Anonymous (2003)_gothic Cathedral Design [6 P.]

Cathedral Design

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Cathedral Design - Ad Triangulum In 1399 a Frenchman, Master Jean Mignot, was placed in charge of Milan Cathedral. He made some serious criticisms of the work done by the Lombard architects up to then. There are extant notes on the argument that resulted. In the heat of passion it was asserted that the science of geometry should not have a place in these matters, since science is one thing and art another. Master Jean replied that art without science is nothing. The Lombard Masters replied: if he [Mignot] invokes, as it were, the rules of geometry, Aristotle says that the movement of man in space which we call locomotion is either straight or circular or a mixture of the two. Likewise the same [writer] says elsewhere that every body is perfected in three [ways], and the movement of this very church rises ad triangulum as has been determined by other engineers. ... All [the measurements] are in a straight line, or an arch, therefore it is concluded that what has been done, has been done according to geometry and to practice... (Alain Erlande-Brandenburg Cathedrals and Castles: Buiding in the Middle Ages, 1993, pp156-9) There is an accompanying illustration of Milan Cathedral analyzed according to the ad triangulum proportional system. Its taken from Cesare Cesariano's 1521 Italian commentary on Vetruvius' Ten Books on Architecture. (Mike Bispham says that this illustration is in Robert Lawlor's Sacred Geometry and in Jean Gimpel's The Cathedral Builders -- note 5 to Platonic Atomism in Midieval Design) Basically it shows how a descending series of nested equilateral triangles fit over the front view of the cathedral, determining heights and placements for towers and flying buttresses. (Won't someone PLEASE translate Cesariano! All I ever see are references to the 1521 manuscript.) Bear all of this in mind while we turn to Chartres Cathedral to reconsider the place of music in the elevations of the Cathedral.

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Cathedral Design

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The following relates to Charpentier's book, The Mysteries of Chartres Cathedral (chapter on the Musical Mystery). Charpentier tells us that there are four horizontals lines on the side walls of the Cathedral at different heights, formed by rows of cornices. (See illustration on page 8 of the photos) The top row forms the base for the arches of the ceiling vault. The next row down is the base for the stained glass windows. Below that is the base for the gallery, and below that is the top of the pillars. 5pt star/vault x x x x x x x x 2 x x x x x x x x x x x x x x x x x x 1

x -- base of ceiling vault x x -- base of windows x x -- base of gallery x x -- top of pillars x x x

The geometry of the elevations of the cathedral will be altogether musical, he says. If you take the width of the floor as a unit of 1, then the hypotenuse of the right triangle formed by connecting the base of the opposite wall with the base of the vault will be 2. 1 to 2 is the interval of the octave. (Further the right triangle that is formed is one half of an equilateral triangle, implying 'ad triangulum'.) The right triangle formed by connecting the base of the opposite wall with the base of the gallery has the proportion 1 to 1.5, or in other words 2 to 3, equal to the musical fifth. The right triangle formed by connecting the base of the opposite wall with the top of the pillars has the proportion 1 to 1.2, or in other words 5 to 6, the musical third. Here Charpentier runs into a problem, because the right triangle that is formed by connecting the base of the opposite wall with the base of the windows

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Cathedral Design

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has the proportion of 1 to 1.75 or in other words 4 to 7, which does not correspond to a musical interval. 1.75 is, however, very close to 1.732, which is the square root of 3. When we look at the heights as square roots we get the following: width of floor = 1 height to base of vault = square root of 3 height to base of windows = square root of 2 height to base of gallery = square root of 5/2 height to tops of pillars = square root of 11/5 Following the Pythagorean theorem for 1 to 2 rt. triangle: square on the hypotenuse = 4 minus 1, square on the base, equals 3 , the square on the other side. Length of the other side is the square root of 3. Following the Pythagorean theorem for 1 to sqrt 3 rt. triangle: square on the hypotenuse = 3 minus 1, square on the base, equals 2 , the square on the other side. Length of the other side is the square root of 2. When 1.75 is used for the square root of 3 instead of 1.732, the figure that results is 1.4361 instead of the true square root of 2 which is 1.4142, a difference of about one and a half percent. Following the Pythagorean theorem for 2 to 3 rt. triangle: square on the hypotenuse = 9 minus 4, square on the base, equals 5 , the square on the other side. Length of the other side is the square root of 5. Since the base is really 1 not 2, divide by 2. Length of the other side is square root of 5 divided by 2. The square root of 5 divided by 2 is familiar, since it makes up part of the golden section number, phi, equal to sqrt 5/2 + 1/2. Sqrt 5/2 = 1.118 plus .5 = 1.618 = phi Barry Carrol on the Sacred Landscape list has pointed out that the square root of 2 and of 3 are important elements in cathedral design. Here I would like to say that I believe there was

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a general knowledge of right triangles, probably associated with musical ratios, geometrical construction methods, the 1/2 of an equilateral triangle significant to Platonic Atomism, and the 'ad triangulum' proportional system. Consider the following: base 1, hypotenuse sqrt 3, side sqrt 2 base 1, hypotenuse 2, side sqrt 3 base 2, hypotenuse 3, side sqrt 5 base 3, hypotenuse 4, side sqrt 7 base 4, hypotenuse 5, side sqrt 9 (the sacred 3-4-5 triangle) base 5, hypotenuse 6, side sqrt 11 base 6, hypotenuse 7, side sqrt 13 Note that base plus hypotenuse equals the sqrt of the side. 6+7=13, 5+6=11 etc. Given a Druid's cord of 12 knots and 13 sections and a T-square right angle measure, it would not be hard to construct these triangles on the ground. It would then be possible to mark out squares on the ground with prime number areas. These squares could be multiplied by whole-numbers to give large areas with prime numbers as the basic factor. Although some of the elevation in Chartres Cathedral is represented by musical triangles, the whole scheme seems more oriented to displaying proportions based on prime number square roots. Dan Washburn, April 1998

I've had a speculative notion about the smallest rightt triangle in the elevation scheme at Chartres, the one that runs from the base of the opposite wall to the top of the columns. base = 1, hypotenuse = 1.2, height = sqrt 11/5 Proportionally, this is equivalent to base = 5, hypotenuse = 6, height = sqrt 11 As you recall the next larger rt triangle was

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base 1, hypotenuse 1.5, height = sqrt 5/2. Phi, the golden section number is sqrt 5/2 plus 1/2. Hence this right triangle is a reference to Phi. Is there any relation between the Phi reference triangle and the 6 to 5 proportioned triangle? 6/5 times Phi squared = approximately Pi = 3.1416407865 Hence my conclusion is that the architects of Chartres were aware of this formula and chose the smallest triangle as part of a proportional scheme that demonstrated this knowledge. OF asks: > I've never heard of a Druid's cord before. > Could you possibly provide some more information (specifically > the total length if that is defined). Thank you in advance All I know is what I have read in Einar Palsson's book, The Sacred Triangle of Pagan Iceland. He quotes Nigel Pennick writing in his book Sacred Geometry "The laying out of areas required a foolproof method for the production of the right angle. This was achieved by marking off the rope with thirteen equal divisions. Four units then formed one side of the the triangle, three another, and five the hypotenuse opposite the right angle. This simple method has persisted to this day, and was used when tomb and temple building began. It was the origin of the historic "cording of the temple", and from this technique it was a relatively simple task to lay out rectangles and other more complex geometric figures." Palsson writes, "The Druid's Cord of which Nigel Pennick speaks had 13 sections with 12 knots, to produce the 3:4:5 triangle and the 5:4:4 triangle, which was reckoned as the seventh part of a circle. This is interesting to compare with the Icelandic counterpart as I had hypothesized that triangle 3:4:5 of Rangarhverfi had an extended hypotenuse from 5 to 6. In other words: the cord with thirteen sections may have been used for just that purpose - to extend the hypotenuse from 5 to 6; i.e. all thirteen sections being

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used not just for triangle 5:4:4 but also for triangle 3:4:5 (which otherwise only needs 12)." Why an extension from 5 to 6? Perhaps because Pi is equal to 6/5 times Phi squared? Too bad all of this is so speculative. I'm afraid I don't know anything about why it is called a "Druid's Cord" or what the actual lengths were. Dan Washburn

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2004-01-12