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Thick or thin?
The density of the shell has a significant impact on the sound of a drum.
A very thick drum will, in relative terms, hardly resonate at all and will tend to project a clear bright version of the head-tone imparting few of its own qualities into the sound, this is because the vibrations from striking are reflected by the shell more than they are absorbed by it (although a certain amount of absorption will always take place with a wooden drum) At lower tunings, they deliver a 'fat' or 'round' sound. At elevated tunings they tend to choke up more easily, or as Bob Gatzen describes them, a 'Blunt' or 'Sticky' sound.
A thinner drum on the other hand will act like a type of membrane, vibrating and resonating in greater harmony with the heads, in this scenario, because it is absorbing vibration, it starts to vibrate itself and imparts its own vibrations and therefore sound into the overall tone of the drum. At higher tunings, they are very clear in pitch and cut well through the music, they deliver greater volume than thicker shelled drums and lower tunings this may be considered a more 'woody' and warmer sound, overall a thinner shell will deliver a more pure tone than a thicker drum. In general a thinner shelled drum will accept all head types and thicknesses and will deliver the optimum tone for that head type.
One caveat about using thinner shelled drums is that at high volumes, the vibrations within the shell of a drum itself can cause the sound to distort. Pick your shells to match the music you aim to play.
What difference does the depth of the drum make?
The depth of a drum has an effect on the overall sound of the drum, not on the pitch but on the quality of the tone. A longer drum will have a more full and rounded tone because the wavelengths generated will have more distance in which to develop, they will also be louder. However with that in mind, longer drums require more effort to excite them as you are shifting a longer column of air with a greater resistance or backpressure. A longer drum will have less attack, it will be slower to respond but will have a fuller tone and longer resonance, a shorter drum will have the reverse, again it is a trade off that only you can apply to your own drum-buying decision making processes. Most off-the-shelf manufacturers provide only a limited number of depths for each diameter made, however custom makers can cut shells to any size.