Section 8

common multiples of any two of them, be- cause the eye, with a little assistance, is able to resolve them into their constituent factors. It is part of the art of architec- ture to render such assistance, for the eye counts always, con-
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98 THE BEAUTIFUL NECESSITY
sciously or unconsciously, and when it is confronted with a number of units greater than it can readily resolve, it is re- freshed and rested if these units are so grouped and arranged that they reveal themselves as factors of some higher quantity. 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 pur- pose of so subdividing a f agade that the eye estimates the number of its openings without conscious effort and consequent fatigue (Illustration 82). The tracery of Gothic windows forms per- haps the highest and finest architectural expression of number (Illustration 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 reveal- ing the inherent simplicity in which this complexity has its root. Some- times, as in the ea
82
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ANUMLRICAL ANALYSIS OF GOTHIC TEACERjY,
99
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84
THE ARITHMETIC OF BEAUTY
case of the Venetian Ducal Palace, the numbers in- volved are too great for counting, but other and dif- ferent 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 quat- refoils immediately above. Seven is proverbially the perfect number. It is of a quantity 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 pos- sessed 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 architec- ture can variously reveal (Illus- tration 77) . Fifteen, also, is a num- ber of great perfection. It is pos- sible to arrange the first 9 num- bers in the form of a "magic" square so that the sum of each line,
read vertically, horizontally or diagonally, will be 15. Thus:
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lOO THE BEAUTIFUL NECESSITY
4 9 2 = 15
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15 1.5 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 fagade 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 re- minded 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 pat- terns 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 arche- typal 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 ^nd 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 at- tempting to apply the known facts of musical harmony and rhythm to any other art, and since these essays concern them- selves 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
101
THE. NORMAN PORCH CANTHR- BURJ/^AN AacHITtCTURAb BX- PRKSICN OPANOTt ftcKAaMONICS
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102 THE BEAUTIFUL NECESSITY
soundless and invisible warp is woven an infinitely various melodic pattern, made up of tones of different pitch and dura- tion arithmetically related and combined according to the laws of harmony. Architecture, correspondingly, implies the rhyth- mical 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 pro- portion only and divorced from or- nament, will prove pleasing to the eye in the same way that its cor- relative is pleasing to the ear, be- cause 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. Con- taining within itself the mys- tery of number, it thrills re- sponsive to every audible or vis- ible presentment of that mystery.
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 repre- sented by 3: 4 yields the fourth; 4:5 the major third; and 5: 6
87
FROZEN MUSIC
103
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the minor third. These comprise the principal consonant intervals within the range of one octave. The ratios of in- verted intervals, so called, are found by doubling the smaller number of the orig- inal interval as given above : 2 : 3, the fifth, gives 3 : 4, the fourth ; 4 : 5, the ma- jor third, gives 5 : 8, the minor sixth ; 5] 6, the minor third, gives 6: 10, or 3:5, the major sixth.
Of these various consonant intervals as
the octave, fifth, and major third are the most important, in the sense of being the most perfect, and they are expressed by num- bers 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 i, 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 archi- tectural length and breadth, height and width, to be "musical"
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104 THE BEAUTIFUL NECESSITY
should be capable of being expressed by ratios of quantitively small num- bers, 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 assump- tion is borne out by experience), that proportions, the numerical ratios of which the eye resolves too readily, become at last wearisome. The re- lation should be felt rather than fathomed. There should be a per- ception of identity, and also of differ- ence. As in music, where dissonances are introduced to give value to con- sonances which follow them, so in ar- chitecture simple ratios should be em- ployed in connection with those more complex.
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 dis-
HOUSE IN ROME
FROZEN MUSIC 105
turbs 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 i, 4, 7, 10, etc. 'Both are "mystic" numbers, and in Gothic architecture particu- larly, propprtions were frequently determined by numbers to which a mystic meaning was attached. According to Gwilt, the Gothic chapels of Windsor and Oxford are divided longitudi- nally by four, and transversely by seven equal parts. The arcade above the roses in the fagade 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 oc- tave is that it advances by the addition of i to both terms : i : 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 ii-i2ths of the
io6
THE BEAUTIFUL NECESSITY
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 cor- nice below; each order being ac- curate as regards column, entabla- ture, 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 dimin- ish in the proportion of 7, 6, 5, 4, 3, respectively, and so carry out the same idea; on which Ruskin ob- serves that diminution in a building should be after the manner of Nature.
It would be a profitless task to formulate exact rules qf architect- ural 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 math- ematical 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 formulae 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
FROZEN MUSIC
107
YH& PALAZZO VERZI AT VERONA (L0WLI6 PORTION ONLY}. A COMPOSITION FOUND£-D ON THE EQUAL AND REGULAR/ DIVISION OF SPACL, AS MU,SIC IS FOUNDED ON TH& EQUAL AND REX3ULAR> DIVISION OF TIME;' »
91
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obey some simple law of numerical progression. From this basic simplicity complexity will fol- low, but it will be an ordered and har- monious complexity, like that of a tree, or of a symphony. In the same way that a musical com- position 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
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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, de- scribing it as a "truly in- genious thing, and of great utility in the work of de- sign." By this means they developed proportions ac- cording to a definite scheme. They set to work with a division of space an- alogous 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).
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 ar- cade 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 dififerences of scale, and of material tend to confuse the mind. The architect should occupy himself with identities, not dififerences. If he will but bear in mind that architecture is pattern in space, just as music
FROZEN MUSIC
109
is pattern in time, he will come to perceive the essential identity between, say, a Greek rosette and a Gothic rose-window; an ar- cade 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 facade makes harmony in three dimensions; every good roof- line sings a melody against the sky.
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94
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 pur- pose 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 ex- pressiveness, 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 expressive- ness 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 in- ductive 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,