If this is an install, you should have QSys or Xilica or Symetrix type DSP; all of these should have this function. Trying to install a sound console for an installed, non-audio-FOH-engineer user is the wrong tool for the job.
^ This!
Yes! That's precisely what I'm looking for. Are you aware of a hardware/plugin solution with low enough latency for live?
The DiGiCo SD I run doesn't do this....
Perhaps there is a big boy console (read $40-200k) that does this natively (although not that I can think of, off the top of my head).
This is exactly what Waves running an F6 plugin can do, which is a heap cheaper than a large format desk, but still cost prohibitive for what you want to do.
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Again if you have a basic console or controller (X32, SQ, QSYS etc) try keying the music channel comp to your mic, and key the frequency to the area where the music is masking the voice. This will get you pretty darn close to the same result.