E minor as you've played it is called E Aeolian, the natural minor. Raise its 6th (C → C♯) and it becomes E Dorian: still minor, but warmer and jazzier, the sound of a lot of funk and Santana. Lower its 2nd (F♯ → F) and it becomes E Phrygian: dark, Spanish, metal.
On the major side, G with its 7th lowered (F♯ → F) is G Mixolydian: bluesy, rock-and-roll major. In every case one note changes and the rest of position 1 stays exactly as you know it.
R2♭345♭6♭7
R2♭345♭76
R♭2♭345♭6♭7
Em7
A
Em
Fmaj7
Try this
Loop Em7 A. Solo with E Dorian and lean on the C♯ when the A comes. Then loop Em Fmaj7 and use E Phrygian.
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