Leonardo’s Score

The Tuned Geometry

ITEN

Santa Maria delle Grazie · 1494–1498

Leonardo’s Score

The Tuned Geometry

A single module n — the thickness of a ceiling rib — governs the Cenacolo. Three tuned rhythms: in depth 1:4:1 on the coffers and 12:6:12 (that is 2:1:2) on the walls; in width 1+(2:12:2, that is 1:6:1)+1.

Compositional register

From the double square to 4:3

Let Q = 22n. The visible frame is 2:1, that is (4×2)Q = (88×44)n. Dropping the painted partition, the picture plane is 4:3: (4×3)Q = (88×66)n. Origin at the upper vertex nearest the left-hand tapestry. The viewpoint on the plane sits at (2, −1)Q, that is (44, −22)n; the horizon at y = −1 Q = −22 n.

Module Q
22n
Visible frame
2:1 · (4×2)Q = (88×44)n
Picture plane
4:3 · (4×3)Q = (88×66)n
Origin
upper vertex, left tapestry
Viewpoint
(2, −1)Q = (44, −22)n
Horizon
y = −1 Q = −22 n
Tapestry height
2Q = 44n
Tapestries
⅔ of H
Apparent back
≈ ⅓ of L

Leonardo’s frame: a double square

Leonardo’s frame: a double squareOV (2, −1)Qy = −1 Q2Q = 44n4Q = 88n2:1 frame · (4×2)Q = 88n × 44n
Eight squares of side Q. Origin at the nearest upper vertex of the first tapestry on the left. Horizon on the median, y = −1 Q. Diagonals of the second and third upper squares align with two beams; the lower ones inscribe the body of Christ.

Extended 4:3 picture plane, virtually dropping the partition

Extended 4:3 picture plane, virtually dropping the partitionOV (2, −1)Q(44, −22)ny = −1 Q(6×8)Q grid(4×8)Q visiblepartition4Q = 88n3Q = 66n2Q = 44n
(6×8)Q grid. The visible (4×8)Q is lighter; above, the band hidden by the partition. Origin at the upper vertex of the left tapestry; V = (2, −1)Q = (44, −22)n.

The measures

Three rhythms, one room

The two extra units at the head and tail of the width rhythm take the total from 86n to 88n, and give the 4:3 (88:66) between width and depth of the represented portion of the room. Visually they are the thickening of the perimeter beams: t + x = 2n + n = 3n. In depth the walls remain 12:6:12 = 66n, the ceiling 1:4:1 = 66n.

Tapestry a
12n
Gap s
6n
Coffer c
4n
Rib n
n
Beam t
2n
Lane k
12n
Perimeter t+x
3n
Width L
86n + 2n = 88n

Rhythm plate

Rhythm plateWalls 12 : 6 : 1212n6n12n6n12n6n12n66nCeiling in depth 1 : 4 : 166nCeiling in width 1+(2:12:2)+13n12n2n12n2n12n2n12n2n12n2n12n3n88n12:6:12 (2:1:2) · rhythm 1:4:11+(1:6:1)+1 · ratio 2:1the two extra units thicken the perimeter beams · 88:66 = 4:3
Walls 12:6:12 (2:1:2), ceiling in depth 1:4:1, in width 1+(2:12:2, 1:6:1)+1. The two extra units fuse into the perimeter beams.

Modular accord

Modular accord12:6:12 (2:1:2)66nrhythm 1:4:166n4×12n + 3×6n = 66n = 14n + 13×4n7×2n + 6×12n = 86n · +2n = 88n
4 × 12n + 3 × 6n = 14n + 13 × 4n = 66n. In width: 7 × 2n + 6 × 12n = 86n, plus 2n of thickening = 88n.

Platonic solid

The prism 88n × 66n × 66n

Width 88n, height and depth 66n. A 4:3 ratio between width and the other two. The observer stands at d = P/2 = 33n, eye-level 22n. The back wall falls at scale 1/3.

Width L
88n
Height H
66n
Depth P
66n
Distance d
33n
Eye height
22n
Back scale
s = 1/3

Axonometry of the room

Last Supper prism 88n × 66n × 66nhorizon 22nhidden bandobserverd = 33n88n66nW 88n · H 66n · D 66n · façade 4×3 Q · observer at 33n, height 22n
Prism 88n × 66n × 66n. The façade is 4×3 Q: the upper 1Q band is hidden by the partition. The observer stands 33n from the picture plane, eye-level 22n.

Ceiling structure

Coffered ceiling structure88n66n7×2n + 6×12n = 86n · +2n = 88n
6 lanes × 13 coffers. k = 6t = 12n, t = 2n. Perimeter 3n = 2n + n. 7t + 6k = 86n; +2n = 88n = 4/3 × 66n.

Proof by exclusion

Only d = 33n

Same room 88n × 66n × 66n, same horizon. Only the distance changes. At 22n the back is 1/4; at 44n it is 2/5; at 66n it is 1/2. Only 33n returns the third visible in the painting.

Perspectival section

Observer distance

d = 33n · s = 0.333 · back 29.3n × 22.0n

Perspectival hypothesis at distance 33nVpartition2:1 frame4Q = 88n · 4:33Q = 66n2Q = 44nConfirmed: the back wall falls at 1/3
The picture plane is 88n × 66n (4:3). The visible 2:1 frame (88n × 44n) is emphasised; above it the 22n band hidden by the partition. At d = 33n the back wall is 1/3.

Vectors

SVG plates

Each plate opens full screen: use Save / Download there. Or take the complete package. Sections cover 22n, 33n, 44n and 66n, each also with the double square overlay.

Download all plates (.zip)