What some call overtones, I often call TONE. Try NOT getting rid of the ring (try to get a ring that sounds sort of musical of course) & see how the drums sound with a band behind them. Very often, what sounds like too much ring alone, ends up sounding great when everyone is playing. And when you get rid of the overtones, (Remos are my fave method) sometimes the drums sound dead once the band starts. So I alternate between using them & not using them depending on venue and volume.
In SEVERE LOUD VOLUME SITUATION I USE NO MUFFLING AT ALL OR ELSE NO ONE WILL HEAR YOU! those tones will be what cuts through This is just my opinion—ask 12 drummers about tuning muffling & overtones & you'll get 13 answers..... use your judgment & find your sound. For less sustain ('ringing') try pinstripes, and Remos If there is still ringing, I use thin snare heads.
But I use pinstripe tom heads, so the toms are not that lively already (playing hard rock). But in really live rooms I still use Remos on toms sometimes. It works so much better than tape etc.—because they float over the head when you hit the drum, then settle on just the edges. This doesn't affecting the main note, just the higher overtones.
Physics. A deeper note wave form on head is moving little on the edges of head—the deepest note moves most in the center of head. The 2nd deepest moves most 1/2 way between head & center and has a nodal point in center of head. The highest overtones have multiple waves across the head. Kill one, and it restricts most on the same harmonic. Killing the wave that is just about 1" wide at edge, severely reduces that highest harmonic. But THAT MAY NOT BE A GOOD THING ! ! ! Use with caution! Muffling with tape & stuff is more random & often affects the main note in negative ways. Always test spots before putting on tape. I use my thumb to lightly touch the head to find a sweet spot to put a piece of tape.
If you're doing MAJOR muffling just put a thin towel or t shirt over head—more even than big wads of paper & napkins or wallets, etc. BUT ALWAYS MUFFLE LAST ! ! TUNE AS BEST YOU CAN FIRST ! ! ! My snare, has thin heads tuned tight —and is lively. But once I put on my groove guide trigger (a 1/2" square felt) against the head, that slightly muffles the highest tones. Then I rarely need to do anything else. New heads usually help. How old are the heads now? Change the bottom too. Don't forget to tune the bottom head—I put the drum on carpet to muffle other head, then lift off to check them together.
Take your time tuning (don't tune when the band is there; "C'mon dude - ain't you done tuning yet?..." as if we have a 'tuner' like guitarists have ! ! !) A band will always rush you. With new heads—I often spend serious time finding the best tuning. Although new heads often tune up to a decent sound really quickly, I try different tensions and such. If Remo pinstripes ring too much— you are either going for a dead sound, or your tuning needs improving. For fat quick bass drum sound, try something inside to dampen. A blanket may be too much for some people, but others prefer it. I find foam on the bass drum head works great (1" wide self sticking strips of window insulation inside the drum in a circle about 2" from edge of head). And don't forget bottom heads. Make sure they are not tuned too high.
If Heads Go Out of Tune Try
Lug Locks—little
pieces of nylon that fit on the rod where the washers go. They butt against
the side of the rim, keeping the rod from turning unless you apply
pressure with a drum key. They are like 3/4" X 1/2" with
a hole in the center for the rod. Ask at your drum shop.
They work quite well.
For a great tuning guide—Go to drummingweb tuning bible:
http://www.drumweb.com/profsound.html