Definition and background:
The edge of an object that is typically circular in shape. Specifically in music, several instruments use the term rim as part of their nomenclature.1. MouthpieceThe rim of the mouthpiece of a brass instrument is the part of the mouthpiece that touches the lips of the musician. The rim is generally fairly flat to distribute the pressure of the mouthpiece to the lips and create a comfortable feel for the embouchure of the musician. A more rounded rim is preferred by some professionals as it will typically provide a greater degree of flexibility in technical passages. This flexibility is often offset by the tendancy to tire the musician more quickly. The degree of flatness or roundness of the rim is referred to as the rim contour.2. DrumThe rim of a drum is a round metal hoop that holds the drum head tight to the shell of the drum. Typically, the drum rim is held in place by tension rods that can be tightened or loosened to adjust the tension on the drum head. The amount of tension on the drum head determines the actual percussive sound produced. Composers have long used the effect of directing the musician to play " on the rim " or use the drum sticks to perform the written rhythms by striking the rim with the sticks instead of the drum head. A similar effect called a rim shot, is created by striking both the drum rim and drum head simultaneously with the sticks. This is called a "shot" because of its loud percussive sound, similar to the sound of a gun shot.3. Cymbal / gong / tam-tam The outer edges of a cymbal, gong, or tam-tam are called rims. The rim are the thinnest part of the cymbal and, when struck with a mallet or stick, produces a different sound from the thicker inner section. The gong and tam-tam are built differently from cymbals with the rims or edges curled in. This changes how they vibrate and means that the primary vibrations are located in the center of the gong as opposed to the rim of the cymbal.
The metal hoop that keeps the drumhead in place. It fastens down onto the drumhead by screws (or tension rods) that go through holes in the rim. A rim must be check occasionally for proper roundness.
The outermost reinforcement of the edge of the bell.
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Wikipedia - Glossary of Musical Terminology
Virginia Tech Multimedia Music Dictionary
ORB -- Medieval Music Glossary