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]