The shape of a chord

How to practice

1. Start with the home triangle. Load the I preset. The polygon is a narrow wedge in the upper hemisphere — three pitches, all close to home. This is what the tonic triad looks like. Click any pitch to toggle it; watch the shape shrink to a single line, then to a single dot.

2. Compare the modes. Load I, then i. The two triangles have the same outline but the third has rotated — major's 3̂ sits at angle 120°, minor's ♭3̂ sits at angle 270°. Both are "imperfect consonances" but the major's third is brighter (LCM 20); the minor's is softer (LCM 30).

3. Build V⁷ from scratch. Clear the chord. Add 5̂. Add 7̂ — feel the pull. Add 2̂. Add 4̂. By the time you have all four notes you have a stretched quadrilateral that touches the 7̂ ring (the leading tone) and the 4̂ ring (the chordal seventh). That's the dominant 7. Internal LCM jumps from 60 (the triad) to 420 (the seventh) — you can hear the density shift.

4. Look at vii°⁷. Load vii°⁷. Four pitches, equally spaced minor-thirds apart, sitting near the perimeter. The polygon is almost a perfect square. Internal LCM is 27,000 — by far the densest chord in tonal harmony. That's why it sounds urgent: the geometry confirms what the ear already knows.

5. Cross-compare I and IV. Load I, note the shape. Now load IV. Same shape, rotated one fifth counter-clockwise. That visual rotation is the same rotation the ear hears as "moving to the subdominant." The shape isn't changing — its position relative to home is.

Why the shapes are stable across keys

The polar plot is always centered on the tonic. When you change the tonic, the geometry stays — and so does the shape of every chord-role. A I in C major and a I in G major produce the same polygon, because in both cases the chord is built on the tonic. That's what makes the shapes worth memorizing: each shape is the chord role, regardless of the absolute pitches.