Brahms was at once innovative and conservative. It seems a strange thing to say. What I mean is that he knew how to tease fresh sounds, fresh harmonic events, out of established contrapuntal “best practices.” Rather than simply throwing something weird at the page, completely out of left-field, his method was to achieve the surprising or unusual phenomenon organically, so that it could be justified from a contrapuntal (and/or harmonic) perspective.
The fiery portion of the 6th movement from his Requiem contains a curious move from C-minor to E minor. You can hear the shift to E-minor at 4:13 in this performance:
How shall we explain this?
One could say that Brahms was simply exploiting a third-relation. Well, C-minor and E-minor certainly do stand in a chromatic third relationship to each other. But this doesn’t necessarily justify the move. Why should he have exploited that relationship?
One could also say that E-minor is not foreign to C-major (that is, it occurs diatonically), and so it represents a kind of borrowed harmony. Usually, however, one borrows from the minor mode while writing in the major mode, not the other way around. But again, while E- and A-natural do indeed come from the major mode, this says nothing in the way of justification.
I think the final answer is that the “visit” to E-minor represents an unfolding of an auxiliary note, namely the leading-tone with respect to the subdominant: E-natural. Indeed, much of the movement is spent gravitating toward the subdominant. There are many appearances of the Phrygian II in C-minor, which is also the diatonic VI with respect to the subdominant F-minor. There is even a portion that unfolds G-flat minor (spelled enharmonically as F-sharp minor) – this would be the (minor) Phrygian ii with respect to the subdominant. These harmonies create the sense of gravitation toward the subdominant I mentioned because they require accidentals in the key of C-minor. They do not “belong” to C-minor, they “belong” to F-minor.