On a club LS4 as well as an LS4 I used to own, we had the same problem. Sounds like a bassoon player sitting in the area of the empennage. Turned out that sealing the top of the vertical stab with foam around the elevator pushrod solved the problem. My 29 hasn't yet displayed this musical tendency, but I have a Nixon/Murray exhaust vent which provides an optimized path for high pressure cockpit air to escape...

P3