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'Running like a fugue'...
Could you paraphrase this for me please?
What exactly does it mean in the following sentence?
All the time running like a fugue throughout this was the presence of Brian Epstein, who was a very difficult person.
A "fugue" is a complex piece of music in which one or more themes (and variants on those themes) are played or sung simultaneously, in accordance with certain musical conventions. For instance, theme A may appear in the soprano line, while the bass sings an inverted version of theme B; and meanwhile the tenor may be singing a very slow variant on theme C.
In your example, however, "fugue" is used (somewhat loosely) to mean "a constantly reappearing theme or melody in a piece of music". Brian Epstein's presence is likened to that reappearing theme.
("Obbligato" might have been a more accurate simile.)