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It's much more significant, however, when the band modulates to a key that's similar but not the exact same pitch collection. One of my favorite examples of this is the Hampton Tweezer 10/20/13. The jam begins in A (as all Tweezer jams do) and gradually goes through A minor with traces of A dorian (which for all intents and purposes sound the same as A minor). But then during the spacey section, Mike initiates a move to E minor. And it feels like a seismic shift.
Now, E minor is v of A minor (the minor dominant, if you will). But, E minor is also the same pitch collection as A dorian, with only 1 sharp (F#). And it's not even the F# that gets played a lot in the early stages of that jam when they switch, it's the G to E motion that really solidifies E minor.
So here, even though the mode has shifted and the band is playing essentially the same pitch collection, there are huge differences in sound and affect. I think one of the most crucial things to remember when we talk about modulation is that almost always, when Phish modulates, they also change the overall affect/mood/feel/whatever you want to call it of the music. They respond to harmonic changes with corresponding changes in style, rhythm, timbre, and feel, and that's what really causes us to "feel" the jam changing. Often it's as simple as Page switching keyboards from piano to Rhodes, Trey adding or subtracting an effects pedal or moving from noodling solo to repetitive groove-based jamming, or as I mentioned in my earlier post, Fishman switching the beat ever so slightly.