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Over the Christmas break I heard this programme on BBC radio:
Shine On You Crazy Diamond

Roger Waters discusses his famous four-note motif B♭ F G E but what wasn't discussed and I wished had been was the first three chords of the song, which are:

Gm G♭ B♭

In the first chord, the minor third of the chord is a B♭ note, which then serves as the major third of the subsequent G♭ major chord.

And I see that Gm is the relative minor of B♭ major. But what function does each chord serve? Is there some sort of cadential preparation going on? It sounds nice, but not like classical harmony to me.

The move from Gm to GbM is definitely not a standard “classical” progression, although it does start to happen more and more in the late Romantic era and in lots of Impressionist and other neo-tonal styles. It’s perhaps especially common in the triadic minimalism of composers like Philip Glass. I don’t think it’s super common in rock music, which might be part of why it sounds so fresh in the Pink Floyd example.

You’re exactly right that the “connective tissue” of the chord change is the Bb common tone. David Lewin called this move SLIDE, and neo-Riemannian theorists often call it P’ or “P prime,” because it’s the opposite of the connection they call P, which is short for parallel. P connects a major triad to its parallel minor counterpart and vice versa. This means the root and the fifth stay the same, and the third moves by a half step. P’ is the opposite, because, as you noted in your example, the third stays the same while both the root and fifth move by a half step. That’s why it’s the opposite of the parallel move.

Anyway, it’s an extremely cool sound, and it contributes a great deal to the moody effect of the beginning of “Shine on.”

EDIT TO ADD:
Sorry, I just realized you also asked about the move from GbM to BbM as well. This move also includes a common tone—notably, the same Bb common tone as the previous move—and half step movements in the other two voices just like in the P’ move. Different voices and directions are involved though. The move is called a “chromatic mediant,” and that’s the name for any two chords with roots a third apart and the same quality (major or minor). There are many questions on the site that explain chromatic mediant a if you want more details.