From Casper, I find out about this interesting analysis of Coltrane's "Giant Steps." Not only does Ian Houghton (or is it Tommy Flanagan, hard to tell) provide a possible explanation for the chord progression, he provides a nifty visual presentation in real time to the music. The explanation does lack some important details, which I will try to deduce.
The chords are arranged around three key areas, B major, G major, and Eb major. For each key, the chords follow the typical jazz (and tonal) progression II - V - I, so in B major the progression is c#m7 - F#7 - B . The key areas are separated by major 3rds (G to B, Eb to G, and B to Eb(D#), known as the chromatic mediant relationship (keys a third apart but sharing the same major or minor mode). Ian (or Tommy) calls this the ditone relationship, and points out that B, G, and Eb equally divide an octave into three parts. The author also points out the quadratone relationship, which is quite unnecessary. The quadratone progression from G to Eb can be explained as a descending ditone.
What the author really misses is how the individual chords progress from one to the next. The first five chords of "Giant Steps," F#7 - D7 - G - Bb7 - Eb, move directly from the V of B major to the V of G major. This cannot be waved away as a "ditone progression" without further explication. Instead, we need to look at what notes are shared between the two chords, and what is changed. F#7 has the notes F# A# C# E, D7 has F# A C D (rearranged so commonalities can be seen). The F# is the only shared note between the two chords. Two notes slide down a half step (A# to A, and C# to C), and one note slides down a whole step (E to D). Viewed from the neo-Riemannian perspective I posted about earlier, the original F#7 chord was transformed by a few operations to become D7, those operations part of a large pattern that governs the ordering of all the chords. Dealing with seventh chords is more complex than triads, though Adrian Childs has developed a neo-Riemannian system for seventh chords: "Moving beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords," Journal of Music Theory Vol. 42/2 (1998), 181-193. Analysts have used other explanations for this progression, from strange altered pivot chords to theories of altered dissonance and consonance patterns. But none so far have provided the nice animation that Ian/Tommy did.
Since music analysis is just an attempt to explain the logic of music, why a melody or chord progression sounds like it makes sense, what is your analysis of "Giant Steps?" Do the first five chords make sense to you? If so, how do the chords seem to belong together?