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A novel technique for the recording of large sets of room impulse responses or head-related transfer functions is presented. The technique uses a microphone or a loudspeaker moving with constant speed. Given a setup (e.g. length of the room impulse response), a careful choice of the recording parameters (excitation signal, speed of movement) is shown to lead to the reconstruction of all impulse responses along the trajectory. In the case of moving element along a circle, the maximal angular speed is given in function of the length of the impulse response, its maximal temporal frequency, the speed of sound propagation and the radius of the circle. As result of this theory, it is shown that head-related transfer functions sampled at $44.1~$kHz can be measured at all angular positions along the horizontal plane in less than one second. The presented theory is compared with a real system implementation using a precision moving microphone holder. The practical setup is discussed together with its limitations.