What does the FAF motto in Brahms’s Symphony No. 3 in F Major mean?

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In Johannes Brahms’s Symphony No. 3 in F Major, the "FAF" motto stands for the German phrase frei aber froh, which translates to "free but joyful."1

Brahms used this three-note motive to unify the movements of the symphony, with the opening chords of the first movement explicitly harmonizing the motto.1