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As you noted, the rim acts as a coupler, connecting the vibrating lips to the instrument. The size of the rim determines the maximine amount of lip that can vibrate, this affects both range and tone in that range. Don't overlook that fact that buzzing your lips with a mpc is different than buzzing with one, so the rim helps to set up the embouchre, and the edge of the rim differentiates between buzzing and non buzzing tissue. The shape of the rim is something of a comprimise, as is generally known wide rims can help endurance, narrow rims can help flexibility, sharp flat rims can help accuracy, round rims can help slurs, The high point of the rim affects how big the mpc feels.

The bite — or sharpness of the inside edge of rim — plays a role in articulation. The sharper the bite the easier to articulate but with a trade off on endurance.