Field
Harmony and voice leading
Chords as places, progressions as paths, and the smallest possible move between them.
Three notes at once, and why these three
A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple.
The shortest move, which is what a chord change is
Between any two chords there is an assignment of voices that moves the least. Compute it and the changes that composers actually use turn out to be the short ones.
A progression is a path, and the map can be drawn
Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.