The behavior of wind instruments, including brass instruments, is primarily determined by the shapes of their air columns, and their interaction with the sound generation mechanism. However, the influence of the surrounding body of the instrument has been a matter of some debate, and papers exploring this question have been published since the early years of the J. Acoust Soc. Am. An apparent correlation between instrument material and playing behavior is disputed by arguments that the structure is stiff and massive compared to the air inside, and that many of the apparent effects are linked to machining differences among materials. The complexity of the instrument body makes this problem well suited for Statistical Energy Analysis (SEA), which treats the air column and the external structure as coupled statistical subsystems that share energy. For trumpets and trombones, the power radiated from the structural vibrations is about 40 dB lower than the energy radiated directly from the air column, with an enhancement at high frequencies due in part to the increasing modal density of the three dimensional structure. The coupling to the structural vibrations themselves from the player’s lips and from the air vibrations are similar to each other in magnitude.