I apologize in advance for my crummy, fat-finger typing.
I'm approachng this question from an engineering and physics perspective. If you look at the tensioned heads as a Helmholtz resonator you will notice a dominant (primary) sound frequency that you can hear depending upon the tuning. There are integral sub-multiples (undertones) and integral mutiples (overtones) of this primary frequency because the whole drum, (heads, shell, hardware, attachments and dampening material) exhibit sympathetic vibrations generated by the air column inside the drum and this heads. Some refer to these viibrations as Eigentones. The stick strike is the primary exciter of this system.
Every system depends upon the mass of its components parts, excitation energy and mechanical energy transfer. The drum as a system generates its own dominant Eigentone and smaller sympathetic Eigentones. Change one, any or all of these factors and you will change the Eigentones of the drum SYSTEM.
Most snare drums (at least in the jazz genre) are tuned to a high pitch. Many jazz drummers (but not all) prefer metal shells with single ply heads, and high tensioning tuning for increased sensitivity. I am a memeber of this group. I have several Ludwig Super Sensitive and Premier snare drums -- and if you have sub-optimal batter and reso head tuning you have an oversupply of dominant, secondary and tertiary Eigentones. This situation makes the drum resonate like a cheap brass bell.
Metal shells are very rigid and offer geat support for high levels of head tension. I bought a tired, old Ludwig Super Sensitive snare drum as a rebuild project. After I stripped everything off the shell for rebuild I started to think of the drum as stated above -- as an Eigentone generating system.
As an engineer / applied physicist, I am an empiricist -- I want numerical data to support my own sensory assesments. My first sensory assessment was that many undamped metal shell drums exhibit irritating, high pitched resonances in the 2KHz - 5 KHz audio passband. Some drummers call this the oingy-boingy basketball resonance effect. I certainly don't like it.
We can apply MoonGel, Gaff tape, dampening rings or multi-layer heads to help ameliorate this problem. But on a jazz or concert snare drum where sensitivity and rapid rebound is king, these solutions carry serious performance degradations.
My empirical experimentation started by using calibrated condensor microphone, tube preamp with phantom power supply, a real-time analyzer, and a Fast Fourier Transform (FFT) analyzer to see in a graphical and numerical format what was happening with the drum Eigentone generation system.
I found that the most aggressive generator of objectionable Eigentones was a poorly (or should I dare say imperfectly) tuned batter / reso head system. The second most aggressive generator of objectionable Eigentones was a 'ringing' undamped shell.
Going back to my Ludwig Super Sensitive rebuild project, I decided to attempt to damp the inside of the shell of the assembled drum with 1/8 inch of Plasti-Dip neoprene coating. This stuff is easily removable by anyone with good manual dexterity and patience. Sounds like a drummer doesn't it?
I tested this drum with identical head tensions and the same heads with and without the Plasti-Kote on the inside of the shell. The Plasti-Kote version of the test showed an average reduction of 15dB on the 2KHz --> 5KHz areas of the audio analysis spectrum. This is very significant -- talk to your sound technician if you dont believe this! This experiment also found that the Plasti-Koted version of the snare drum was not so picky about tuning -- there was a significantly widened window of imperfect tuning in which objectionable Eigentones (ongy-boingy tones) were acceptably low.
In my perspective, this means that an undamped, ringy drum shell is a significant factor in the production of objectionable Eigentones (ongy-boingy tones). My next step in this project is further testing on various composite shells that I am building. These are wound fiberglass / epoxy, wound kevlar / epoxy, wound carbon fiber / epoxy, and mixed wound kevlar / carbon fiber / epoxy. Did I say I was a materials scientist? I also have filthy, sticky hands from these engineering materials.
I am doing this as a personal, non commercial project to enhance drummers' knowlege of the materials science and physics of their drum systems. Music is physics with a human touch... Any comments or questions are warmly welcomed.
Tim