The lesson
Press play on the circle progression. C, F, B°, Em, Am, Dm, G, C. Every root falls a fifth. You've heard it in "Autumn Leaves", "Fly Me to the Moon", "I Will Survive", and anything with a ii–V–I, which is the last three stops. Schillinger called this C5 : cycle of the fifth.
Now thirds. C, Am, F, Dm, B°, G, Em, C. Each chord shares two notes with the next, so it glides. This is where "Heart and Soul" and doo-wop live. That's C3 .
Then steps. C, B°, Am, G, F, Em, Dm, C walking down, or the same rising. "Like a Rolling Stone" rises; a thousand ballads fall. Schillinger calls this C7 , because a step down is a seventh up; we label it a step, since that's what your hand does.
Up or down? "Down" is the direction the ear wants to go: V to I, root falling a fifth, tension releasing. "Up" is the same roots in reverse, and it tends to feel like it's gathering tension. Play both and you'll hear the difference immediately.
Equal slices. Switch to symmetric and the key disappears. Three tonics a major third apart with a major chord on each is the "Giant Steps" sound; four tonics a minor third apart with dominant 7ths is the diminished-cycle trick behind a lot of bebop turnarounds. The chords are all the same quality, so one shape slid around the neck plays the whole thing.
From the Special Theory of Harmony (Book V): diatonic progressions by cycle (chapter 2) and the symmetric systems of two, three, four, six and twelve tonics (chapter 5). The Roman numerals and chord names come from Erasmus's own chord builder.
Practice task
Play C5 down in G major with sevenths, two beats per chord, until the changes come without thinking.
Play the same cycle in A minor (natural minor), then harmonic minor. Hear what the V chord does.
Pick a symmetric cycle, three tonics, major chords, and find one barre shape that plays all three.