Giant Steps and the chromatic group

Most Western harmonies are built on the circle of fifths. There are 12 notes in the chromatic scale, and moving up by a perfect fifth (7 semitones) cycles through all of them before you get back to where you started. Algebraically this is $\mathbb{Z}/12\mathbb{Z}$ with generator 7 --- since $\gcd(7, 12) = 1$, repeated addition of 7 mod 12 hits every element. The circle of fifths works because 7 generates the whole group.

In 1959, John Coltrane walked into a recording session and played this chord progression perfectly. It was unlike what anyone had ever seen before.

The key centers in Giant Steps by Coltrane are B, G, and E$\flat$, separated by major thirds (4 semitones). Starting from B (call it 0):

$$ 0 \xrightarrow{+4} 4 \xrightarrow{+4} 8 \xrightarrow{+4} 0 \pmod{12} $$

That's the subgroup $\langle 4 \rangle = {0, 4, 8} \cong \mathbb{Z}/3\mathbb{Z}$ inside $\mathbb{Z}/12\mathbb{Z}$. Instead of a generator that cycles through all 12 notes, he used one that only produces 3 and just rotated between them.

This matters because in circle-of-fifths harmony, adjacent keys share most of their notes. C major and G major differ by one note so modulations feel smooth. But B major, G major, and E$\flat$ major are about as far apart as you can get --- they share very few notes. Every time the key shifts you feel as if you're listening to a different song.

The chord progression:

$$ \text{B}\Delta \to \text{D}7 \to \text{G}\Delta \to \text{B}\flat 7 \to \text{E}\flat\Delta \to \text{F}\sharp 7 \to \text{B}\Delta $$

Each dominant 7th chord (D7, B$\flat$7, F$\sharp$7) resolves down a fifth to the next key center. So locally, every two-chord motion is a standard V-I resolution, which is the most basic move in jazz. But globally those resolutions are chained across the $\mathbb{Z}/3\mathbb{Z}$ cycle.

Tommy Flanagan, the pianist on the recording, audibly stumbles through his solo. He was one of the best pianists in jazz. Coltrane is the only one who sounds comfortable because he'd been woodshedding the cycle for months before the session. He had to build a completely new set of patterns just for this song.