u.s.tration 90: ARCADE OF THE CANCELLERIA]