The Harmonic Armature: What It Is and How to Use It

The harmonic armature is a set of eight lines across the canvas: the two diagonals, the vertical and horizontal center lines, and four reciprocals that leave the corners at right angles to the diagonals. Painters use the lines, and the points where they cross, to place the main shapes and directions of a picture.

The lines

1. Two diagonals, corner to corner.
2. Two center lines, one vertical and one horizontal, through the middle.
3. Four reciprocals. Each leaves a corner at right angles to the diagonal that doesn't pass through that corner, and runs until it meets the opposite long side.

Drawing it on the canvas

The diagonals and center lines need only a straightedge. For a reciprocal you need the point where it lands: multiply the short side by itself and divide by the long side, then measure that far along the opposite long side, from the same end as the corner you start at. On a 16 × 20 in canvas that's 16 × 16 ÷ 20 = 12.8 in, so the line from the bottom-left corner ends on the right edge 12.8 in up from the bottom.

On a landscape canvas the same distances run along the top and bottom edges. On a square canvas each reciprocal falls on a diagonal, so only the diagonals and center lines are left.

Where the lines cross

• The center, where both diagonals and both center lines meet.
• Four eyes, where each reciprocal crosses a diagonal at right angles. On a 16 × 20 in canvas they sit 6.24 and 9.76 in across, and 7.8 and 12.2 in down.
• Four points on the center lines, where two reciprocals meet: on 16 × 20 in, 6.4 and 13.6 in down the vertical center line, and 3.5 and 12.5 in along the horizontal one. Three lines cross at each.

Where it comes from

The reciprocal is a construction from dynamic symmetry, the system Jay Hambidge set out in Dynamic Symmetry: The Greek Vase (1920). He drew a diagonal, then from another corner a line cutting it at right angles, carried on to the far side. That line, he wrote, is the diagonal of the rectangle's reciprocal: a smaller rectangle of the same shape. On a 16 × 20 in canvas, the reciprocal from the bottom-left corner is the diagonal of a 16 × 12.8 in rectangle along the bottom, the canvas's 4:5 shape turned on its side. Hambidge called the points where these lines cut the main diagonals the "poles or eyes".

Charles Bouleau's The Painter's Secret Geometry (1963; in French, Charpentes) analyzes more than 100 paintings in terms of what he called the armature of the rectangle, with diagonals, divisions, circles and other constructions drawn over each one. The harmonic armature in the drawing grid maker and in Paintsight uses only the eight lines above.

How painters use it

• Directions. Lay a big slope along a diagonal or a reciprocal: a hillside, a road, a limb, the edge of a cast shadow. Each reciprocal crosses a diagonal at right angles, so a second, smaller movement on the crossing line counters the first.
• The focal point. An eye is off center, and two lines lead to it. Put the center of interest on or near one; see the focal point in painting.
• Big masses. Let the edge of a large dark shape follow part of a line, or stop where lines cross.

With a reference photo, open it in the drawing grid maker, enter your canvas size and choose Harmonic armature under Composition guide. Look for lines the photo already has, such as a ridge, a shadow edge or a branch, and see whether one runs near an armature line. If none does, try cropping the photo differently. Then rule only the lines you used, lightly, on the canvas.

A fair warning

Any picture can be made to fit some set of lines if you allow enough of them. Bruce MacEvoy points out that Bouleau's book gives no historical record or x-ray evidence that the painters used its constructions, and that its diagram for Vermeer's Allegory of Painting needs 45 lines, most of which mark no important edge or form. The systems also overlap: on a 4:5 canvas the eyes land within 1% of the golden section crossings (see the rule of thirds and the golden section), so a picture that fits one nearly fits the other. Use the armature to plan a new picture or test a crop, not as proof of how a past painter worked.

Sources

• Jay Hambidge, Dynamic Symmetry: The Greek Vase, Yale University Press, 1920, figures 7, 8 and 36. Internet Archive, from Cornell University Library's copy.
• Charles Bouleau, The Painter's Secret Geometry, Harcourt, Brace & World, 1963. Internet Archive lending copy.
• Bruce MacEvoy, format proportions, handprint.com, revised 2015, for what Bouleau's book covers and a critique of its analyses.
• Reed Enger, "Advanced Composition Techniques", Obelisk Art History, 2019, on how painters used armatures: focal points at intersections, figures on diagonals.