♩
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A chord progression is a sequence of shapes, and the motion between them is a set of voice-leading gestures. Pick a cadence; step through it. The previous chord appears as a faint ghost polygon, the current chord is solid, and arrows show which pitch moved where. Different cadences leave visibly different arrow patterns — once you've seen V→I a few times you'll be able to predict the arrows before you click.
Predominant — far enough from home to want motion.
1. Look at V → I. Two arrows: 7̂ steps up by a half-step to 1̂; 4̂ steps down to 3̂. Two common tones (5̂ and 2̂) are held. The geometry confirms what the ear hears — two dissonances released, two voices held, and the chord moves home.
2. ii → V → I is V → I twice. Step through ii→V→I. Notice that each transition involves the same "fifth descent" in the root — ii's root (2̂) moves down a fifth to V's root (5̂), which moves down a fifth to I's root (1̂). On the polar plot, each chord rotates one fifth clockwise.
3. The Phrygian half-step is unmistakable. Try iv → V (Phrygian). Compare to V → I. The Phrygian motion has a ♭6̂ → 5̂ pull (half-step from above); the leading-tone motion has a 7̂ → 1̂ pull (half-step from below). The arrow direction is opposite on the polar plot — even though both are "half-step cadential pulls," they point opposite ways.
4. The vii°⁷ → i implosion. Step through vii°⁷ → i. The diminished seventh's four pitches are arranged near-symmetrically at the perimeter. Each of the four arrows points inward — every voice resolves by half- or whole-step toward the tonic triad. This is why the chord feels urgent: four simultaneous pulls demanding release.
5. The Tristan slide. Try Half-dim → V⁷. Two quadrilaterals on top of each other, each voice sliding by step to the next. Wagner stretched this single voice-leading event across a whole opera by never letting the V⁷ resolve to I — but the local gesture itself is right here, two chords apart.
Gold arrows are steps (half-step or whole-step voice motion). Purple arrows are leaps (more than two semitones). Good voice leading is mostly gold; leaps appear at chord-tone "skips" like the bass moving in fifths. A green dashed circle marks a common tone — a voice that stayed on the same pitch through the chord change.