This is how I make the alto size mutes using a wooden curtain ring and some velvet ribbon although there are lots of variations. I just found this to be the quickest and easiest. The thickness can be controlled by your choice of material how you do the wrapping. The conductor of my university symphonic band had the sax section use these mutes in order to blend better with the french horns and euphoniums whose parts were often doubled in the saxophones. It also made it so we didn't overpower the other woodwinds.
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What a great idea, and simple! I'm guessing that wrapping with something that's not quite so sound absorbent could do the trick, but soft enough to not harm the finish in the bell. Certainly not expensive and no harm in trying. Thanks jbesax. Much appreciated.

