to dampen the zinc alloy platter without affecting the sound too too much
I'd recommend The Funk Firm's Acromat, only available in 3-mm thickness these days. It is a rigid acrylic mat designed to absorb vibrations generated in the vinyl by the stylus. It can be stuck to the platter with very thin double-sided adhesive pads. Acrylic and vinyl have about the same acoustic impedance, and the mat is filled with micro-bubbles for vibration absorption. I have Achromats on both my tables. There is also a more expensive version bonded to a glass plate which I have not tried

