I'm reading an essay by Dan Adler that opened up a new relationship between math and music for me, and it has changed how I study John Coltrane's "Giant Steps." The trick is to stop reading the tune as a long string of fast chords and start seeing it as a triangle.

Giant Steps doesn't move by fifths the way most standards do. It moves by major thirds. The whole piece orbits three tonal centers: B major, G major and E♭ major. Those three notes sit exactly a major third apart from each other (B down to G, G down to E♭, E♭ back to B), so together they spell an augmented triad and they divide the octave into three equal parts. That is the geometry hiding inside the tune.

Put those three keys at the vertices of an equilateral triangle on the chromatic clock: B, G, E♭. Coltrane's move (what people call the "Coltrane changes") is to travel between the vertices by inserting the dominant of the key he is heading to. So every major chord gets approached by its own V7:

  • D7 resolves to Gmaj7
  • B♭7 resolves to E♭maj7
  • F♯7 resolves to Bmaj7

Lay it out and the opening phrase reads Bmaj7 D7 | Gmaj7 B♭7 | E♭maj7, which is just the triangle walked once around, with a dominant placed on every edge to push you to the next corner.

Why does this make improvising easier? Because it shrinks the problem. On paper Giant Steps looks like a brutal sequence of unrelated changes at tempo. Seen as a triangle it is really only three home keys plus the three dominants that connect them. Instead of fighting a wall of fast changes, you internalize three centers and learn to hear which corner you are standing on and which one you are walking toward.

That reframing also tells you what to practice. Take the three major scales (or arpeggios, or pentatonics) over B, G and E♭ until they are automatic, then drill the V7 links (D7, B♭7, F♯7) that carry you from one corner to the next. Once the triangle is in your ear, you stop spelling chords in real time and start moving around a shape you already know.

For the full, and better, explanation, including the math behind it, read this article by Dan Adler.

Peace.