Take the section of the standard timber, the cai, as the module of the building: it is 15 fen high and 10 fen wide, and every member is dimensioned in fen.
1 cai = 15 fen (height) × 10 fen (width)
Rules · Roofs · Proportion · Materials
Thatch a third, tile a quarter
Raise a thatched roof by a third and a tiled roof by a quarter: the rougher and more absorbent the covering, the steeper the roof must be.
thatch: rise = ⅓ · tile: rise = ¼
The Kaogong ji gives the rule in four characters for each covering. Commentators disagree whether the fraction measures the rise against the whole depth of the building or against half of it, but either way the principle is the one every roofer knows: water must run off faster than it can soak in, and straw, which holds water in its fibres, needs a steeper slope than fired clay, which sheds it. It is perhaps the oldest written statement of the relation between pitch and material.
Related rules
There are eight grades of cai, from 9 × 6 cun down to 4½ × 3 cun; use the first for great halls of nine or eleven bays and the smallest for pavilions and ceiling coffers.
grades 1–8: 9×6, 8¼×5½, 7½×5, 7.2×4.8, 6.6×4.4, 6×4, 5¼×3½, 4½×3 cun
For a great hall, measure the distance between the front and back eave purlins and raise the ridge purlin above them by one third of it; for lesser halls, by about a quarter.
rise = ⅓ × depth between eave purlins (halls); ≈ ¼ (+ a small addition) for ting-tang halls
Draw a straight line from the ridge purlin to the eave purlin and lower the first purlin below the ridge from it by one tenth of the rise; from that point draw a new line to the eave purlin and lower the next purlin by one twentieth; continue, halving each time.
drop₁ = H⁄10 · drop₂ = H⁄20 · drop₃ = H⁄40 · …
Give each rafter span a fixed slope, increasing from the eaves to the ridge: half-pitch (5 jǔ) for the eave span, then about 6½ or 7, 7½, and 9 tenths for the span below the ridge.
rise of span k = run × (0.5, 0.65, 0.75, 0.9) for four spans
From the intermediate bracket set of the second bay the rafter ends advance progressively outward, so that at the corner the eave projects further (by 4 cun for a hall of one bay, 5 for three bays, 7 for five bays) and rises with the corner beam.
Cite this page Chris Paton, “Thatch a third, tile a quarter”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/kaogongji-roof-pitch. Text, drawings and models licensed under CC BY-NC 4.0.