As several others have observed, 42 is the answer to life, the universe, everything, and this Junto assignment. So I put on my two Zaphod Beeblebrox thinking caps and approached this piece as a mathematical problem. A piece composed in 3/4, 2/4 and 7/4 will converge every 42 beats: 14 measures of 3/4, 21 measures of 2/4, and 6 measures of 7/4.
More math: I decided to use three chords during each 42 beat phrase, and make the chords change on measure boundaries even though the measures don’t line up. The 7/4 line plays each chord for 2 measures. The 2/4 line plays each chord for 7 measures The 3/4 line plays 5 measures of the first chord, 4 of the second, and 5 of the third. Careful selection of the individual notes around the chord changes allowed this to work without dissonance.
Created in Live with Ableton instruments. Playing each time signature are: cello on 3/4, bass on 2/4, and guitar, violin and flute on 7/4. Live handled the multiple simultaneous time signatures pretty well.